text:
\section{\sc Introduction}

In any life testing experiment it is a common practice to terminate the experiment before all specimens under observation fail.  In a type-I censoring scheme the test is terminated at a prefixed time point whereas in a type-II censoring scheme the experiment continues until a certain number of failures occurs.  In a practical scenario it might be necessary to remove some of the experimental units during the experiment.  Different progressive censoring schemes allow removal of experimental units during the experiment.  A progressive type-II censoring scheme can be briefly described as follows.  It is assumed that $n$ items are put on a test.  Suppose $k$, $R_1, \ldots, R_k$ are non-negative integers such that $n-k = R_1 + \ldots + R_k$. At the time of the first failure say $t_1$, $R_1$ units are chosen at random from the remaining $(n-1)$ items and they are removed from the experiment.  Then
at the time of the second failure, say $t_2$, $R_2$ units are chosen at random from the remaining $n-2 - R_1$ units and they are removed.  The process continues, finally at the time of the $k$-th failure all the remaining $R_k$ items are removed from the experiment, and the experiment stops.

During the second world war, due to the demand of the highly reliable military equipment, engineers started laboratory investigation instead of long awaited field experiments.  Due to tremendous pressure on cost and time, various schemes were  introduced to reduce the cost of testing and multi-stage censoring was one of them. As it has been mentioned in Herd (1956) that in many laboratory evaluations, due to limited allocation of funds there are attempts to study the factors  contributing to either the reliability or the unreliability of the components or the whole systems under study, as well as to estimate the reliability of the items.  In some investigations in order to measure these auxiliary variables, some system needed to be disassembled or subject to measurement which are destructive in nature or might change the expected lifetime.  In certain evaluations, systems are removed from the main experimental setup in order to measure certain specific characteristics.  Most of the cases systems are subjected to interaction of human operator and to asses the effect of the human operator on the efficiency of the active system, sometimes it requires to withdraw few systems from the main experimental set up.  During the reliability study of production prototype systems, the addition of engineering modifications also require analyses on multistage censoring.

In Montanari and Cacciari( 1988) experimenters conducted testing to evaluate endurance of XLPE-insulated  cable to
electrical and thermal stress along with the ageing mechanism.  Some specimens were removed from the test at selected times or at a time of breakdowns for the measurements of electrical, mechanical, chemical properties along with micro structural analyses in order to evaluate aging process.  These measurements are destructive in nature.  In this experiment the data obtained consist of failure time and censored time of specimens.
Ng et al. (2009) presented a clinical study where multistage censored data arise quite naturally.  In a study with plasma cell Myeloma at National Cancer Institute on 112 patients, few patients were dropping out at the end of certain intervals whose survival was ensured at that time but no further follow up was available.  In his thesis, Herd (1956) discussed estimation under multistage censoring scheme, and refer the scheme as "multi-censored samples".  Later on different multistage censoring schemes were referred as progressive censoring schemes.  Cohen (1963) studied the importance of progressive censoring scheme in reliability  experiment and Cohen (1966) discussed several cases where censored data occurred quite naturally.

To be more precise, in favor of progressive censoring scheme the following points can be mentioned.  In practical scenario experimental units might get damaged due to some unrelated reasons than the normal failure mechanism.  To incorporate these information into inference study, we can rely on progressive censoring schemes.  Sometimes the multistage censoring is also intentionally done to use censored units from one experiment to another related experiment due to budgetary constraints.

The progressive censoring scheme has received considerable attention in the literature.  Mann (1971) and Lemon (1975) studied on estimation for Weibull parameters under progressive censoring scheme.  Viveros and Balakrishnan (1994) provided interval estimation on progressively censored data.  Ng et al. (2004) studied on optimal progressive censoring plan when the underlined distribution is Weibull distribution whereas  Kundu (2008) provided Bayesian inference of the Weibull population under progressive censoring scheme.  Wang et al. (2010) studied the inference of certain life time distributions under progressive type-II right censored scheme.
Recently published book by
Balakrishnan and Cramer (2014) provided an excellent overview of the different developments on different aspects of the progressive
censoring scheme, which have taken place on this topic
over the last 20 years.  Although, type-II progressive censoring scheme is the most popular one, several other progressive
censoring schemes have also been introduced in the literature.  In this article we have restricted our attention to the
type-II progressive censoring scheme, although most of our results can be extended for other progressive censoring schemes also.

Although, extensive work has been done on the progressive censoring scheme for one group, not much work has been done when
two or more groups are present.  Rasouli and Balakrishnan (2010) introduced a joint progressive censoring (JPC) scheme, which
can be used to compare the lifetime
distributions of two products manufactured in different units under the same environmental conditions.   The JPC scheme
proposed by Rasouli and Balakrishnan (2010) can be briefly described as follows.  Suppose $m$ units of Product A (Group 1) and
$n$ units of product B (Group 2) are put on a test simultaneously  at time zero.
It is assumed that $R_1, \ldots, R_k$ are $k$ non-negative integers such that $R_1 + \ldots + R_k = n+m-k$.
At the time of the first failure, which may be either
from Group 1 or Group 2, $R_1$ items from the remaining $n+m-1$ remaining items have been selected at random, and they have been
removed from the experiment.  These $R_1$ censored units consist of $S_1$ units from Group 1 and $R_1-S_1$ units from Group 2.  Here $S_1$ is random.  The time and group of the first failed item are recorded.  Similarly,
at the time of the second failure, $R_2$ items from the remaining $n+m-R_1 - 2$ items have been chosen at random and they have been removed
from the experiment.  Among censored $R_2$ units random number of $S_2$ units come from Group 1.  The time and group of the second failed items are recorded.   The process continues till the $k$-th failure takes place, when all the remaining items are removed and the experiment ends.

Based on the assumptions that the lifetime distributions of the
experimental units of the two populations follow exponential distribution with different scale parameters, Rasouli and Balakrishnan (2010)
provided the exact distributions of the maximum likelihood estimators (MLEs) of the unknown parameters and suggested several confidence
intervals.  Some of the related work on JPC scheme are by Shafay et al. (2014), Parsi and Bairamov (2009) and  Doostparast et al. (2013).  In most of these
cases it has been assumed that the lifetime distributions of the items in two groups follow exponential distribution.

The exponential distribution has the constant hazard rate, which may not be very reasonable in a practical scenario.  Again when two similar kind of products are tested, it is quite expected to have some common parameters from the underlined distributions.
To justify these scenario, in this paper it is assumed that the lifetime
distributions of the individual items of the two different groups follow Weibull distribution with the same shape parameter,
but different scale parameters.

Our aim is to compare the lifetime distributions of the two populations.  The problem is a typical two-sample problem.  It can appear in a accelerated life testing problem, or in estimating the stress-strength
parameter of a system.

First we consider the MLEs of the unknown parameters based on the data obtained from a JPC scheme as
proposed by Rasouli and Balakrishnan (2010).  It has been shown that the MLEs exist under a very general condition and they
are unique.  The MLE of the common shape parameter can be obtained by solving one non-linear equation, and given the MLE of the
common shape parameter, the MLEs of the scale parameters can be obtained in explicit forms.  Although, the performances of the MLEs
are quite satisfactory, the associated confidence intervals are not very easy to obtain.  Hence, it seems the Bayesian inference is a
natural choice in this case.  It may be treated as an extension of the work of Kundu (2008),  where the
Bayesian inference of the unknown Weibull parameters for one sample problem was considered, and in this case the results have
been generalized to two sample problem.  Clearly, the generalization is a non-trivial generalization.  Although, the whole development in this paper is for two groups, but the results can be easily generalized to more than two groups also.

For the Bayesian inference we need to assume some priors on the unknown parameters.
If the common shape parameter is known, the most convenient but a fairly general conjugate prior on the scale parameters can be the Beta-Gamma prior, as it was suggested by Pena and Gupta (1990).  In this case the explicit form of the Bayes estimates of the scale parameters can be obtained.  When the shape parameter is unknown, the conjugate prior does not exist.  In this case following the approach of Berger and Sun (1993) or Kundu (2008), it is assumed that the prior on the shape parameter has a support on $(0, \infty)$, and it has a log-concave density function.  It may be mentioned that many well known distribution functions for example log-normal, Weibull, gamma etc. may have log-concave probability density function.  Based on the prior distributions, the joint posterior density function can be obtained.  As expected the explicit expressions of the Bayes estimates cannot be obtained in explicit forms.  We propose to use importance sampling technique to compute the Bayes estimate of any function of the unknown parameters, and also to construct the associated highest posterior density (HPD) credible interval.  Monte Carlo simulations are performed to see the performances of the proposed method, and one data analysis has been performed for illustrative  purposes.

In many practical situations it is known apriori that one population is better than the other in terms of the expected lifetime of the experimental
units. In accelerated life testing if units of one sample are put on higher stress than the units of other sample, it is quite expected to get shorter life time of the specimens under higher stress.  We incorporate this information by considering the order restricted classical and Bayesian inference of the unknown parameters based on joint progressively censored samples.  We obtain the MLEs of the unknown parameters based on the order restriction.  The construction of the confidence intervals
of the unknown parameters can be obtained using bootstrap method.  For Bayesian inference, we propose a new order restricted Beta-Gamma prior.  In this case the explicit expressions of the Bayes estimates cannot
be obtained and we propose to use importance sampling technique to compute the Bayes estimates and the associated HPD credible intervals.  We
re-analyze the data set based on the order restriction.

Rest of the paper is organized as follows.  In Section 2, we present the notations, preliminaries and the priors.  Maximum likelihood
estimators are presented in Section 3.  Posterior analysis are provided in Section 4.  In Section 5, we present the order restricted
inferences of the unknown parameters.  In Section 6, we
present the simulation results and data analysis.  Finally we conclude the paper in Section 7.

\section{\sc Notations, Model Assumptions and Priors}

\subsection{\sc Notations and Model Assumptions}
$$
\matrix{
\hbox{PDF}: & \hbox{Probability density function}   \cr
\hbox{HPD}: & \hbox{Highest posterior density}  \cr
\hbox{CDF}: & \hbox{Cumulative distribution function}  \cr
\hbox{MLE}: & \hbox{Maximum likelihood estimator}  \cr
\hbox{i.i.d.}: & \hbox{independent and identically distributed} \cr
F_1: & \hbox{CDF of the lifetime distribution of the items of Group 1}  \cr
F_2: & \hbox{CDF of the lifetime distribution of the items of Group 2}  \cr
T_1: & \hbox{Random variable with CDF $F_1$}  \cr
T_2: & \hbox{Random variable with CDF $F_2$}  \cr
k_1: & \hbox{Number of failures from Group 1}  \cr
k_2: & \hbox{Number of failures from Group 2}  \cr
k: & \hbox{Total number of failures:} \ \ k = k_1+k_2   \cr
\hbox{GA}(\alpha,\lambda): & \hbox{Gamma random variable with PDF:    }
\displaystyle (\lambda^{\alpha}/\Gamma(\alpha)) x^{\alpha-1}
e^{-\lambda x}; \ \  x > 0.  \cr
\hbox{WE}(\alpha,\lambda): & \hbox{Weibull random variable with PDF:   }
\displaystyle \alpha \lambda x^{\alpha-1} e^{-\lambda x^{\alpha}}; \ \  x > 0.  \cr
\hbox{Bin}(n,p): & \hbox{Binomial random variable with probability mass function:    }
{n\choose{i}} p^i (1-p)^{n-i}   \cr
\hbox{Beta}(a,b): & \hbox{Beta random variable with PDF:    } \displaystyle
(\Gamma(a+b)/\Gamma(a) \Gamma (b)) p^{a-1} (1-p)^{b-1}; \ \  0 < p < 1.   \cr
}
$$

Suppose $m$ and $n$ independent units are placed on a test with the corresponding lifetimes being identically distributed
with PDF $f_1(\cdot)$ and $f_2(\cdot)$, and CDF $F_1(\cdot)$ and $F_2(\cdot)$, respectively.  It is assumed that the lifetime
distribution of the items from Group 1 and Group 2, follow WE$(\alpha,\lambda_1)$ and WE$(\alpha,\lambda_2)$, respectively.
For a given $(R_1, \ldots, R_k)$, as described before, in a JPC scheme, the observations are as follows;
\begin{equation}
\{(t_1, \delta_1, s_1), \ldots, (t_k, \delta_k, s_k)\}.   \label{data}
\end{equation}
Here for $j = 1, \ldots, k$, $\delta_j$ = 1, if the failure at $t_j$ occurs from Group 1, otherwise $\delta_j$ = 0, and $s_j$ denotes the number
of items from Group 1, which have been removed at the time $t_j$.  Therefore, the likelihood function can be written as
\begin{equation}
L(data|\alpha, \lambda_1, \lambda_2) \propto
\alpha^k \lambda_1^{k_1} \lambda_2^{k_2} \prod_{j=1}^k t_j^{\alpha-1}
e^{-\lambda_1 U(\alpha)} e^{-\lambda_2 V(\alpha)}, \ \ \ \ \label{lh-2}
\end{equation}
here $\displaystyle k_1 = \sum_{j=1}^k \delta_j$ and $k_2 = k - k_1$,
$$
C_1 = \{j; \delta_j = 1\}, \ \ \ C_2 = \{j; \delta_j = 0\},
$$
$$
U(\alpha) = \sum_{j=1}^k s_j t_j^{\alpha} + \sum_{C_1} t_j^{\alpha}, \ \ \
V(\alpha) = \sum_{j=1}^k w_j t_j^{\alpha} + \sum_{C_2} t_j^{\alpha},
$$
where $w_j = R_j - s_j$.

\subsection{\sc Prior Assumptions: Without Order Restriction}

The following prior assumptions are made on the common shape parameter $\alpha$ and on the scale parameters $\lambda_1$ and
$\lambda_2$, when there is no order restriction on $\lambda_1$ and $\lambda_2$.  If we denote $\lambda = \lambda_1 + \lambda_2$, then similarly as in Pena and Gupta (1990), it is assumed that
$\lambda \sim$ GA($a_0,b_0$), with $a_0 > 0$, $b_0 > 0$ and $p = \lambda_1/(\lambda_1+\lambda_2) \sim$ Beta$(a_1,a_2)$, with
$a_1 > 0$, $a_2 > 0$, and they are independently distributed.  The joint PDF of $\lambda_1$ and $\lambda_2$ can be obtained
as follows
\begin{eqnarray}
\pi_1(\lambda_1, \lambda_2|a_0,b_0,a_1,a_2) & = & \frac{\Gamma(a_1+a_2)}{\Gamma(a_0)} (b_0 \lambda)^{a_0-a_1-a_2} \times
\frac{b_0^{a_1}}{\Gamma(a_1)} \lambda_1^{a_1-1} e^{-b_0 \lambda_1}    \nonumber  \\
&  & \times  \frac{b_0^{a_2}}{\Gamma(a_2)} \lambda_2^{a_2-1} e^{-b_0 \lambda_2}; \ \ \ 0 < \lambda_1, \ \lambda_2 < \infty.
\label{prior-1}
\end{eqnarray}
It is known as the Beta-Gamma PDF, and it will be denoted by BG$(a_0,b_0,a_1,a_2)$.

The above Beta-Gamma prior is a very flexible prior on the scale parameters.  Depending on the values of the hyper-parameters,
the joint prior on $\lambda_1$ and $\lambda_2$ can take variety of shapes.  Moreover, for a given $\alpha$, it is a conjugate
prior on $(\lambda_1, \lambda_2)$.  The correlation between $\lambda_1$ and $\lambda_2$ can be both positive and negative
depending on the values of $a_0$, $a_1$ and $a_2$.  If $a_0  = a_1 + a_2$, $\lambda_1$ and $\lambda_2$  are independent.  If
$a_0 > a_1 + a_2$, then they are positively correlated, and for $a_0 < a_1 + a_2$, they are negatively correlated.  The
following results will be useful for further development, and they can be established very easily.

\noindent {\sc Result 1:} If $(\lambda_1, \lambda_2) \sim$ BG$(a_0,b_0,a_1,a_2)$, then for $i = 1,2$,
\begin{equation}
E(\lambda_i) = \frac{a_0 a_i}{b_0(a_1+a_2)} \ \ \ \hbox{and} \ \ \
V(\lambda_i) = \frac{a_0 a_i}{b_0(a_1+a_2)} \times \left \{ \frac{(a_i+1)(a_0+1)}{a_1+a_2+1} - \frac{a_0 a_i}{a_1 + a_2}
\right \}.    \label{mean-var}
\end{equation}
Moreover, the generation from a Beta-Gamma distribution is quite simple using the property that
$(\lambda_1, \lambda_2) \sim$ BG$(a_0,b_0,a_1,a_2)$, if and only, $\lambda_1+\lambda_2$ has a gamma distribution and
$\displaystyle \frac{\lambda_1}{\lambda_1+\lambda_2}$ has a beta distribution and they are independently distributed, see for
example Kundu and Pradhan (2011).

No specific form of prior has been assumed here on the common shape parameter $\alpha$.  Following the idea of
Berger and Sun (1993), it is assumed that $\pi(\alpha)$, the prior on $\alpha$, has a support on the positive real line, and
it has a log-concave PDF.  Moreover, $\pi(\alpha)$ and $\pi(\lambda_1, \lambda_1|b_0, a_),a_1,a_2)$ are independently
distributed.  It may be mentioned that many well known distribution has log-concave PDF.  For example, log-normal, gamma, Weibull
have log-concave PDFs.  $\pi(\alpha)$ also has its hyper parameters.  We do not make it explicit here,  whenever it is needed, we will make it explicit.

\subsection{\sc Prior Assumptions: Order Restricted}

If we have a order restriction on the scale parameters as $\lambda_1 < \lambda_2$, then we make the following prior assumption
on $\lambda_1$ and $\lambda_2$.
\begin{eqnarray}
\pi_2(\lambda_1, \lambda_2|a_0,b_0,a_1,a_2) & = & \frac{\Gamma(a_1+a_2)}{\Gamma(a_0) \Gamma(a_1)\Gamma(a_2)}
b_0^{a_0}  \lambda^{a_0-a_1-a_2} e^{-b_0 (\lambda_1+\lambda_2)}   \times  \nonumber  \\
&  & \left (\lambda_1^{a_1-1} \lambda_2^{a_2-1} + \lambda_1^{a_2-1} \lambda_2^{a_1-1} \right ); \ \ \  0 < \lambda_1 < \lambda_2 < \infty.
\label{prior-2}
\end{eqnarray}
We will call it as the ordered Beta-Gamma PDF, and it will be denoted by OBG($a_0, b_0, a_1, a_2$).  Note that (\ref{prior-2})
is the PDF of the ordered random variable $(\lambda_{(1)}, \lambda_{(2)})$, where $(\lambda_{(1)}, \lambda_{(2)}) =
(\lambda_1, \lambda_2)$ if $\lambda_1 < \lambda_2$, $(\lambda_{(1)}, \lambda_{(2)}) =
(\lambda_2, \lambda_1)$ if $\lambda_2 < \lambda_1$ and $(\lambda_1, \lambda_2)$ follows $(\sim$) BG($a_0, b_0, a_1, a_2$).
It may be noted that the generation from a ordered Beta-Gamma distribution is also quite straight forward.  First we
generate sample from a Beta-Gamma distribution and then by ordering them, we obtain a random sample from a
ordered Beta-Gamma distribution.
We assume the same prior $\pi(\alpha)$ on $\alpha$ as before and $\alpha$ and $(\lambda_1, \lambda_2)$ are assumed to be
independently distributed.

\section{\sc Maximum Likelihood Estimators}

Based on the observations described in ({\ref{data}), the log-likelihood function without the additive constant becomes:
\begin{equation}
l(data|\alpha,\lambda_1,\lambda_2)  =  k \ln \alpha +  k_1 \ln \lambda_1 +  k_2 \ln \lambda_2 + (\alpha-1)
\sum_{j=1}^k \ln t_j - \lambda_1 U(\alpha) - \lambda_2 V(\alpha),   \label{ll-1}
\end{equation}
here $k_1$, $k_2$, $U(\alpha)$ and $V(\alpha)$ are same as defined before.
The following result provides the uniqueness of the MLEs of $\lambda_1$ and $\lambda_2$ for a given $\alpha$.

\noindent {\sc Theorem 1:} If $k_1 > 0$ and $k_2 > 0$, then for a fixed $\alpha > 0$, $g_1(\lambda_1, \lambda_2) =
\displaystyle l(data|\alpha,\lambda_1,\lambda_2)$
is a unimodal function of $(\lambda_1, \lambda_2)$.

\noindent {\sc Proof:} Note that $\displaystyle g_1(\lambda_1, \lambda_2)$ is a concave function as the Hessian matrix of $g_1(\lambda_1, \lambda_2)$
is a negative definite matrix.    Now the result follows because
for fixed $\lambda_1 (\lambda_2)$, $g_1(\lambda_1, \lambda_2)$ tends to $-\infty$, as $\lambda_2 (\lambda_1)$ tends to 0, or
$\infty$.   \hfill \vrule height7pt width 5pt depth 0pt

For known $\alpha$, the MLEs of $\lambda_1$ and $\lambda_2$, say $\widehat{\lambda}_1(\alpha)$ and $\widehat{\lambda}_2(\alpha)$,
respectively, can be obtained as follows:
\begin{equation}
\widehat{\lambda}_1(\alpha) = \frac{k_1}{U(\alpha)} \ \ \ \ \hbox{and} \ \ \
\widehat{\lambda}_2(\alpha) = \frac{k_2}{V(\alpha)}.     \label{mle-rl}
\end{equation}
When $\alpha$ is unknown, first the MLE of $\alpha$ can be obtained by maximizing the profile log-likelihood function
of $\alpha$ without the additive constant, and that is
\begin{equation}
p_1(\alpha)  =  l(data|\alpha, \widehat{\lambda}_1(\alpha),\widehat{\lambda}_2(\alpha))  =
k \ln \alpha - k_1 \ln U(\alpha) - k_2 \ln V(\alpha) + (\alpha-1) \sum_{j=1}^k \ln t_j.
\end{equation}
The following result will provide the existence and uniqueness of the MLE of $\alpha$.

\noindent {\sc Theorem 2:} If $k_1 > 0$ and $k_2 > 0$, $p_1(\alpha)$ is a unimodal function of $\alpha$.

\noindent {\sc Proof:} See in the appendix A.

Therefore, combining Theorem 1 and Theorem 2, it is immediately obtained that for $k_1 > 0$ and $k_2 > 0$, the MLEs of
$\alpha$, $\lambda_1$ and $\lambda_2$ are unique.  It is quite simple to compute the MLE of $\alpha$ as $p_1(\alpha)$ is a unimodal
function.  Use the bisection or Newton-Raphson method to compute the MLEs of
$\alpha$, and once the MLE of $\alpha$ is obtained the MLEs of $\lambda_1$ and $\lambda_2$ can be obtained
from (\ref{mle-rl}).  Although, the MLEs can be obtained quite efficiently in this case, the exact distribution of the MLEs
is not possible to obtain.  Hence, the construction of confidence intervals of the unknown parameters may not be very simple.
The Fisher information matrix may be used to construct the asymptotic confidence intervals of the unknown parameters and it
is provided in Appendix B.  Alternatively, we propose to use the bootstrap method to construct the confidence intervals of the
unknown parameters in this case.  Since the exact confidence intervals cannot be obtained, the Bayesian inference seems to be a natural choice in this case.

\section {\sc Bayes Estimates and Credible Intervals}

In this section we provide the Bayes estimates of the unknown parameters, and the corresponding credible sets based on JPC
scheme as described before.  We mainly assume the squared error
loss function, although any other loss function can be easily incorporated.
Now to compute the Bayes estimates of the unknown parameters, we need to assume some specific form of the prior
distribution of $\alpha$, and it is assumed that $\pi(\alpha)$ has a Gamma($a, b)$ distribution.  Hence,
the joint posterior density function of $\lambda_1, \lambda_2$ and $\alpha$ can be written as
\begin{eqnarray}
\pi(\lambda_1, \lambda_2, \alpha| Data) & \propto & (\lambda_1+\lambda_2)^{a_0-a_1-a_2} \lambda_1^{a_1+k_1-1} \lambda_2^{a_2+k_2-1}
e^{-\lambda_1(b_0+U(\alpha))} e^{-\lambda_2(b_0+V(\alpha))}   \nonumber  \\
&  & \alpha^{k+a-1} e^{-b \alpha}   \prod_{i=1}^k t_i^{\alpha}.   \label{post}
\end{eqnarray}
We re-write (\ref{post}) in the following manner
\begin{eqnarray}
\pi(\lambda_1, \lambda_2, \alpha| Data) & \propto & (\lambda_1+\lambda_2)^{a_0-a_1-a_2} \lambda_1^{a_1+k_1-1} \lambda_2^{a_2+k_2-1}
e^{-(\lambda_1+\lambda_2)(b_0+W(\alpha))}    \nonumber  \\
&  & \alpha^{k+a-1} e^{-\alpha(b - \sum_{i=1}^k \ln t_i)} \times e^{-\lambda_1(U(\alpha) - W(\alpha))} e^{-\lambda_2(V(\alpha) - W(\alpha)}.   \label{post-2}
\end{eqnarray}
Here $W(\alpha) = \min\{U(\alpha), V(\alpha)\}$.  The posterior density function
of $\alpha$, $\lambda_1$ and $\lambda_2$ can be written as
\begin{equation}
\pi(\alpha, \lambda_1, \lambda_2|Data) \propto \pi_1^*(\lambda_1,\lambda_2|data, \alpha) \times
\pi_2^*(\alpha|data) \times g(\alpha,\lambda_1,\lambda_2|data).
\end{equation}
Here $\pi_1^*(\lambda_1, \lambda_2|\alpha, Data)$ is the PDF of a BG$\displaystyle (a_0+k_1+k_2,b_0+W(\alpha),a_1+k_1, a_2+k_2)$, and
\begin{equation}
\pi_2^*(\alpha) \propto \frac{\alpha^{k+a-1} e^{-\alpha((b - \sum_{i=1}^k \ln t_i))}}{(b_0+W(\alpha))^{a_0+k}}, \ \ \ \
g(\alpha, \lambda_1, \lambda_2|Data) = e^{-\lambda_1(U(\alpha) - W(\alpha))} e^{-\lambda_2(V(\alpha) - W(\alpha))}
\end{equation}
Therefore, the Bayes estimate of $h(\alpha,\lambda_1,\lambda_2)$, any function of $\alpha$, $\lambda_1$, $\lambda_2$
with respect to squared error loss function is
\begin{equation}
E_{\pi(\alpha,\lambda_1,\lambda_2|Data)}(h(\alpha,\lambda_1,\lambda_2))  =
\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} h(\alpha,\lambda_1,\lambda_2)
\pi(\alpha, \lambda_1, \lambda_2|Data) d\alpha d \lambda_1 d \lambda_2  = \frac{K_1}{K_2},    \label{post-exp}
\end{equation}
provided it exists.  Here
\begin{equation}
K_1 =
\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} h(\alpha,\lambda_1,\lambda_2) \times
\pi_1^*(\lambda_1,\lambda_2|Data, \alpha) \times
\pi_2^*(\alpha|data) \times g(\alpha,\lambda_1,\lambda_2|Data) d\alpha d \lambda_1 d \lambda_2.    \label{exp-k1}
\end{equation}
and
\begin{equation}
K_2 =
\int_0^{\infty} \int_0^{\infty} \int_0^{\infty}
\pi_1^*(\lambda_1,\lambda_2|Data, \alpha) \times
\pi_2^*(\alpha|data) \times g(\alpha,\lambda_1,\lambda_2|Data) d\alpha d \lambda_1 d \lambda_2.    \label{exp-k2}
\end{equation}
It is clear that (\ref{post-exp}) cannot be obtained in closed form.  We may use Lindley's approximation to compute
(\ref{post-exp}), but we may not be able to compute the associated credible interval using that.  We propose to use
importance sampling technique to compute simulation consistent Bayes estimate and the associated credible interval.
The details will be explained later.  We use the following result for that purpose.

\noindent {\sc Theorem 3:} The density function $\pi_2^*(\alpha|Data)$ is log-concave.

\noindent {\sc Proof:} It can be obtained along the same line as in Theorem 1, the details are avoided.    \hfill \vrule height7pt width 5pt depth 0pt

Therefore, it is quite simple to generate samples from $l(\alpha|Data)$ using the method of Devroye (1984) or Kundu (2008), and
for a given $\alpha$, random samples from  $l(\lambda_1, \lambda_2|Data, \alpha)$ can be easily generated using
the method of Kundu and Pradhan (2011).  The following algorithm can be used to compute the Bayes estimate and also the
associated HPD credible interval of $h(\alpha,\lambda_1,\lambda_2)$.

\noindent {\sc Algorithm}

\begin{itemize}

\item Step 1: Generate $\alpha$ from $\pi^*_2(\alpha|data)$.

\item Step 2: For a given $\alpha$, generate $\lambda_1$ and $\lambda_2$ from $\pi^*_1(\lambda_1,\lambda_2|data,\alpha)$.

\item Step 3: Repeat the procedure $N$ times to generate $(\alpha_1, \lambda_{11}, \lambda_{21}), \ldots, (\alpha_N, \lambda_{1N},
\lambda_{2N})$.

\item Step 4: To obtain Bayes estimate of $h(\alpha, \lambda_1, \lambda_2)$, compute $(h_1,\ldots, h_N)$, where $h_i=h(\alpha_i, \lambda_{1i},  \lambda_{2i})$ as well as compute $(g_1, \ldots , g_N)$, where $g_i=g(\alpha_i, \lambda_{1i},  \lambda_{2i})$.
\item Step 5:  Bayes estimate of $h(\beta, \lambda_1, \lambda_2)$ can be approximated as $\frac{\sum_{i=1}^{N}g_ih_i}{\sum_{j=1}^{N}g_j} =\sum_{i=1}^{N}{v}_i h_i$ where ${v}_i=\frac{g_i}{\sum_{j=1}^{N}g_j}$.
\item Step 6: To compute $100(1-\beta)\%$ CRI of $h( \alpha, \lambda_1, \lambda_2)$ , arrange ${h_i}$ in ascending order to obtain $(h_{(1)},\ldots h_{(N)})$ and record the corresponding ${v_i}$ as $(v_{(1)}, \ldots, v_{(N)})$.  A $100(1-\gamma)\%$ CRI can be obtained as $(h_{(j_1)}, h_{(j_2)})$ where $j_1, j_2$ such that

\begin{eqnarray}
\label{eq}
j_1 < j_2, \quad  j_1 , j_2 \in  \{ 1, \ldots , N \} \quad  \mbox{and} \quad \sum_{i=j_1}^{j_2}v_i \leq 1-\beta <\sum_{i=j_1}^{j_2+1}v_i
\end{eqnarray}

The $100(1-\beta)\%$ highest posterior density (HPD) CRI can be obtained as $(h_{(j^*_1)}, h_{(j^*_2)})$, such that $h_{(j^*_2)} -h_{(j^*_1)}\leq  h_{(j_2)} - h_{(j_1)}$ and $j^*_1, j^*_2$ satisfying (\ref{eq}) for all $j_1, j_2$ satisfying (\ref{eq}).

\end{itemize}

\section{\sc Order Restricted Inference}

In this section we consider the inference on the unknown parameters under the restriction $\lambda_1 < \lambda_2$.  In many practical
situations experimenter may have the information that one population has a smaller expected lifetime than the other.  In our case, this
leads to the above restriction.  Therefore, our problem can be stated as follows.  Based on the same set of assumptions as in Section 2.2
and with $\lambda_1 < \lambda_2$, the problem is to estimate the unknown parameters $\alpha$, $\lambda_1$ and $\lambda_2$ using the data (\ref{data}).

\subsection{\sc Maximum Likelihood Estimators}

In this case also we proceed along the same line as before.  For a given  $\alpha$, the log-likelihood function (\ref{ll-1})
is concave as a function of $\lambda_1$ and $\lambda_2$ and it has a unique maximum.  The maximum value of the function (\ref{ll-1})
is obtained at the point ($\widehat{\lambda}_1(\alpha), \widehat{\lambda}_2(\alpha)$).  Clearly if $\widehat{\lambda}_1(\alpha) <
\widehat{\lambda}_1(\alpha)$, then the order restricted MLEs of $\lambda_1$ and $\lambda_2$, say $\widetilde{\lambda}_1(\alpha) =
\widehat{\lambda}_1(\alpha)$ and $\widetilde{\lambda}_2(\alpha) = \widetilde{\lambda}_2(\alpha)$, respectively.  On the other hand
if $\widehat{\lambda}_1(\alpha) \ge \widehat{\lambda}_1(\alpha)$, then the maximum value of $g_1(\lambda_1, \lambda_2)$ will be on
the line $\lambda_1 = \lambda_2$ under the order restriction $\lambda_1 < \lambda_2$.  Therefore, in this case
$$
\widetilde{\lambda}_1(\alpha) = \widetilde{\lambda}_2(\alpha) = \hbox{arg max } g_1(\lambda, \lambda).
$$
Hence for a given $\alpha$, the order restricted MLEs of $\lambda_1$ and $\lambda_2$ become
\begin{equation}
(\widetilde{\lambda}_1(\alpha), \widetilde{\lambda}_2(\alpha)) =
\left \{ \matrix{(\widehat{\lambda}_1(\alpha), \widehat{\lambda}_2(\alpha)) & \hbox{if} & \widehat{\lambda}_1(\alpha) <
\widehat{\lambda}_2(\alpha)  \cr
\left (\frac{k}{\sum_{j=1}^k(R_j+1)t_j^{\alpha}}, \frac{k}{\sum_{j=1}^k(R_j+1)t_j^{\alpha}} \right )
& \hbox{if} & \widehat{\lambda}_1(\alpha) \ge \widehat{\lambda}_2(\alpha).  \cr   \label{rl1rl2}
}  \right .
\end{equation}
The MLE of $\alpha$, say $\widetilde{\alpha}$ can be obtained by maximizing $p_2(\alpha) = l(data|\alpha, \widetilde{\lambda}_1(\alpha), \widetilde{\lambda}_2(\alpha))$ with respect to $\alpha$.  The following result provides the existence and uniqueness of the MLE of
$\alpha$.

\noindent {\sc Theorem 4:} If $k_1 > 0$ and $k_2 > 0$, $p_2(\alpha)$ is a unimodal function of $\alpha$.

\noindent {\sc Proof:} The result follows along the same line as in Theorem 2 by observing the fact that $p_2(\alpha)$ is log-concave
in both the region, and $p_2(\alpha)$ is a continuous function of $\alpha$.  \hfill \vrule height7pt width 5pt depth 0pt

Once the MLE of $\alpha$ is obtained, the MLEs of $\lambda_1$ and $\lambda_2$ can be obtained from (\ref{rl1rl2}) explicitly.  We
propose to use bootstrap method to construct the confidence intervals of the unknown parameters.

\subsection{\sc Bayes Estimates and Credible Intervals}

In this section we will provide the order restricted Bayesian inference of the unknown parameters based on the prior assumptions as
provided in Section 2.4.  Similarly as before, for specific implementation we assume that $\pi(\alpha)$ has a Gamma$(a,b)$ distribution.
The joint posterior density function of $\alpha$, $\lambda_1$ and $\lambda_2$ for
$\alpha > 0, 0 < \lambda_1 < \lambda_2$, can be written as
\begin{eqnarray}
\pi(\lambda_1, \lambda_2, \alpha| Data) & \propto & (\lambda_1+\lambda_2)^{a_0-a_1-a_2} \left ( \lambda_1^{a_1-1} \lambda_2^{a_2-1}
+ \lambda_1^{a_2-1} \lambda_2^{a_1-1} \right ) \lambda_1^{k_1} \lambda_2^{k_2}
  \nonumber  \\
&  & e^{-\lambda_1(b_0+U(\alpha))} e^{-\lambda_2(b_0+V(\alpha))}  \alpha^{k+a-1} e^{-b \alpha}   \prod_{i=1}^k t_i^{\alpha}.    \label{post-3}
\end{eqnarray}
We re-write (\ref{post-3}) as follows
\begin{eqnarray}
\pi(\lambda_1, \lambda_2, \alpha| Data) & \propto & (\lambda_1+\lambda_2)^{a_0-a_1-a_2} \left ( \lambda_1^{a_1+J-1} \lambda_2^{a_2+J-1}
+ \lambda_1^{a_2+J-1} \lambda_2^{a_1+J-1} \right ) e^{-(\lambda_1+\lambda_2)(b_0+W(\alpha))}
  \nonumber  \\
&  & \lambda_1^{k_1-J} \lambda_2^{k_2-J} e^{-\lambda_1(U(\alpha)-W(\alpha))} e^{-\lambda_2(V(\alpha)-W(\alpha))}
\alpha^{k+a-1} e^{-\alpha(b - \sum_{i=1}^k \ln t_i)},     \label{post-4}
\end{eqnarray}
here $J = \min\{k_1, k_2\}$ and $W(\alpha)$ is same as defined before.

The posterior density function of $\alpha$, $\lambda_1$ and $\lambda_2$ in this case can be written as
\begin{equation}
\pi(\alpha, \lambda_1, \lambda_2|Data) \propto \pi_1^*(\lambda_1,\lambda_2|data, \alpha) \times
\pi_2^*(\alpha|data) \times g(\alpha,\lambda_1,\lambda_2|data).
\end{equation}
Here $\pi_1^*(\lambda_1, \lambda_2|\alpha, Data)$ is the PDF of a OBG$\displaystyle (a_0+2J,b_0+W(\alpha),a_1+J, a_2+J)$, and
\begin{equation}
\pi_2^*(\alpha) \propto \frac{\alpha^{k+a-1} e^{-\alpha((b - \sum_{i=1}^k \ln t_i))}}{(b_0+W(\alpha))^{a_0+2J}}
\end{equation}
and
\begin{equation}
g(\alpha, \lambda_1, \lambda_2|Data) = \lambda_1^{k_1-J} \lambda_2^{k_2-J} e^{-\lambda_1(U(\alpha) - W(\alpha))} e^{-\lambda_2(V(\alpha) - W(\alpha))}.
\end{equation}
Since $\pi_2^*(\alpha|Data)$ is a log-concave function, and the generation from a $\pi_1^*(\lambda_1, \lambda_2|\alpha)$ can be
performed quite conveniently, we can use the same importance sampling technique as in Section 4, to compute the Bayes estimate
and the associated HPD credible interval of any function of $\alpha$, $\lambda_1$ and $\lambda_2$.

\section{\sc Numerical Experiments and Data Analysis}

\subsection{\sc Numerical Experiments}

In this section we have performed some simulation experiments to see the effectiveness of the proposed methods and also to
observe how the ordered restricted inference behaves in practice.  We have considered $m$ = 20, $n$ = 25, $\lambda_1$ = 1.0,
$\lambda_2$ = 0.5.  We have taken different effective sample sizes, $k$ = 20, 25, different censoring schemes, and different
$\alpha$ values, $\alpha$ = 1 and 2.

In each case we obtain the MLEs of the unknown parameters.  We compute the average estimates (AE) and the mean squared errors (MSE) based
on 10,000 replications.  The results are reported in Table \ref{table-1} and Table \ref{table-2}.  In each case we also compute
90\% symmetric percentile bootstrap confidence interval based on 500 bootstrap samples.  We repeat the process 1000 times, and
obtain the average lengths (AL) of the confidence intervals and their coverage percentages (CP).  The results are reported in Table \ref{table-3} and Table \ref{table-4}.

We further compute the Bayes
estimates and the associated 90\% credible intervals based on both informative priors (IP) and non-informative priors (NIP).
In case of non-informative
priors it is assumed that $b_0 = a_0 = a_1 = a_2 = a = b$ = 0.  For informative priors, when $\alpha$ = 1,
then $b_0$ = 1, $\displaystyle a_0 = \frac{3}{2} b_0$, $a_1$ = 2, $a_2$ = 4, $a$ = 2, $b$ = 2, and when $\alpha$ = 2,
$b_0$ = 1, $\displaystyle a_0 = \frac{3}{2} b_0$, $a_1$ = 2, $a_2$ = 4, $a$ = 4, $b$ = 2.  We compute the Bayes estimates and the associated credible intervals based on 1000 samples.  We report the average Bayes
estimates and the corresponding MSEs in Table \ref{table-1} and Table \ref{table-2}.  The average lengths of the credible intervals
and the associated coverage percentages are reported in Table \ref{table-3} and Table \ref{table-4}.  We use the following notation for a particular progressive censoring scheme.  For example $k=6$ and $R = (4, 0_{(5)})$ means $R_1$ = 4, $R_2 = R_3 = R_4 = R_5 = R_6=0$.

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=1,\lambda_1=0.5, \lambda_2=1$}
\label{table-1}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter}
&\multicolumn{2}{c}{MLE} & \multicolumn{2}{l}{Bayes IP}  & \multicolumn{2}{l}{Bayes NIP}\\
&&\multicolumn{1}{c}{AE} & \multicolumn{1}{c}{MSE}
&\multicolumn{1}{c}{AE}  &\multicolumn{1}{l}{MSE} & \multicolumn{1}{l}{AE} &\multicolumn{1}{l}{MSE} \\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ & 1.097 &0.063 & 1.075 &0.050 &1.095& 0.064\\
 & $\lambda_1$ &0.554&0.057&0.536 &0.035 &0.543 &0.047\\
 & $\lambda_2$ &1.102&0.147&1.066&0.086&1.086&0.123\\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ & 1.101 &0.067&1.062&0.048 &1.066 &0.055\\
 &$\lambda_1$ & 0.562&0.062&0.535&0.031&0.557&0.057\\
 & $\lambda_2$ &1.123 &0.185&1.064&0.077&1.081&0.127\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &1.105&0.072&1.068&0.052&1.064&0.060\\
 & $\lambda_1$ &0.569&0.074&0.548&0.035&0.551&0.052\\
 & $\lambda_2$ &1.126&0.191&1.077&0.092&1.085&0.117\\
 \midrule
 k=25,R=(7,0$_{(23)}$,10) &$\alpha$ & 1.077 & 0.044 & 1.063&0.043&1.064&0.040\\
   & $\lambda_1$ & 0.532&0.036&0.523 & 0.025&0.532& 0.036\\
   & $\lambda_2$ & 1.062&0.095 &1.042 & 0.068&1.044&0.085\\
   \midrule
 k=25,R=(0$_{(11)}$,7,0$_{(12)}$,10) & $\alpha$ & 1.078&0.044& 1.062&0.036&1.075&0.044\\
 & $\lambda_1$ & 0.537 & 0.038 & 0.524&0.025&0.530&0.034\\
 & $\lambda_2$ & 1.071&0.099& 1.053&0.069 &1.064&0.101\\
 \midrule
 k=25,R=(0$_{(23)}$,7,10)& $\alpha$ & 1.080 & 0.049 & 1.073& 0.043&1.079&0.047\\
 & $\lambda_1$ & 0.540& 0.040 & 0.529&0.026&0.535&0.035\\
 & $\lambda_2$ & 1.071 & 0.099&1.059&0.075&1.069&0.098\\
 \bottomrule

\end{tabular}
}
\end{center}

\end{table}

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=2,\lambda_1=0.5, \lambda_2=1$}
\label{table-2}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter}
&\multicolumn{2}{c}{MLE} & \multicolumn{2}{l}{Bayes IP}  & \multicolumn{2}{l}{Bayes NIP}\\
&&\multicolumn{1}{c}{AE} & \multicolumn{1}{c}{MSE}
&\multicolumn{1}{c}{AE}  &\multicolumn{1}{l}{MSE} & \multicolumn{1}{l}{AE} &\multicolumn{1}{l}{MSE} \\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ &2.191 &0.252 & 2.127 &0.157&2.185&0.243\\
 & $\lambda_1$ &0.555&0.057&0.545&0.035&0.550&0.049\\
 & $\lambda_2$ &1.097&0.143&1.055&0.078&1.104&0.131\\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &2.209&0.264&2.108&0.145&2.147&0.228\\
 &$\lambda_1$ &0.563&0.063&0.536&0.034&0.556&0.049\\
 & $\lambda_2$ &1.118&0.165&1.097&0.099&1.081&0.126\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &2.207&0.273&2.120&0.157&2.160&0.251\\
 & $\lambda_1$ &0.561&0.067&0.548&0.035&0.541&.046\\
 & $\lambda_2$ &1.122&0.183&1.064&0.082&1.112&0.148\\
 \midrule
 k=25,R=(7,0$_{(23)}$,10) & $\alpha$ & 2.149 & 0.169 &2.133&0.151&2.139&0.171\\
 & $\lambda_1$ & 0.534 & 0.037 &0.523&0.025&0.524&0.034\\
 & $\lambda_2$ & 1.065&0.101&1.056&0.073&1.061&0.094\\
 \midrule
 k=25,R=(0$_{(11)}$,7,0$_{(12)}$,10) & $\alpha$ & 2.150& 0.176 &2.127&0.135&2.146&0.192\\
 & $\lambda_1$ & 0.5380& 0.036 &0.523&0.023&0.541&0.035\\
 & $\lambda_2$ & 1.066&0.102&1.045&0.068&1.047&0.088\\
 \midrule
 k=25,R=(R=(0$_{(23)}$,7,10) & $\alpha$ & 2.156 &0.193 &2.122&0.144&2.132&0.191\\
  & $\lambda_1$ &0.537 & 0.038 &0.527&0.025&0.545&0.039\\
  & $\lambda_2$ &1.071&0.101&1.048&0.062&1.049&0.093 \\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=1,\lambda_1=0.5, \lambda_2=1$}
\label{table-3}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter} & \multicolumn{2}{l}{90\% HPD CRI IP}  & \multicolumn{2}{l}{90\% HPD CRI NIP} &\multicolumn{2}{l}{90\% Bootstrap CI}\\
&&\multicolumn{1}{c}{AL} & \multicolumn{1}{l}{CP}
&\multicolumn{1}{c}{AL}  &\multicolumn{1}{l}{CP} &\multicolumn{1}{c}{AL}  &\multicolumn{1}{l}{CP}\\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ & 0.621 & 86.6\% &0.663&82.8\% &0.804 & 82.2\%\\
 & $\lambda_1$ &0.560&90.0\%&0.627&86.8\% & 0.814 & 87.0\%\\
 & $\lambda_2$&0.894&89.6\%&1.018&87.6\% &1.358&83.8\% \\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &0.603&84.6\%&0.624&85.0\% & 0.818&83.0\%\\
 &$\lambda_1$ &0.565&87.6\%&0.635&87.2\% &0.907&85.6\% \\
 & $\lambda_2$ &0.900&91.6\%&0.977&89.2\% &1.454&82.8\%\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &0.615&87.0\%&0.624&82.0\% & 0.854 & 83.6\%\\
 & $\lambda_1$ &0.582&92.0\%&0.649&87.2\% & 0.900&87.2\%\\
 & $\lambda_2$ &0.903&91.4\%&0.981&89.4\%& 1.463 &86.0\%\\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=2,\lambda_1=0.5, \lambda_2=1$}
\label{table-4}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter} & \multicolumn{2}{l}{90\% HPD CRI IP}  & \multicolumn{2}{l}{90\% HPD CRI NIP} &\multicolumn{2}{l}{90\% Bootstrap CI}\\
&&\multicolumn{1}{c}{AL} & \multicolumn{1}{l}{CP}
&\multicolumn{1}{c}{AL}  &\multicolumn{1}{l}{CP} &\multicolumn{1}{c}{AL} &\multicolumn{1}{l}{CP}\\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ & 1.198 & 90.8\% &1.309&87.4\% &1.612&80.2\%\\
 & $\lambda_1$ &0.562&88.2\%&0.639&85.0\%&0.827&87.2\%\\
 & $\lambda_2$&0.897&90.8\%&0.993&89.6\% &1.293&87.8\%\\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &1.166&89.4\%&1.209&82.2\% &1.690&77.2\%\\
 &$\lambda_1$ &0.569&91.4\%&0.614&85.0\% &0.903&88.4\%\\
 & $\lambda_2$ &0.910&90.4\%&1.000&88.4\% &1.492&85.6\%\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &1.171&89.0\%&1.258&81.8\% &1.758&81.0\%\\
 & $\lambda_1$ &0.572&90.6\%&0.652&85.0\% &0.902&87.8\%\\
 & $\lambda_2$ &0.907&91.8\%&0.988&89.0\%&1.468&84.4\%\\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

From Table \ref{table-1} and Table \ref{table-2}, it is clear that as the
effective sample size increases in all the cases the average biases and the MSEs decrease.  The performances of the Bayes estimators with respect to the NIPs  are slightly better than the MLEs both in terms of biases and MSEs in most of the cases investigated here.  Also the Bayes estimators
with respect to IPS are performing better than the Bayes estimators with respect to NIPs in terms of the biases and MSEs, as expected.
In Table \ref{table-3} and Table \ref{table-4} we report the performances of the percentile bootstrap confidence intervals and the Bayes credible intervals.  In most of the cases the coverage percentages are very close to the nominal values.  The average credible lengths based on the non-informative priors are larger than the informative priors, but they are slightly lower than the average bootstrap confidence intervals.

Therefore, it is clear in this case that the Bayesian inference is preferable than the classical inference.
If we have some prior information on the unknown parameters then we should use the Bayes estimates and the
associated credible intervals with respect to the informative priors, otherwise we should use the non-informative priors.

Further, we perform the order restricted inference of the unknown parameters.  In this case we have taken the same set of
parameter values, the sample sizes and the censoring schemes mainly for comparison purposes.  For the Bayesian inference we have considered the same set of
hyper parameters also.  The average estimates and the associated MSEs are reported in Table \ref{table-5} and Table \ref{table-6}.
The confidence and credible intervals are reported in Table \ref{table-7} and Table \ref{table-8}.

From Tables \ref{table-5} and
\ref{table-6}, it is clear that the performances of the Bayes estimators based on the non-informative priors behave slightly better than the MLEs in terms of MSEs
particularly for the scale parameters.  In this case also the performances of the Bayes estimators with respect to the informative
priors perform much better than the non-informative priors, as expected.  From Tables \ref{table-7} and \ref{table-8}, it is observed
that coverage percentages of the bootstrap confidence intervals are very similar to the Bayes estimators based on the non-informative
priors.  The coverage percentages of the bootstrap confidence intervals are slightly closer to the nominal value than the Bayes
estimator with respect to the non-informative priors in most of the cases considered here.

From above discussion we can conclude that when the order restriction is imposed on the scale parameters, and if we have some prior information
about the unknown parameters, then the Bayesian inference with respect to the informative priors is preferable, otherwise non-informative
priors may be used.

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=1,\lambda_1=0.5, \lambda_2=1$ \\(With  Order Restriction)}
\label{table-5}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter}
&\multicolumn{2}{c}{MLE} & \multicolumn{2}{l}{Bayes inf-prior}  & \multicolumn{2}{l}{Bayes noninf-prior}\\
&&\multicolumn{1}{c}{AE} & \multicolumn{1}{c}{MSE}
&\multicolumn{1}{c}{AE}  &\multicolumn{1}{l}{MSE} & \multicolumn{1}{l}{AE} &\multicolumn{1}{l}{MSE} \\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ &1.093& 0.063& 1.080&0.051&1.078&0.056\\
 & $\lambda_1$ &0.551&0.052&0.526&0.024&0.511&0.031\\
 & $\lambda_2$ &1.113&0.155&1.041&0.067&1.103&0.110\\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &1.102&0.064&1.057&0.055&1.077&0.057\\
 &$\lambda_1$ &0.553&0.058&0.527&0.026&0.506&0.034\\
 & $\lambda_2$ &1.124&0.178&1.051&0.076&1.095&0.109\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &1.102&0.072&1.078&0.054&1.081&0.066\\
 & $\lambda_1$ &0.560&0.062&0.535&0.028&0.504&0.028\\
 & $\lambda_2$ &1.129&0.184&1.057&0.080&1.085&0.102\\
 \midrule
 k=25,R=(7,0$_{(23)}$,10) &$\alpha$ &1.075&0.043&1.081&0.052&1.066&0.045\\
   & $\lambda_1$ &0.529&0.033&0.523&0.020&0.508&0.022\\
   & $\lambda_2$ &1.068&0.986&1.032&0.062&1.056&0.081\\
   \midrule
 k=25,R=(0$_{(11)}$,7,0$_{(12)}$,10) & $\alpha$ &1.079&0.045&1.053&0.035&1.079&0.048\\
 & $\lambda_1$ &0.535&0.034&0.529&0.022&0.501&0.022\\
 & $\lambda_2$ &1.072&0.099&1.034&0.058&1.073&0.080\\
 \midrule
 k=25,R=(0$_{(23)}$,7,10)& $\alpha$ &1.082&0.050&1.061&0.042&1.070&0.046\\
 & $\lambda_1$ &0.536&0.036&0.526&0.021&0.508&0.023\\
 & $\lambda_2$ &1.075&0.101&1.029&0.058&1.065&0.075\\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=2,\lambda_1=0.5, \lambda_2=1$\\ (With  Order Restriction)}
\label{table-6}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter}
&\multicolumn{2}{c}{MLE} & \multicolumn{2}{l}{Bayes inf-prior}  & \multicolumn{2}{l}{Bayes noninf-prior}\\
&&\multicolumn{1}{c}{AE} & \multicolumn{1}{c}{MSE}
&\multicolumn{1}{c}{AE}  &\multicolumn{1}{l}{MSE} & \multicolumn{1}{l}{AE} &\multicolumn{1}{l}{MSE} \\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ &2.185&0.247&2.123&0.160&2.161&0.239\\
 & $\lambda_1$ &0.548&0.051&0.528&0.028&0.505&0.033\\
 & $\lambda_2$ &1.107&0.147&1.056&0.075&1.091&0.110\\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &2.197&0.266&2.108&0.153&2.146&0.219\\
 &$\lambda_1$ &0.555&0.057&0.524&0.025&0.508&0.032\\
 & $\lambda_2$ &1.119&0.167&1.046&0.074&1.092&0.113\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &2.198&0.271&2.111&0.163&2.145&0.249\\
 & $\lambda_1$ &0.558&0.058&0.525&0.025&0.521&0.034\\
 & $\lambda_2$ &1.129&0.171&1.047&0.069&1.089&0.106\\
 \midrule
 k=25,R=(7,0$_{(23)}$,10) & $\alpha$ &2.149&0.175&2.127&0.171&2.121&0.157\\
 & $\lambda_1$ &0.527&0.033&0.517&0.021&0.508&0.023\\
 & $\lambda_2$ &1.061&0.094&1.034&0.063&1.072&0.082\\
 \midrule
 k=25,R=(0$_{(11)}$,7,0$_{(12)}$,10) & $\alpha$ &2.149&0.176&2.115&0.126&2.102&0.150\\
 & $\lambda_1$ &0.536&0.033&0.514&0.020&0.502&0.020\\
 & $\lambda_2$ &1.073&0.101&1.024&0.063&1.058&0.073\\
 \midrule
 k=25,R=(R=(0$_{(23)}$,7,10) & $\alpha$ &2.169&0.201&2.126&0.144&2.154&0.180\\
  & $\lambda_1$ &0.540&0.039&0.530&0.022&0.502&0.023\\
  & $\lambda_2$ &1.079&0.102&1.039&0.060&1.052&0.072\\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=1,\lambda_1=0.5, \lambda_2=1$\\ (With Order Restriction)}
\label{table-7}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter} & \multicolumn{2}{l}{90\% HPD CRI inf-prior}  & \multicolumn{2}{l}{90\% HPD CRI noninf-prior}& \multicolumn{2}{l}{90\% Bootstrap CI}\\
&&\multicolumn{1}{c}{AL} & \multicolumn{1}{l}{CP}
&\multicolumn{1}{c}{AL}  &\multicolumn{1}{l}{CP}&\multicolumn{1}{c}{AL}  &\multicolumn{1}{l}{CP}\\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ &0.600&83.8\% &0.624&82.2\%&0.801&80.8\%\\
 & $\lambda_1$ &0.502&88.6\%&0.515&86.2\%&0.740&89.2\%\\
 & $\lambda_2$&0.711&83.4\%&0.722&81.4\%&1.312&84.0\% \\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &0.587&83.6\%&0.612&82.4\% &0.813&81.4\%\\
 &$\lambda_1$ &0.515&88.5\%&0.537&87.2\% &0.763&89.8\%\\
 & $\lambda_2$ &0.685&80.0\%&0.731&81.4\%&1.368&86.6\%\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &0.591&81.4\%&0.630&81.0\%&0.865&83.0\%\\
 & $\lambda_1$ &0.517&91.4\%&0.534&82.4\%&0.829&89.8\%\\
 & $\lambda_2$ &0.683&82.6\%&0.728&77.0\%&1.433&85.8\%\\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

\begin{table}[h]
\caption{ $m=20,n=22,\alpha=2,\lambda_1=0.5, \lambda_2=1$\\ (With Order Restriction)}
\label{table-8}
\begin{center}
\scalebox{.7}{
\begin{tabular}{llllllll
\toprule
\multicolumn{1}{c}{Censoring scheme} & \multicolumn{1}{c}{Parameter} & \multicolumn{2}{l}{90\% HPD CRI inf-prior}  & \multicolumn{2}{l}{90\% HPD CRI noninf-prior}& \multicolumn{2}{l}{90\% Bootstrap CI}\\
&&\multicolumn{1}{c}{AL} & \multicolumn{1}{l}{CP}
&\multicolumn{1}{c}{AL}  &\multicolumn{1}{l}{CP}&\multicolumn{1}{c}{AL} &\multicolumn{1}{l}{CP}\\
\midrule
k=20,R=(7,0$_{(18)}$,15) & $\alpha$ &1.141&87.8\% &1.280&84.6\%&1.618&80.0\%\\
 & $\lambda_1$ &0.499&89.2\%&0.517&86.8\% &0.743&88.8\%\\
 & $\lambda_2$&0.724&83.4\%&0.746&79.8\% &1.314&85.4\%\\
 \midrule
 k=20,R=(0$_{(9)}$ ,7,0$_{(9)}$,15) &$\alpha$ &1.135&88.4\%&1.230&82.0\%&1.639&82.4\% \\
 &$\lambda_1$ &0.492&89.2\%&0.526&84.8\%&0.780&88.6\%\\
 & $\lambda_2$ &0.703&82.8\%&0.740&80.2\%&1.378&88.0\%\\
 \midrule
 k=20,R=(0$_{(18)}$,7,15) & $\alpha$ &1.157&86.6\%&1.239&84.4\%&1.702&82.4\%\\
 & $\lambda_1$ &0.509&89.6\%&0.527&85.2\%&0.800&88.8\%\\
 & $\lambda_2$ &0.694&84.2\%&0.704&82.8\%&1.416&86.8\%\\
 \bottomrule
\end{tabular}
}
\end{center}
\end{table}

Now comparing the results between the ordered inference and the unordered inference, it is observed that the estimators obtained using
ordered information are slightly better than the unordered ones both the in terms of biases and MSEs.  Moreover, the confidence
intervals and the credible intervals based on the ordered inference are slightly smaller than the unordered ones.  Hence, if we have
some knowledge about the ordering on the scale parameters, it is better to use that information.

\subsection{\sc Data  Analysis}

In this section we present the analysis of real data sets to show how the different methods work in practice.  The data represent the strength measured in GPA for single carbon fibers and can be obtained from Kundu and Gupta (2006).  Single fibers were tested under tension at different gauge lengths.  Data set 1 are measurements on single fiber of gauge length 20 mm and Data set 2 are obtained from single fiber of gauge length 10 mm.  They are presented below for an easy reference:\\
Data  set 1:\\
1.312,1.314,1.479,1.552,1.700,1.803,1.861,1.865,1.944,1.958,1.966,1.997,2.006,2.021,
2.027,2.055,
2.063,2.098,2.140,2.179,2.224,2.240,2.253,2.270,2.272,2.274,2.301,2.301,2.359,2.382,2.382,2.426,
2.434,2.435,2.478,2.490,2.511,2.514,2.535,2.554,2.566,2.570,2.586,2.629,2.633,2.642,2.648,2.684,
2.697,2.726,2.770,2.773,2.800,2.809,2.818,2.821,2.848,2.880,2.954,3.012,3.067,3.084,3.090,3.096,
3.128,3.233,3.433,3.585,3.585.\\
Data set 2:\\Rightarrow.901,2.132,2.203,2.228,2.257,2.350,2.361,2.396,2.397,2.445,2.454,2.474,2.518,2.522,2.525,2.532,
2.575,2.614,2.616,2.618,2.624,2.659,2.675,2.738,2.740,2.856,2.917,2.928,2.937,2.937,2.977,2.996,
3.030,3.125,3.139,3.145,3.220,3.223,3.235,3.243,3.264,3.272,3.294,3.332,3.346,3.377,3.408,3.435,
3.493,3.501,3.537,3.554,3.562,3.628,3.852,3.871,3.886,3.971,4.024,4.027,4.225,4.395,
5.020.

Kundu and Gupta (2006) subtract 0.75 from both the data sets and fit Weibull distributions for both the modified data sets separately.  The MLEs of shape parameters and scale parameters and Kolmogorov-Smirnov (K-S) distances between empirical distribution functions and fitted distribution functions along with $p$ values are provided in Table \ref{data-test1} for both the data sets.  The K-S distances and $p$ values clearly indicate both the data sets after subtracting 0.75 are well fitted by Weibull distributions.

Along with the classical K-S test we also compute the Bayesian counterpart $p_b(data)$, the posterior predictive $p$ value based on the K-S discrepancies, see for example Gelman et al. (1996).   We compute the $p_b(data)$ based on
the non-informative prior and the results are reported in Table \ref{data-test1-bayesian}.

\begin{table}[h]
\caption{MLEs of parameters, K-S distance and $p$ values for the fitted Weibull models to modified Data set 1 and 2}
\label{data-test1}
\begin{center}
\begin{tabular}{lllll
\toprule
\multicolumn{1}{c}{data } &\multicolumn{2}{c}{MLE from complete sample} & \multicolumn{1}{c}{K-S distance} & \multicolumn{1}{c}{p value} \\
&\multicolumn{1}{c}{shape parameter }&\multicolumn{1}{c}{scale parameter } & &\\
\midrule
Data set1 & $\alpha= 3.843$& $\lambda=0.088$& 0.046& 0.998 \\
 \midrule
Data set 2 & $\alpha =3.909 $& $\lambda=0.025  $ & 0.079 & 0.815\\
\bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{table}[h]
\caption{Bayes estimates of parameters, expected posterior K-S and $p$ values for the fitted Weibull models to modified Data set 1 and 2}
\label{data-test1-bayesian}
\begin{center}
\begin{tabular}{lllll
\toprule
\multicolumn{1}{c}{data } &\multicolumn{2}{c}{Bayes estimate from complete sample} & \multicolumn{1}{c}{expected posterior } & \multicolumn{1}{c}{$p_b(data)$} \\
&\multicolumn{1}{c}{shape parameter }&\multicolumn{1}{c}{scale parameter } &\multicolumn{1}{l}{K-S distance} &\\
\midrule
Data set1 & $\alpha= 3.930$& $\lambda=0.082$& 0.061& 0.877 \\
 \midrule
Data set 2 & $\alpha =4.051 $& $\lambda=0.022  $ & 0.090 & 0.703\\
\bottomrule
\end{tabular}
\end{center}
\end{table}

 To test whether two data sets have common shape parameter, we perform the K-S test as well as conventional likelihood ratio test (L-R).  The MLE of the common shape parameter, two scale parameters along with $p$ values  are recorded in Table \ref{data-test2}.
 Similarly  we compute the posterior predictive $p$ values $p_b(data)$ to test the equality of the shape parameters based on the K-S test.
Results are provided in Table \ref{data-test2-bayesian}.  We have obtained the expected $p$ value of the LR test based on the posterior
density and it is 0.41.  Based on these values, it is reasonable to assume that both data sets are from Weibull distributions with equal shape
parameter.

 \begin{table}[h]
\caption{MLEs of parameters,  p value of L-R and K-S test for  common shape parameter on fitted Weibull models to modified Data set 1 and 2}
\label{data-test2}
\begin{center}
\begin{tabular}{llll
\toprule

\multicolumn{1}{c}{shape parameter } &\multicolumn{1}{c}{scale parameter} &  \multicolumn{1}{c}{p value(L-R test)}& \multicolumn{1}{c}{p value(K-S test)} \\
\toprule
 $\alpha= 3.876$ & $\lambda_1=0.0861$   & 0.895 & (data set1) 0.998\\
 & $\lambda_2=0.026$ & & (data set 2) 0.852\\
\bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{table}[h]
\caption{Bayes estimate of parameters,  expected posterior K-S distance and $p$ values for  common shape parameter on fitted Weibull models to modified Data set 1 and 2}
\label{data-test2-bayesian}
\begin{center}
\begin{tabular}{llll
\toprule
\multicolumn{1}{c}{shape parameter } &\multicolumn{1}{c}{scale parameter} &\multicolumn{1}{c}{expected posterior K-S distance} & \multicolumn{1}{c}{$p_b(data)$} \\
\toprule
 $\alpha= 4.074$ & $\lambda_1=0.0881$ & (data set1) 0.102  & (data set1) 0.552\\
 & $\lambda_2=0.030$ & (data set 2) 0.134 &(data set 2) 0.351\\
\bottomrule
\end{tabular}
\end{center}
\end{table}

 For illustrative purpose we have generated a joint progressive type-II censored sample with $k=20$, $R_i=4$ for $i=1,\ldots,19$ and $R_{20}=36$.  The data set is as follows:
(1.312,1,2) , (1.314,1,2), (1.479,1,1) ,(1.552,1,1), (,1.700,1,3), (1.861,1,1) ,(1.865,1,1), (1.901,0,4), (1.944,1,2), (1.966,1,3), (1.997,1,1), (2.006,1,2), (2.027,1,1), (2.055,1,3), (2.098,1,2), (2.132 ,0,3),(2.140,1,2), (2.179 ,1,2), (2.203,0,3), (2.257,0,14).

First we obtain the MLEs of the unknown parameters without any order restriction and the associate $90\%$ confidence intervals.  The Bayes estimators (BEs) and the associated $90\%$ confidence and credible intervals without any order restriction are obtained based on non-informative priors setting $a_0=0,b_0=0, a_1=0, a_2=0$ and $a=0, b=4$.  We set non-informative prior of $\alpha$ as $GA(0,4)$ so that the posterior density $\pi^*_2(\alpha|data)$ is integrable based on the given data.  The results are presented in Table \ref{data-no-1} and Table \ref{data-no-2}. The corresponding results based on order restriction are presented in Table \ref{data-not-1} and Table \ref{data-not-2}.

\begin{table}[h]
\caption{The MLEs and BEs of the unknown parameters for the real data set}
\label{data-no-1}
\begin{center}
\begin{tabular}{lll
\toprule
\multicolumn{1}{c}{Parameter} &\multicolumn{1}{c}{MLE} & \multicolumn{1}{c}{BE} \\
\midrule
$\alpha$ & 4.495 & 3.896\\
$\lambda_1$ &0.071 & 0.098\\
$\lambda_2$ & 0.016 &0.028\\
 \bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{table}[h]
\caption{The CIs and CRIs of the unknown parameters for the  real data set}
\label{data-no-2}
\begin{center}
\begin{tabular}{lllll
\toprule
\multicolumn{1}{c}{Parameter} &\multicolumn{2}{c}{ 90\% HPD CRI} & \multicolumn{2}{l}{90\% Bootstrap CI} \\
& \multicolumn{1}{l}{LL} & \multicolumn{1}{l}{UL}
&\multicolumn{1}{l}{LL}  &\multicolumn{1}{l}{UL} \\
\midrule
$\alpha$ & 3.472 & 4.338 &3.461 &6.693 \\
$\lambda_1$ & 0.070&0.167 &0.030&0.117  \\
$\lambda_2$ &0.007 & 0.049&0.0004 & 0.037  \\
 \bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{table}[h]
\caption{The MLEs and BEs of the unknown parameters for the  real data set with order restriction}
\label{data-not-1}
\begin{center}
\begin{tabular}{lll
\toprule
\multicolumn{1}{c}{Parameter} &\multicolumn{1}{c}{MLE} & \multicolumn{1}{c}{BE} \\
\midrule
$\alpha$ & 4.495 & 3.728\\
$\lambda_1$ & 0.071 & 0.088\\
$\lambda_2$ &0.016 &0.022\\
 \bottomrule
\end{tabular}
\end{center}
\end{table}

\begin{table}[h]
\caption{The CIs and CRIs of the unknown parameters for the  real data set with order restriction}
\label{data-not-2}
\begin{center}
\begin{tabular}{lllll
\toprule
\multicolumn{1}{c}{Parameter} &\multicolumn{2}{c}{ 90\% HPD CRI} & \multicolumn{2}{l}{90\% Bootstrap CI} \\
& \multicolumn{1}{l}{LL} & \multicolumn{1}{l}{UL}
&\multicolumn{1}{l}{LL}  &\multicolumn{1}{l}{UL} \\
\midrule
$\alpha$ & 3.442 & 4.360 & 3.461 & 6.774\\
$\lambda_1$ & 0.078 & 0.112 & 0.028 & 0.117  \\
$\lambda_2$ & 0.007 & 0.033 &  0.004 & 0.038  \\
 \bottomrule
\end{tabular}
\end{center}
\end{table}

Based on  the joint type-II progressively censored data, among
non-informative Bayes estimates and MLEs, non-informative Bayes estimates are more close to the estimates based on complete data sets (see Table \ref{data-test2})
Also the length of the HPD credible intervals are less than that of the Bootstrap CIs.

\section{\sc Conclusions}

In this paper we consider the analysis of joint progressively censored data for two populations.  It is assumed that the lifetime
of the items of the two populations follow Weibull distributions with the same shape parameter but different scale parameters.  We
obtain the MLEs of the unknown parameters, and showed that they always exist and they are unique.  We also obtain the Bayes estimate
and the associated credible interval of a function of the unknown parameters based on a fairly general prior distribution both on the
scale and shape parameters.  We further consider the order restricted inference on the unknown parameters.  Based on an extensive
simulation experiment it is observed that the Bayes estimators with respect to the informative priors perform significantly better
than the corresponding Bayes estimators based on non-informative priors for point estimation in terms of average bias and mean squared errors.  Also non-informative prior based estimators perform slightly better than MLEs
in terms of average bias and mean squared errors.   The credible intervals based on an informative prior perform better
than the other two methods in terms of average lengths and coverage percentage.  Also it is observed that if the ordering
information on the scale parameters is available, it is better to use it.

It may be mentioned that in this paper we have mainly considered
two populations.  It will be interesting to extend the results  for more than two populations also.  Su (2013) developed joint progressive censoring scheme for multiple  populations.  For multiple Weibull populations with common shape parameter, we can assume multivariate gamma-Dirichlet prior for scale parameters and a log-concave prior for common shape parameter as described in this article.  More work is needed along that
direction.

\noindent {\sc Acknowledgements:} The authors would like to thank the Editor-in-Chief Prof. D. Dey for providing
several important suggestions which have helped to improve the manuscript significantly.

\section*{\sc Appendix A: Proof of Theorem 1}

To prove Theorem 1, first we show the following result.  Suppose $a_j \ge 0$, for $j = 1, \ldots, k$, $\displaystyle g(\alpha) = \sum_{j=1}^k
a_j t_j^{\alpha}$, then -$\displaystyle \ln(g(\alpha))$ is a concave function.  Note that
$$
g'(\alpha) = \sum_{j=1}^k a_j t_j^{\alpha} \ln t_j \ \ \  \hbox{and} \ \ \ \
g''(\alpha) = \sum_{j=1}^k a_j t_j^{\alpha} (\ln t_j)^2.
$$
Moreover,
$$
\left (\sum_{j=1}^k a_j t_j^{\alpha} (\ln t_j)^2 \right ) \left ( \sum_{j=1}^m a_j t_j^{\alpha} \right ) - \left (\sum_{j=1}^m a_j t_j^{\alpha} \ln t_j
\right )^2 = \sum_{1 \le i < j \le k} a_i a_j (\ln t_i - \ln t_j)^2 \ge 0.
$$
Therefore,
$$
-\frac{d^2 \ln g(\alpha)}{d \alpha^2}  = - \frac{g''(\alpha) g(\alpha) - (g'(\alpha))^2}{(g(\alpha))^2} \le 0.
$$
Now from the above observation, it immediately follows that $p(\alpha)$ is a concave function.  The result follows by observing the
fact that $p(\alpha) \rightarrow -\infty$, as $\alpha \rightarrow 0$ or $\alpha \rightarrow \infty$.    \hfill \vrule height7pt width 5pt depth 0pt

\section*{\sc Appendix B: Fisher Information Matrix}

If we denote the matrix ${\bf A}$ as
$$
{\bf A} = - \left [ \matrix{\frac{\partial^2 l}{\partial \alpha^2} & \frac{\partial^2 l}{\partial \alpha \partial \lambda_1}
&  \frac{\partial^2 l}{\partial \alpha \partial \lambda_2}  \cr
\frac{\partial^2 l}{\partial \lambda_1  \partial \alpha} & \frac{\partial^2 l}{\partial \lambda_1^2} &
\frac{\partial^2 l}{\partial \lambda_1  \partial \lambda_2} \cr
\frac{\partial^2 l}{\partial \lambda_2  \partial \alpha} & \frac{\partial^2 l}{\partial \lambda_2  \partial \lambda_1} &
\frac{\partial^2 l}{\partial \lambda_2^2} \cr} \right ] = \left [ \matrix{a_{11} & a_{12} & a_{13}  \cr a_{21} & a_{22} & a_{23}
\cr a_{31} & a_{32} & a_{33} \cr} \right ],
$$
then
\begin{eqnarray*}
a_{11} & = & \frac{k}{\alpha^2} + \lambda_1 \left [ \sum_{j=1}^k s_j t_j^{\alpha} (\ln t_j)^2 + \sum_{C_1} t_j^{\alpha} (\ln t_j)^2
\right ] + \lambda_2 \left [ \sum_{j=1}^k w_j t_j^{\alpha} (\ln t_j)^2 + \sum_{C_2} t_j^{\alpha} (\ln t_j)^2
\right ]  \\
a_{12} & = & a_{21} =  \left [ \sum_{j=1}^k s_j t_j^{\alpha} \ln t_j + \sum_{C_1} t_j^{\alpha} \ln t_j \right ]   \\
a_{13} & = & a_{31} =  \left [ \sum_{j=1}^k w_j t_j^{\alpha} \ln t_j + \sum_{C_2} t_j^{\alpha} \ln t_j \right ]  \\
a_{22} & = & \frac{k_1}{\lambda_1^2}, \ \ a_{33} = \frac{k_1}{\lambda_1^2}, \ \ a_{23} = a_{32} = 0.
\end{eqnarray*}

redpajama_set_name: RedPajamaArXiv


text:
Q: UFW with NAT : Need to block all NAT requests except to one IP with out cutting NAT server from internet. Is this possible? I have a small cluster of kiosks using a single RPi for bootp, dhcp, dns, radius, and nat to the internet.  The RPi4 running RPi version of Ubuntu 18.04.
What I am struggling with is building a configuration a UFW rule(s) to block all NAT requests from private network out to the internet except for one IP address.
My first attempt to block the external interface failed, as it blocked the NAT server from getting outside.
is there a way to build a rule like this? For example, allow all NAT to 1.1.1.1. Internal LAN with NAT interface is using eth0, the WAN is using eth1.
ufw deny in on eth0 to eth1 outgoing
ufw allow in on eth0 to [eth1 ip 1.1.1.1] port 443 proto tcp

redpajama_set_name: RedPajamaStackExchange


text:
Bombay HC to BJP: Are you willing to pull down illegal additions to Maharashtra office?
According to the PIL, the plot has been shown in the development plan for the city as recreation garden and is generally known as Jawaharlal Nehru Garden
mumbai Updated: Feb 10, 2016 17:28 IST
The Bombay high court on Wednesday sought to know from the ruling Bharatiya Janata Party (BJP) if it was willing to pull down the purportedly illegal additions made to its state office opposite Mantralaya.
"Are you willing to bring down the area of the structure to 2,682 square feet?" the division bench of justice Abhay Oka and justice CV Bhadang sought to know from BJP counsel SU Kamdar. The senior advocate has now sought time to inform the court if the ruling party was willing to demolish additions made to its state office.
Accordingly, the court has now posted a public interest litigation filed by citizen associations from Nariman Point and Churchgate area raising concern over alleged encroachments over a plot of land near Mantralaya for further hearing on Thursday. According to the PIL, the plot has been shown in the development plan for the city as recreation garden and is generally known as Jawaharlal Nehru Garden.
The citizens association has contended that though the plot is reserved for recreation ground, it has been encroached upon and now hosts offices of the BJP and the Janata Dal, as also that of the Maharashtra Tourism Development Corporation (MTDC) and the Employment Exchange.
Apart from the construction being carried out at the BJP office, the PIL has also taken strong objection to the BJP reserving a huge public parking space for its office-bearers.
The court has also sought to know that if the state unit of the BJP had entered into any agreement with the state government and if it was paying any rent for the land allotted to it by the government. Kamdar is expected to answer these questions also on Thursday.
On Thursday, the counsel for the petitioner body pointed out that originally the state government has allotted 1,200 square feet of land for the state office of the ruling party and it was enhanced to 1,482 square feet in October 1995. But now the office of the party is spread over 9,700 square feet, housing 14 cabins, a conference hall and canteen, bathroom and toilet facilities. The counsel argued that the structure could not be regularised, as it has not been a legal structure, for being constructed on a plot reserved for garden.
Advocate general Shrihari Aney, however, said the land was allotted to the BJP for putting up office and, therefore, it cannot be said the structure is illegal. He pointed out that even the development plan for the city shows the recreation garden with the structure and there were no plans or other documents to show that the structure was either legal or illegal.
Aney said the state government did not agree that the entire structure was unlawful and the BMC should be permitted to take action as regards the unlawful portion.
Kamdar, too, disputed the claim of the petitioner body that the BJP state office was spread over 9,700 square feet and claimed that it was about 4,000 square feet. It was then the court sought to know from him, if the political party was willing to bring down the size of its office to 2,682 square feet — 1,200 square feet originally allotted plus addition of 1,482 square feet made in 1995.
First Published: Feb 10, 2016 17:28 IST
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Although it was created with the purpose of being a fast and filling alternative for people who do not have the time to sit down for a full meal during the day, street food has really come into its own. There is the same level of expectation from street food as there is from classy restaurants. Therefore, knowing something about each type of dish might come in handy. Let's take shawarma and gyros, for example. Do you know the difference between them?
Shawarma is an Arabic dish served as street food or in bistros and ethnic restaurants all over the world. It is particularly popular in Arab and Balcanic countries. It is made with meat, lamb, chicken, turkey, veal, beef, or a combination of meats. The meat is grilled on a turning spit then shredded and served on a plate, in a wrap, or in a sandwich. Among the most common toppings used are tabbouleh, fattoush, and hummus. Dressings vary from amba (pickled mango with chili) or tahini (sesame sauce). Cardamom, cinnamon, and nutmeg are among the spices used for this dish.
A Gyro is a Greek dish served as street or bistro food. It is traditionally made with pork or chicken, and with veal, beef, or lamb outside of Greece. The meat is usually ground and spiced with salt, paprika, pepper, parsley, dried garlic, thyme, rosemary, marjoram, and oregano. It is then placed on a spit and slow-cooked in its own fat as the spit turns. If the meat does not contain enough fat for this, layers of fat may be added. The meat is usually eaten in a piece of pita with tzatziki sauce, tomato, and onions.
So what is the difference between shawarma and a gyro?
These two dishes look the same and they have the same origin, the Turkish Doner Kebab. However, they are two local takes on the 18th-century Turkish dish. Therefore, the Arabs called it "shawarma" and the Greeks called it "gyro," both terms coming from the verb "to turn." This is a reference to the turning motion of the meat on the spit.
Greeks prefer to use chicken and pork, although lamb, veal, and beef are used as well. Arabs chose chicken, turkey, lamb, beef, and combinations of these meats. The spices used also differ as they both correspond to their own regions. Shawarma has cardamom, cinnamon, and nutmeg. Gyros, on the other hand, have parsley, oregano, paprika, parsley, dried garlic, onions, thyme, marjoram, and rosemary.
Shawarma is served on a plate, in a wrap, or in a sandwich. Traditionally, a gyro is served on pita bread (Greek flatbread). Specific Arab salads like tabbouleh and fattoush, and toppings such as tahini and amba complement the meat of a shawarma. A gyro, however, is served with tzatziki, tomatoes, and onions.


Let's set the record straight. No one is wastin' away in Margaritaville because it's his own damn fault.
The pathway it took for a couple of ounces of tequila and a shaker of salt — often lost — to become one of the most popular cocktails in the United States is a full-frontal group effort that includes folklore, a lot of bartenders speaking Spanish, illegal booze, maybe a famous stage star or two, a whole assembly of women named Margaret and one convenience store's Slurpee machine.
As is often the case with anything this muddled, confused, exaggerated and disputed, Texas is in the middle of it.
First of all, what is a margarita? The generic description simply says it's a cocktail made with tequila, a liqueur or other sweetener and lime juice. That description is, of course, comparable to a fuse on a box of dynamite. Everything might start there, but it quickly gets explosive. There are more opinions, recipes and versions of the mighty margarita than there are memory lapses the morning after.
Natalia Castravet is the manager at Lucky Lounge, 8305 Stewart Road in Galveston, the place many claim is the go-to destination for great margaritas. In fact, Lucky Lounge was the first-place winner at the Texas Tequila & Margarita Festival a few years back.
Like everything about margaritas, throwing in passion is a practice without restrictions. Consider that Lucky Lounge's award-winning recipe calls for blueberry and lemon basil syrup.
One thing cocktail researchers seem to agree on is that this cocktail can trace its roots back more than a century and it was indeed identified by what is often a woman's name.
There's a cocktail called a daisy that first started appearing in the late 19th century and has all the DNA that made up the early margaritas decades later. The most common cocktail in this family was the brandy daisy, made with two and a half ounces of brandy, two tablespoons of fresh lemon juice and a tablespoon of grenadine. Sound familiar?
The drink was immensely popular, and by the 1930s and '40s, the brandy daisy had begat the gin daisy, the Irish daisy, the rum daisy and the Scotch daisy, all with the same recipe other than the change in liquor. And yes, during Prohibition, when tequila found a ready market in alcohol-starved areas near Mexico, a daisy bloomed with that as well. In the 1930 "My New Cocktail Book" by G.F. Steele, a daisy recipe using tequila was included, and in 1936, a drink specifically named tequila daisy was mentioned in the Syracuse Herald, a New York newspaper.
Somewhere around that time, the margarita was born, although the first known publication of a margarita recipe wasn't until December 1953.
The birth certificate leaves the parentage questionable.
The date was 1948. Because the famous singer's actual name was Margaret, Cruz named the drink after her using the Spanish spelling of Margaret.
Another story goes south with the origin, crediting a Mexican restaurant owner named Carlos (Danny) Herrera. Visiting his restaurant in 1938, the stunningly beautiful Ziegfeld showgirl, Marjorie King, informed him that she was allergic to all forms of alcohol except tequila. She refused to drink it straight, so Herrera concocted a drink with tequila, lime juice and salt. Since Marjorie is a derivative of the name Margaret, he named the drink after her, using the Spanish spelling.
And then there is the tale of a Dallas socialite named Margaret Sames, who was said to have invented the drink during a house party in Acapulco in 1948 — the same year Cruz created a drink for Peggy Lee. Being named Margaret, Sames used the Spanish version of her name for her drink. Several years before, another story goes, actress Rita Hayworth was performing in Tijuana, when a smitten bartender invented the drink for her. The actress' real name was Margarita Cansino.
There are more stories, including one that lacks the panache of all the others. The Spanish word for daisy is margarita.
One thing is certain about all this confusion of origin, and it's exactly what some people think. There's a woman to blame.
Castravet believes you're either drinking tequila or just a drink, she said.
But shunning frozen also is shunning something truly Texas.
Mariano Martinez, the owner and operator of Mariano's Mexican Cuisine in Dallas, was doing a booming business selling a signature frozen margarita in the 1970s. The problem was, the drink was so popular, the bar was selling hundreds of them a night using one small blender. The drinks were inconsistent, the blender was constantly overheating and the bartenders were standing in line waiting to get the drinks made.
The museum does not have a shaker of salt. It's lost.
Place all ingredients in a cocktail shaker with ice and shake. Strain into a margarita glass.
Bring water and fresh blueberries to a boil. Add sugar and keep over medium heat, pressing down on the blueberries. Add the fresh basil leaves and the zest of 1 whole lemon. Simmer until slightly thickened — about 15 minutes.
Strain the mixture through a fine sieve or cheese cloth, pressing the ingredients to get all the liquid out. Discard solids. After syrup is cool, pour into a sealable container and store in the refrigerator until chilled. Will keep 2-4 weeks.


There are no new sailings this week. The deal for Canadian residents on the 9-night Alaskan sailing with a stop in Sitka remains bookable as well as a handful of sailings on the Disney Dream and Disney Wonder for members of the US Military and Florida Residents.
For those hoping to hear about the Summer 2015 Disney Cruise itineraries, the wait continues… In the mean time, there is an interesting video playing over on DisneyCruise.com under Destinations that mentions Hawaii, Mexican Riviera, Canada & New England, Europe, Bahamas, and Caribbean.
PSA: DO NOT CALL Disney Cruise Line until the dates are officially released. You will just be clogging up the phone lines for people who are trying to call and book available sail dates. The itineraries will likely be announced Tuesday. That's announced… Following past patterns, bookings should open up for Gold & Platinum Castaway Club members on Wednesday. On Thursday, bookings will open to Silver Castaway Club members and first time cruisers.
This entry was posted in Itineraries, Special Offers and tagged 2014, Alaska, CAN, Canadian Resident Rate, Disney Dream, Disney Wonder, Florida Resident Rate, FLR, Military Rates, MTO by Scott Sanders. Bookmark the permalink.
Thanks Scott. I can't help but think that Disney Cruise Line is shooting themselves in the foot. As a Platinum it is not fare to make long time cruisers wait and then set out a 18 month cushion. Somebody is not thinking logically. I can't see how this is a win for all. You have to take care of those that take care of you. For me and my family, I have never been on any other cruise line. AND STILL I AM PATIENT. But then again I can only speak for me.
They seem to be systematically killing off the dummy bookings. There are way too many cruises that have rooms booked for people that are camping on that will move to another date. This prevents people from booking due to the cruise being full, or the price is driven up based on available inventory for the particular sailing. This alone should be a reason for DCL to start selling onboard rebooking vouchers with an 18-month expiration date.
In my opinion, releasing just the summer allows DCL to control the supply and demand to internally justify their recent cruise fares. Less cruises available could lead to fuller ships. Release high then discount via OGT/IGT/VGT rates to fill ship if needed.
Do you have an idea if they are going to be offering more 9 nights cruises in 2015 for Alaska? I have read on the disboards that someone thought they would have more 9 night Alaska cruises, just wondering your thoughts on it.
Thanks- enjoy reading your blog!
I have not found any evidence to support this, but I would not rule it out.
Scott I understand the reason for DCL to start selling onboard rebooking vouchers with an 18-month expiration date.
But even the OGT/IGT/VGT rates are very high and will deter others with big families or not financially equipped to do a short notice or last minute cruise. I understand what you are saying, but I don't see this working because everybody is not like the few that have. They will have to make changes to fill their ships in the future. I even heard they have a low volume due to the changes. For me I am a die hard Disney fan and we can't see ourselves doing any other cruise lines, but then again I speak for my self only.
So I think I know the answer to this, but does this mean they are only releasing summer and not all of 2015?!? I'm a DVC member and taking a trip in nov 2015 to WDW and want to add a cruise too. Getting frustrated that I can't confirm my plans because the cruises aren't out yet and of course that means the points aren't out either. Also wanted to see the points and compare Hawaii or the Caribbean at thanksgiving.
Based on DCL's social media teasers, It certainly sounds like just the summer itineraries will be released.
SCOTT I just heard that the schedule for late 2015 will be out tomorrow morning. But keep in mind they are saying LATE 2015 summer schedule. According to my educational guessing, I think SUMMER ends in September. For me I hoping on OCTOBER. But for now I don't see that happening. I don't think this is going to work for ones wanting to book toward the end of the year. But then again Scott thinks this is a good ideal. Well we'll have to see. I'll be checking you blog tomorrow morning Scott. Thank you for your honesty.
I'm confused; I thought as silver, gold, and platinum members, we got early booking privileges. But I saw on Disney's Facebook page that the summer 2015 dates are being released tomorrow. So, are they not giving us 2 days to book before the general public anymore?
The itineraries will be released tomorrow and as long as the previous pattern holds booking will begin on Wednesday for Gold & Platinum followed by everyone else on Thursday.
So, will it be the Disney Magic out of New York? We left out of NY to the Bahamas a couple of years ago and would love to see her reimagined!
Not really out if NYC.
My family and I are hoping to book our first disney cruise for October 2015. Any idea when the final dates will be released if they are not included in this week's announcement?
Maybe June, not sure just yet on when the rest will be announced.
We were just on the Fantasy in March and it will be our last Disney cruise. The cost of the suite we usually book has doubled in price for 2015. $22,000 or more is too expensive for a 7-night cruise.


A project using light therapy to kill harmful bacteria reduced post-surgical infections by almost 40 per cent at Vancouver General Hospital, medical authorities say.
Vancouver General Hospital says a pilot project using light therapy to kill harmful bacteria has reduced post-surgical infections by almost 40 per cent.
The hospital reported Wednesday on the 12-month experiment using a process called nasal photodisinfection.
"They have this treatment right before they go into the operating room for our high-risk surgeries and we've had a very high degree of acceptance by the patients," said Dr. Elizabeth Bryce, Regional Medical Director of Infection Control at Vancouver Coastal Health.
Bacteria from hotspots on the body, especially the nose, can migrate simply by touch and cause an infection when patients are highly vulnerable, after surgery.
"So if we can decrease the amount of bacteria in your body and in your nose, then we reduce your risk of an infection," Bryce said.
Nasal photodisinfection involves applying a blue dye to the inside of a patient's nose before surgery. The dye attaches itself to germs and when a light is shone inside, potentially "bad" bacteria are destroyed.
"Overall, we saved about $1.9 million. More importantly, we freed up about 550 patient-days and those patient-days we can use for additional surgeries," said Bryce, who noted that the project led to a 39-per-cent reduction in infections.
The technology was developed by a Vancouver-based company, Ondine Biomedical.
"Definitely 10,000, 20,000 people that are dying of hospital-acquired infections and we're hoping to take a big chunk of those and get them back into the land of the living and healthy," said Ondine CEO Carolyn Cross.
Cross estimates that in B.C. alone, $90 million is spent each year dealing with preventable surgical-related infections in hospitals.