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The Annals of Mathematical Statistics
VOL. 36 · NO. 6 | December, 1965
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Ann. Math. Statist. 36 (6), (December, 1965) DOI: 10.1214/aoms/1177699791
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Miscellaneous front pages, Ann. Math. Statist., Volume 36, Number 6 (1965)
Ann. Math. Statist. 36 (6), (December, 1965)
On a Theorem of Hoel and Levine on Extrapolation Designs
J. Kiefer, J. Wolfowitz
Ann. Math. Statist. 36 (6), 1627-1655, (December, 1965) DOI: 10.1214/aoms/1177699793
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Recent results [5] of Hoel and Levine (1964), which assert that designs on $\lbrack -1, 1\rbrack$ which are optimum for certain polynomial regression extrapolation problems are supported by the "Chebyshev points," are extended to cover other nonpolynomial regression problems involving Chebyshev systems. In addition, the large class of linear parametric functions which are optimally estimated by designs supported by these Chebyshev points is characterized.
The Moments of the Number of Crossings of a Level by a Stationary Normal Process
Harald Cramer, M. R. Leadbetter
In this paper we consider the number $N$ of upcrossings of a level $u$ by a stationary normal process $\xi(t)$ in $0 \leqq t \leqq T$. A formula is obtained for the factorial moment $M_k = \varepsilon\{N(N - 1) \cdots (N - k + 1)\}$ of any desired order $k$. The main condition assumed in the derivation is that $\xi(t)$ have, with probability one, a continuous sample derivative $\xi'(t)$ in the interval $\lbrack 0, T\rbrack$. This condition involves hardly any restriction since an example shows that even a slight relaxation of it causes all moments of order greater than one to become infinite. The moments of the number of downcrossings or total number of crossings can be obtained analogously.
The Ergodicity of Series Queues with General Priorities
Bruce McK. Johnson
Markovian Sequential Replacement Processes
Howard M. Taylor III
A sequential control process is a dynamic system which is observed periodically and classified into one of a number of possible states. After each observation one of a number of possible decisions is made. These decisions are the "control"; they determine the chance laws of the system. A replacement process is a control process with an additional special action, called replacement, which instantaneously returns the system to some initial state. Let $\mathbf{X}$ denote the state space of the system, assumed to be a Borel subset of finite dimensional Euclidean space. The case where $\mathbf{X}$ is finite has been treated by Derman [8], and thus $\mathbf{X}$ is considered infinite here. Let $\mathscr{B}$ be the $\sigma$-algebra of Borel sets in $\mathbf{X}$. Let $\{X_t; t = 0, 1, 2, \cdots\}$ be the sequence of states and $\{\Delta_t; t = 0, 1, 2, \cdots\}$ be the sequence of decisions. In a replacement problem it is assumed that there is a distinguished state $x_0 \epsilon \mathbf{X}$ with $X_0 = x_0$ with probability one. For any time $t$ let $S_t$ be the history of states and decisions up to and including time $t$. Let $\mathbf{A}$ be the set of possible actions, excluding replacement, where: $A1^\circ$ It is assumed that the action space $\mathbf{A}$ is a finite set with $n((\mathbf{A})$ elements. Since $\mathbf{A}$ is finite, assume $\mathbf{A} = \{1, 2, \cdots, n(\mathbf{A})\}$. Let $k_0 \not\varepsilon \mathbf{A}$ denote the replacement action. The action $k_0$ instantaneously returns the system to state $x_0$, and it may be followed by some action $k \varepsilon \mathbf{A}$ which "acts on" the state $x_0$. The pair $(k_0, k)$ itself constitutes a possible action. A decision at time $t$ is either a choice of an element $k \varepsilon \mathbf{A}$ or a choice of a pair $(k_0, k)$ with $k \varepsilon \mathbf{A}$. Let $\mathbf{A}_0$ be the total action space, where: $\mathbf{A}_0 = \mathbf{A} \cup \{ (k_0, k); k \varepsilon \mathbf{A}\}.$ There are $2n(\mathbf{A})$ elements in $\mathbf{A}_0$. Let $\Xi = \{\xi; \xi = \langle\xi_1, \cdots, \xi_{2n(\mathbf{A})}\rangle, \xi_j \geqq 0, \sum\xi_j = 1\}$ be the simplex of all probability distributions on $\mathbf{A}_0$. A sequential control rule is a function $D(s_{t - 1}, x) = \langle D_1(s_{t - 1}, x), \cdots, D_{2n(\mathbf{A})} (s_{t - 1}, x)\rangle$ of histories $s_{t - 1}$ and present states $x$ with values in $\Xi$. The interpretation is: At a history of $S_{t - 1} = s_{t - 1}$ and a present state $X_t = x$, decision $j \varepsilon \mathbf{A}_0$ is taken with probability $D_j(s_{t - 1}, x)$. In order that the integrals later to be written have meaning it is necessary to restrict attention to control rules $D(s_{t - 1}, x)$ which are Biare functions of their arguments. Let $\mathbf{R}$ be the space of all such control rules. A sequential control process is not specified until a "law of motion" is given. $A2^\circ$ It is assumed that for every $x \varepsilon \mathbf{X}$ and $k \varepsilon \mathbf{A}$ there exists a probability measure $Q(\cdot; x, k)$ on $\mathscr{B}$ such that for some version $\mathrm{Pr}\{X_{t + 1} \varepsilon B \mid S_{t - 1}, X_t = x, \Delta_t = k\} = Q(B; x, k)$; for every $B \varepsilon \mathscr{B}$ and history $S_{t - 1}$. For every $B \varepsilon \mathscr{B}$ and $k \varepsilon \mathbf{A}, Q(B; \cdot, k)$ is assumed to be a Baire function on $\mathbf{X}$. It is assumed that $Q(\cdot, x, k)$ is absolutely continuous with respect to some $\sigma$-finite measure $\mu$ on $\mathscr{B}$, and possessing a density $q(\cdot, x, k)$, also assumed to be a Baire function in $x$. Since $X_0 = x_0$ a.s., once a rule $R \varepsilon \mathbf{R}$ is specified, the sequences $\{X_t, t = 0, 1, 2, \cdots\}$ and $\{(X_t, \Delta_t); t = 0, 1, 2, \cdots\}$ are stochastic processes. The previous Assumption $A2^\circ$ imposes a structure similar to that of a Markov process in that the law of motion does not depend on the past history, but only on the present state. In a manner similar to Derman [9], the process $\{(X_t, \Delta_t); t = 0, 1, 2, \cdots\}$ will be called a Markovian sequential replacement process. It is not true that $\{X_t; t = 0, 1, \cdots\}$ nor even $\{(X_t, \Delta_t); t = 0, 1, \cdots\}$ will always be Markov processes; whether they are or not will depend on the rule $R$. Two assumptions particular to the development in this paper and insuring the ergodicity of the process are: $A3^\circ$ For every $x \varepsilon \mathbf{X}$ and $k \varepsilon \mathbf{A}$ it is assumed that $\lim_{x' \rightarrow x} \int |q(y; x, k) - q(y; x', k)| \mu(dy) = 0.$ $A4^\circ$ For every compact set $G \subset \mathbf{X}$ it is assumed that $\sup_{x \varepsilon G}\int_G q(y; x, k) \mu (dy) < 1$ for all $k \varepsilon \mathbf{A}$. The last assumption, $A4^\circ$, is stronger than needed, as may be seen in the examples in Section 4. However, it is easily verified and seems natural in many applications of the theory. Let $w(x, k)$ be the immediate cost whenever the system is in state $x \varepsilon \mathbf{X}$ and decision $k \varepsilon \mathbf{A}$ is made. It often occurs that the cost in an actual situation is a random variable whose distribution is determined by knowledge of the state and decision. In such a case, with some loss in generality, attention is restricted to $w(x, k)$ representing the expected one stage cost under the appropriate distribution. Let $K(x)$ be the cost of replacing a system in state $x$. If $w_0(\cdot, \cdot)$ is the cost function defined on $\mathbf{X} \times \mathbf{A}_0$ then the relationship is: $w_0(x, k) = w(x, k)\quad\text{for} k \neq k_0$ and $w_0(x, (k_0, k)) = K(x) + w(x_0, k)\quad\text{for} k \varepsilon \mathbf{A}.$ $A5^\circ$ Assume that $K(\cdot)$ is bounded and continuous with $0 \leqq K(x) \leqq M$ for all $x \varepsilon \mathbf{X}$. For every $k \varepsilon \mathbf{A}$ assume that $w(\cdot, k)$ is a non-negative continuous function on $\mathbf{X}$ with $\lim \inf_{x \rightarrow \infty} w(x, k) \gg 0$ (For the limiting operation here, a neighborhood of $\infty$ is the complement of a compact set.). The notation $a \gg 0$ means that $a$ is much greater than zero, but not necessarily infinite. One needs $\lim \inf_{x \rightarrow \infty} w(x, k)$ large enough so that Lemmas 3.2, 3.3 and 3.4 will hold. Intuitively, one needs the cost of continuing sufficiently large for some states so as to ensure that the expected time to a replacement action is finite. It should be noted that $\sup_{x \varepsilon X} \min_{a \varepsilon \mathbf{A}_0} w_0(x, a) \leqq M_0$ where $M_0 = M + \min_{k \varepsilon \mathbf{A}} w(x_0, k)$. Let $P_t(B, a \mid x, R) = \mathrm{Pr}\{X_t \varepsilon B, \Delta_t = a \mid X_0 = x, R\}$ for $B \varepsilon \mathscr{B}, x \varepsilon X$ and $a \varepsilon \mathbf{A}_0$. Let the appropriate density be labeled $p_t(\cdot, \cdot \mid x, R)$ where $p_t(y, a \mid x, R)\mu(dy) = \mathrm{Pr}\{X_t \varepsilon dy, \Delta_t = a \mid X_0 = x, R\}.$ Two common measures of effectiveness of a Markovian sequential decision process are the expected total discounted future cost and the average cost per unit time. The first, abbreviated to "discounted cost" assumes a discount factor $\alpha \varepsilon (0, 1)$, with the interpretation that a unit of value $n$ periods hence has a present value of $\alpha^n$. For a starting state of $X_0 = x_0$ the objective is to choose a rule $R$ so as to minimize $\psi(x_0, \alpha, R) = \sum^\infty_{t = 0} \alpha^t \int x\sum_{a \varepsilon \mathbf{A}_0} w_0(x, a)p_t(x, a \mid x_0, R)\mu(dx).$ The second criterion, abbreviated to "average cost" examines the function $\varphi(x_0, R) = \lim \inf_{T \rightarrow \infty} T^{-1} \sum^{T - 1}_{t = 0} \int_{\mathbf{X}} \sum_{a \varepsilon \mathbf{A}_0} w_0(x, a)p_t(x, a \mid x_0, R)\mu(dx)$ Section 2 presents the solution of the problem under the discounted cost measure. Building upon the work of Blackwell [4] and Karlin [12], Derman [9] has shown that an optimal non-randomized stationary rule exists for the case where $\mathbf{X}$ is denumerable. Blackwell [5] recently has given a complete discussion of the general use. The rule is characterized by a functional equation of the dynamic programming type. Iterative methods for solving such functional equations are now almost commonplace. Section 3 uses the known results in the discounted cost model: (a) to show the existence of a non-randomized stationary solution in the average cost case, (b) to show the existence of a functional equation characterizing the solution in the average cost case, and (c) to show that the average cost solution is the limit, in some sense, of the discounted cost solutions as the discount factor approaches unity. Section 4 presents some applications of the theory. The attempt is to show how the work of several authors fits into this general theory of control of replacement processes. For example, while supporting one claim in a quality control paper by Girshick and Rubin [10], the theory also provides a counter example for another of their claims.
A Dynamic Stochastic Approximation Method
Vaclav Dupac
This investigation has been inspired by a paper of V. Fabian [3], where inter alia the applicability of stochastic approximation methods for progressive improvement of production processes is discussed; his non-formal discussion includes the case where the optimum of the production process moves during the optimization process. In the present paper, the last case is treated in a formal way. A modified approximation scheme is suggested, which turns out to be an adequate tool, when the position of the optimum is a linear (or nearly linear) function of time. The domain of effectiveness of the unmodified approximation scheme is also investigated. In this context, the incorrectness of a theorem of T. Kitagawa is pointed out. The considerations are performed for the Robbins-Monro case in detail; they can all be repeated for the Kiefer-Wolfowitz case and for the multidimensional case, as indicated in Section 4. Among the properties of the method, only the mean convergence and the order of magnitude of $E\lbrack (x_n - \theta_n)^2\rbrack$ are investigated; (here $x_n$ denotes the estimated and $\theta_n$ the true position of the optimum at time $n$.) A lemma, due to K. L. Chung [1] is used repeatedly: LEMMA. Let $b_n, n = 1, 2, \cdots$, be real numbers such that for $n \geqq n_0$, \begin{equation*}\tag{1}b_{n + 1} \leqq (1 - c/n^s)b_n + c'/n^t,\end{equation*} where $0 < s < 1, c > 0, c' > 0, t$ real. Then \begin{equation*}\tag{2}\lim \sup_{n \rightarrow \infty} n^{t - s}b_n \leqq c'/c.\end{equation*} The lemma remains true, if the inequalities (1) and (2) are reversed and, simultaneously, $\lim \sup_{n \rightarrow \infty}$ is changed into $\lim \inf_{n \rightarrow \infty}$. (In Chung's paper, a further assumption $t > s$ is made, but it is easily seen, that both versions of the lemma hold true also when $t \leqq s$; this fact is used in Section 3.) Throughout the paper, $K_0, K_1, K_2, \cdots$ denote positive constants, numbered in order of appearance.
Peakedness of Distributions of Convex Combinations
Frank Proschan
Admissibility and Bayes Estimation in Sampling Finite Populations. I
V. P. Godambe, V. M. Joshi
Admissibility and Bayes Estimation in Sampling Finite Populations. II
V. M. Joshi
Admissibility and Bayes Estimation in Sampling Finite Populations. III
Rate of Convergence in the Compound Decision Problem for Two Completely Specified Distributions
J. F. Hannan, J. R. Van Ryzin
Simultaneous consideration of $n$ statistical decision problems having identical generic structure constitutes a compound decision problem. The risk of a compound decision problem is defined as the average risk of the component problems. When the component decisions are between two fully specified distributions $P_0$ and $P_1, P_0 \neq P_1$, Hannan and Robbins [2] give a decision function whose risk is uniformly close (for $n$ large) to the risk of the best "simple" procedure based on knowing the proportion of component problems in which $P_1$ is the governing distribution. This result was motivated by heuristic arguments and an example (component decisions between $N(-1, 1)$ and $N(1, 1))$ given by Robbins [4]. In both papers, the decision functions for the component problems depended on data from all $n$ problems. The present paper considers, as in Hannan and Robbins [2], compound decision problems in which the component decisions are between two distinct completely specified distributions. The decision functions considered are those of [2]. The improvement is in the sense that a convergence order of the bound is obtained in Theorem 1. Higher order bounds are attained in Theorems 2 and 3 under certain continuity assumptions on the induced distribution of a suitably chosen function of the likelihood ratio of the two distributions.
A Robust Version of the Probability Ratio Test
Peter J. Huber
A statistical procedure is called robust, if its performance is insensitive to small deviations of the actual situation from the idealized theoretical model. In particular, a robust procedure should be insensitive to the presence of a few "bad" observations; that is, a small minority of the observations should never be able to override the evidence of the majority. (But at the same time the discordant minority might be a prime source of information for improving the theoretical model!) The classical probability ratio test is not robust in this sense: a single factor $p_1(x_j)/p_0(x_j)$ equal (or almost equal) to 0 or $\infty$ may upset the test statistic $T(x) = \prod^n_1 p_1(x_j)/p_0(x_j)$. This leads to the conjecture that appropriate robust substitutes to both fixed sample size and sequential probability ratio tests might be obtained by censoring the single factors at some fixed numbers $c' < c''$. Thus, one would replace the test statistic by $T'(x) = \prod^n_1 \pi(x_j)$, where $\pi(x_j) = \max (c', \min (c'', p_1(x_j)/p_0(x_j)))$. The problem of robustly testing a simple hypothesis $P_0$ against a simple alternative $P_1$ may be formalized by assuming that the true underlying distribution lies in some neighborhood of either of the idealized model distributions $P_0$ or $P_1$. The present paper exhibits two different types of such neighborhoods for which the above mentioned test, to be called censored probability ratio test, is most robust in a well defined minimax sense. The problem solved here originated through the earlier paper Huber (1964), over the question how to test hypotheses about the mean of contaminated normal distributions.
Alternative Efficiencies for Signed Rank Tests
Jerome Klotz
Asymptotic efficiency curves for the one sample Wilcoxon and normal scores tests are obtained by comparing the exponential rate of convergence to zero of the type I error $(\alpha)$ while keeping the type II error $(\beta)$ fixed $(0 < \beta < 1)$. A wider than usual view of test performance consistent with small sample results is obtained. The Pitman efficiency value is derived as a special case when the alternative approaches the null hypothesis. Comparisons of the signed rank tests relative to $\bar{x}$ or $t$ for normal location alternatives yield small efficiency values for extreme alternatives. The relative performance of the Wilcoxon is seen to be slightly better than the normal scores for normal alternatives with larger location parameter values despite the local (Pitman) optimality of the normal scores. Similar results hold for two other non-normal alternatives considered.
Small Sample Power of the One Sample Wilcoxon Test for Non-Normal Shift Alternatives
Harvey J. Arnold
The power of the one sample Wilcoxon test is computed for the hypothesis that the median is zero against various shift alternatives for samples drawn from several different non-normal distributions. A recursive scheme given by Klotz [2] simplifies the problem of power computations and allows investigating samples to size $n = 10$ on a large computer. The power for selected type I errors $\alpha$ are compared with the power of a best signed-rank procedure obtained by sorting the probabilities in decreasing order for all possible sample configurations for fixed $n$ and adding up the probabilities associated with the most probable 100 $\alpha$% of the configurations. The non-normal distributions selected for study are the $t$ distribution with degrees of freedom $\frac{1}{2}$, 1, 2, and 4. The one sample Wilcoxon test, found to be powerful for normal shift alternatives by Klotz [2], deteriorates badly in power for the long-tailed distributions studied as does the one sample $t$ test. However, the Wilcoxon test remains more powerful than the $t$ test. The sign test is still more powerful than either Wilcoxon or $t$. No asymptotic results are obtained.
Properties of the Median and Other Order Statistics of Logistic Variates
Michael E. Tarter, Virginia A. Clark
Recently, increasing use has been made of the logistic distribution. In addition to the many investigations of the logistic as a regression curve, several authors have investigated the properties and suggested the use of the logistic as a density function. One reason for this is the property that the inverse of the logistic distribution function can be expressed as the logarithm of a rational function. A general background for the work in this field can be gained by reading references [1], [2], [3], [7], and [10]. In this paper several additional properties of a sample of logistic variates will be described. In Section 2, it will be shown that the logistic distribution and its moment generating function can be expressed as a Maclaurin series where the coefficients are simple functions of Bernoulli numbers. In Section 3, the moment generating function of the median is given. In Section 4, the variance of the median will be determined and the relative efficiency of the median of logistic variates to the mean for various sample sizes as well as the asymptotic efficiency is given. In Section 5, the cumulant generating function of the median of logistic variates is derived and the cumulants, themselves, given in terms of zeta values. In Section 6, the variance and covariance of any two order statistics are given. The order statistic variance-covarance matrix can be used to determine the BLUE estimators of the population location and scale parameters. These may be expected to have high efficiency for the estimation of logistic parameters [4]. Although the maximum likelihood estimate of the population location parameter $\hat\theta$ satisfies the relationship, $\frac{1}{2} = \lbrack\sum^n_{i = 1} F(x_i - \hat{\theta})\rbrack/n,$ iterative methods must be used to solve for $\hat{\theta}$. Therefore the computational advantage of the BLUE estimates as well as their probable efficiency would tend to make them useful in the case of the logistic distribution.
Multivariate-Normal Classification with Covariances Known
Bob E. Ellison
The admissibility of the minimum distance rule (= maximum likelihood rule) and of a restricted maximum likelihood rule are proved in [5] for a zero-one loss function in a classification problem in which information about the means of the $k$ alternative multivariate normal distributions is based on samples. That classification problem is a special case of the problem of deciding in which of $k$ given linear manifolds the mean of a normally distributed vector lies when the covariance matrix is known. The admissibility of the minimum distance rule (= maximum likelihood rule) in the more general problem is proved in [3] and [4]. The proof is similar to that given in [5] for the special case; the choice of the prior distribution used in the proof is dictated by Lemmas 2 and 4 of [5]. The purpose of this paper is to present the admissibility proof for the more general problem. The more general problem includes classification problems in which information about the means of the $k$ alternative multivariate normal distributions is based on samples, and the means are linearly restricted. The admissibility of classification rules in such problems has received attention recently (see, e.g., the abstracts by Das Gupta, [2], and Srivastava, [7]). A problem more general yet than that treated in this paper is the problem of deciding in which of $k$ given linear manifolds the mean of a normally distributed vector lies when the covariance matrix is a possibly different known matrix for each of the $k$ alternatives. A parametric family of admissible classification rules for that problem can be obtained by simply replacing $\not\Sigma$ by $\not\Sigma_j$, the known covariance matrix for the $j$th alternative, in the statistic $t_j(x \mid h)$ given by Equation (4) of Section 5, $j = 1, \cdots, k$. However, such a family of admissible classification rules is of little interest per se, since other such families are easily generated as Bayes procedures relative to parametric families of prior distributions. What is of considerably more interest is the question of whether or not certain "natural" rules are admissible. The maximum likelihood rule, which is not identical with the minimum distance rule in this problem, is a "natural" rule. The maximum likelihood rule is not contained in the family of admissible rules obtained by the replacement of $\not\Sigma$ in (4), and whether or not it is in general admissible in this problem is not known to me. If the covariance matrix is unknown, but an independent estimate of it is available, it is "natural" to use the estimate in place of the true covariance matrix in the minimum distance rule (= maximum likelihood rule). Whether or not the "natural" rule is in general admissible in this problem is not known. The problem considered in this paper is stated in Section 2. In Section 3 the minimum distance and maximum likelihood rules for the problem are defined; the rules are seen to be equivalent. In Section 4 the problem is reparametrized, and Bayes procedures relative to prior distributions of the new parameters are obtained in general. The minimum distance rule is obtained as a Bayes procedure in Section 5, and its admissibility is deduced. Examples of applications are given in Section 6.
A Class of Tests with Monotone Power Functions for Two Problems in Multivariate Statistical Analysis
Govind S. Mudholkar
The problem of testing the general multivariate linear hypothesis, also known as MANOVA, and the problem of testing independence between two sets of multivariate normal random variables are the two problems considered in this paper. In [3] two sufficient conditions for the power function of an invariant test of the general linear hypothesis to be a monotone increasing function of each of the noncentrality parameters have been obtained. In [2] one of these two conditions has been extended to invariant tests of the hypothesis of independence between two sets of variates. These conditions are in terms of, respectively, the convexity and the symmetry of certain sections of the acceptance regions of the tests; and their verification is, in general, nontrivial. In this paper it is shown that the power functions of the members of a class of invariant tests based on statistics "generated" by symmetric gauge functions of convex increasing functions of the maximal invariants are monotone increasing functions of the relevant noncentrality parameters. In this process we have explained an interesting tie-up between the monotonicity properties of the invariant tests for the two problems (Theorem 2) and have obtained extension of some results on the symmetric gauge functions and convexity in the matrix theory (Theorem 3 and Theorem 4). The terms "invariance," "invariant tests," and "maximal invariants" have been used throughout this paper in connection with the relevant groups of transformations mentioned in the Section 2 without their explicit mention each time.
Some Tests for the Intraclass Correlation Model
M. S. Srivastava
On a Class of Partially Balanced Incomplete Block Designs
S. S. Shrikhande
The purpose of this note is to show that the existence of any one of a particular family of four partially balanced incomplete block designs (pbibd) implies the existence of the remaining three designs. A sufficient condition for the existence of this family is given and a nonexistence result is also obtained.
The Relationship Algebra and the Analysis of Variance of a Partially Balanced Incomplete Block Design
Junjiro Ogawa, Goro Ishii
The analysis of variance of a partially balanced incomplete block design is investigated in connection with its relationship algebra. This gives a somewhat clearer insight into the structure of the partition of the total sum of squares than before. A. T. James [5] dealt with the same problem with the balanced incomplete block design and this article should be regarded as a generalization of his work to the partially balanced incomplete block design. The definitions and necessary notations concerning a partially balanced incomplete block design (PBIBD) and its association algebra are briefly given in Section 1. Although the properties of the association algebra have been known in another expression [4], they are presented in Section 2 in the form fitting to our discussions on the relationship algebra. In Section 3, the definition and properties of the relationship algebra of a PBIBD are given and these are believed to be new. In Section 4, the analysis of variance of a PBIBD is considered and the partition of the sum of squares due to treatments adjusted pertinent to the association under consideration is given. Finally, Section 5 is devoted to the analysis of variance of a PBIBD of triangular type and a numerical illustration.
On Hotelling's Weighing Designs Under Autocorrelation of Errors
K. S. Banerjee
Some results are indicated in this note to show the efficiencies of weighing designs when least square estimates are used and errors are correlated.
Asymptotic Distributions for the Coupon Collector's Problem
Leonard E. Baum, Patrick Billingsley
Convergence Rates for the Law of Large Numbers for Linear Combinations of Exchangeable and $^\ast$-Mixing Stochastic Processes
D. L. Hanson, L. H. Koopmans
Let $\xi = \{\xi_k \mid k = 0, \pm 1, \cdots\}$ be a sequence of real valued random variables with $E\xi_k \equiv 0$ and let $\{a_{n, k} \mid n = 1, 2, \cdots; k = 0, \pm 1, \cdots\}$ be a doubly indexed sequence of real numbers such that \begin{equation*}\tag{1}\sum_k |a_{n,k}| \leqq 1\quad\text{for} n = 1, 2, \cdots,\end{equation*} and \begin{equation*}\tag{2}\lambda(n) = \max_k |a_{n,k}| \rightarrow 0\quad\text{as} n \rightarrow \infty.\end{equation*} (In (1) we have replaced the upper bound $A < \infty$ of Condition 2 [3] by 1 which clearly entails no loss of generality.) The following result was established in [3]: Theorem A. If $\xi$ is an independent sequence of random variables and if the moment generating functions, $f_k(t)$, of the $\xi_k$'s exist and satisfy the condition; \begin{equation*}\tag{3} for every \beta > 0 there exists T_\beta > 0 such that |t| \leqq T_\beta implies\end{equation*} $f_k(t) \leqq \exp (\beta|t|) uniformly in k,$ then the random variables $S_n = \sum^\infty_{k = -\infty} a_{n,k}\xi_k$ are defined almost surely as limits of the partial sums for each $n$ and for every $\epsilon > 0$ there exists $\rho < 1$ (which depends on $\epsilon$ and $T_\beta$ but not on the particular $a_{n,k}$'s) such that $P\lbrack |S_n| \geqq \epsilon\rbrack \leq 2\rho^{1/\lambda(n)}.$ This result was used to establish exponential convergence rates for the law of large numbers for arbitrary subsequences of linear stochastic processes with absolutely convergent coefficients and for the convergence in probability to zero of Toeplitz means of independent random variables, extending previous results in [1] and [4]. In the present paper we investigate upper bounds on $P\lbrack |S_n| \geqq \epsilon\rbrack$ for two types of discrete parameter stochastic processes, $\xi$, closely related to independent sequences; exchangeable processes and $^\ast$-mixing processes. In Section 2 we establish a basic theorem for exchangeable processes, analogous to Theorem A, which enables us to state conditions leading to upper bounds which tend to zero with $n$ at virtually any sub-exponential rate. In Section 3 these bounds are shown to be sharp in certain important special cases by exhibiting a mixture of normal random variables which actually attains them under the given conditions. In Section 4, the exponential bounds obtained in [3] for independent sequences is shown to carry over to $^\ast$-mixing processes, thus extending a result in [2].
A Uniform Ergodic Theorem
George Burke
A Random Set Process in the Plane with a Markovian Property
Paul Switzer
On an Operator Limit Theorem of Rota
Norton Starr
A Note on the Sequential $t$-Test
J. Sacks
Hypergeometric Functions in Sequential Analysis
Adnan F. Ifram
In several sequential probability ratio tests [9] [12], density ratios may be expressed in terms of hypergeometric functions whose asymptotic behavior is indirectly available in the literature, and is useful in establishing the almost sure termination of these tests [6] [7] [8] [10]. The results of this paper are new for the sequential ordinary and multiple correlation coefficient tests [4] [6]. In addition, they complete the results of [8] and [10] for the sequential $F$-test [2] [6] as well as those of [7] for the sequential $\chi^2$- and $T^2$-tests [5] [6]. The generalized hypergeometric function $_pF_q$ is defined by: \begin{equation*} \tag{1.1}_pF_q(a_1, \cdots, a_p; c_1, \cdots, c_q; z) = 1 + (a_1 \cdots a_p/c_1 \cdots c_q)z/1! \end{equation*} $ + \lbrack a_1(a_1 + 1) \cdots a_p(a_p + 1)/c_1(c_1 + 1) \cdots c_q(c_q + 1) \rbrack z^2/2! + \cdots $ for $p, q \geqq 0$ and $c_i > 0, i = 1, \cdots, q$. We shall need in the sequel three such functions: $_2F_1(a, b; c; z)$, which is convergent for $|z| < 1, _1F_1(a; c; z)$, and $_0F_1(\quad ; c; z)$, which are convergent for all $z$.
A Lemma for Multiple Inference
William Knight
Multiple comparison methods have been constructed for an analysis of variance to test simultaneously a set of hypotheses so that with a predetermined joint confidence all hypotheses rejected are false. A general description of these is given by Scheffe ([6], Chapter 3). A lemma is presented to facilitate the construction of similar methods in other areas.
On the Distribution of the Latent Vectors for Principal Component Analysis
T. Sugiyama
The distribution of the latent vectors of a sample covariance matrix was found by T. W. Anderson [1] (1951) when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I.$ The asymptotic distribution for arbitrary $\Sigma$, also, was obtained by T. W. Anderson [3] in 1963. The elements of each latent vector are the coefficients of a principal component (with sum of squares of coefficients being unity). The object of the paper is to obtain the exact distribution of the latent vectors when the observations are obtained from bivariate normal distribution.
A Note on Incomplete Block Designs with $b = v$
M. Bhaskar Rao
Ann. Math. Statist. 36 (6), 1877, (December, 1965) DOI: 10.1214/aoms/1177699822
The Limit of the $n$th Power of a Density
Robert J. Buehler
Principles of Professional Statistical Practice
W. Edwards Deming
Acknowledgment of Priority: Some New Distribution-Free Statistics
C. B. Bell, K. A. Doksum
Review: Richard E. Barlow, Frank Proschan, Larry C. Hunter, Mathematical Theory of Reliability
J. J. McCall
Abstracts of Papers
Publications Received
Miscellaneous back pages, Ann. Math. Statist., Volume 36, Number 6 (1965)