A Study of Some Properties and Goodness-of-Fit of a Gompertz-Rayleigh Model

Department of Statistics, College of Science and Technology, Adamawa State Polytechnic, Yola, Adamawa State, Nigeria. Department of General Studies Education, School of General Education, Federal College of Education, Yola, Adamawa State, Nigeria. Department of Statistics, Ahmadu Bello University, P.M.B. 1045, Zaria, Kaduna State, Nigeria. Department of Statistics, Shehu Idris College of Health Science and Technology, Makarfi, Kaduna State, Nigeria. Department of Mathematical Science, Abubakar Tafawa Balewa University, Bauchi, Bauchi State, Nigeria.


Introduction
Rayleigh [1] proposed a continuous distribution using the amplitude of sound resulting from different significant sources. This distribution has been applied in several areas such as in life testing experiments, reliability analysis, applied statistics and clinical studies. It is also a special case of the two-parameter Weibull distribution when the shape parameter becomes 2. The history and other useful features of the Rayleigh model could be seen in Siddiqui [2], Hirano [3] and Howlader and Hossian [4].
Other generalized distributions away from Rayleigh distribution have shown that most compound distributions proposed in the literature fit various kind of data better than their standard counterparts. For instance, Weibull-Exponential distribution performed better than the Exponential distribution (Oguntunde et al. [12]), Weibull-Frechet distribution exhibited a very higher level of flexibility compared to the standard Frechet distribution (Afify et al. [13]), Lomax-Exponential distribution performed better than the exponential distribution (Ieren and Kuhe, [14]), others examples include Weibull-Lindley distribution (Ieren et al. [15]), Gompertz-Lindley distribution (Koleoso et al. [16]), Lomax-inverse Lindley distribution (Ieren et al. [17]), transmuted Lindley-Exponential distribution (Umar et al. [18]), Power Gompertz distribution (Ieren et al. [19]) and many others. Therefore the proposed model would be a more flexible distribution with greater skewness and flexibility using the Gompertz-G family.
This article is presented in sections as follows. The definition of the new distribution and its plots are provided in section 2. Section 3 derived some Statistical properties of the new distribution. The estimation of unknown parameters of the proposed model using maximum likelihood estimation (MLE) is provided in section 4. In section 5, the proposed distribution with some competing distributions is applied to a real life dataset. Finally, a brief summary with some useful conclusions is given in section 5.

The Gompertz-Rayleigh Distribution (GomRD)
For any continuous distribution with cdf ( ) G x and pdf, ( ) g x Alizadeh et al. [20] proposed the Gompertz generalized (denoted "Gompertz-G) family of distributions that provides greater flexibility in modeling of real data sets. According to this methodology, the cdf and pdf of the Gompertz-G family are defined for any continuous distribution as: Solving the integral above, equation (3) yields the cdf and pdf of the Gompertz-G family as respectively, where g(x) and G(x) are the pdf and cdf of any continuous distribution to be generalized respectively and 0   and 0   are the two additional shape parameters.
Making substituting of equation (1) and (2) in (4) and (5) above and simplifying, we obtain the cdf and pdf of the GomRD for a random variable X as: and The graph of the pdf and cdf of the GomRD using some arbitrary parameter values are shown as follows:

Fig. 2. CDF of GomRD for selected parameter values
From the above cdf plot, the cdf increases when x increases, and approaches 1 when x becomes large, as expected.

Quantile function, median and simulation
Hyndman and Fan [21] defined the quantile function for any distribution in the form     Using the cdf of the proposed model and applying simple algebra derive the quantile function of the GomRD as follows: Simplifying equation (8) above gives: Collecting like terms in equation (9) and simplifying, the quantile function of the GomRD is obtained as: where u is a uniform variate on the unit interval (0,1).
The median of X from the GomRD is simply obtained by setting u=0.5 and this substitution of 0.5 u  in Equation (10) which gives: Similarly, random numbers can be simulated from the GomRD by setting   Q u X  and this process is called simulation using inverse transformation method. This means for any , , 0 "where u is a uniform variate on the unit interval (0,1).
Also, using equation (10), the quantile based measures of skewness and kurtosis are derived as follows: Kennedy and Keeping [22] defined the Bowley's measure of skewness based on quartiles as: And Moors [23] presented the Moors' kurtosis based on octiles by: "where   . Q is calculated by using the quantile function from equation (10).

Reliability analysis of the GomRD
This part of the article presents the survival (or reliability) function and the hazard (or failure) rate function of GomRD as follows: The Survival function relates the likelihood of success for a unit or an individual after a given time. Mathematically, the survival function is given by: Using the cdf of the GomRD in (15) and collecting like terms gives the survival function for the GomRD as: The graph of the survival function of the GomRD using different parameter values is shown below.

Fig. 3. Survival function of GomRD for selected parameter values
The graph of the survival function in Fig. 3 above reveals that the probability of survival for any random variable or event following a GomRD decreases as the variable or event advances in age or time, which means as time goes on, probability of life decreases. This implies that the GomRD could be used to analyze random variables whose survival rate decreases as they advance in age.
Hazard function is a mathematical function that relates the likelihood that a component will fail for an interval of time. The hazard function is defined as: Considering the pdf and cdf of GomRD, an expression for the hazard rate of GomRD is simplified and stated as: The graph of the hazard function for some parameter values is as given below: Fig. 4 presents the graph of the hazard function of GomRD and it reveals that the failure rate for any random variable or event following a GomRD increases as the variable or event advance in time or age, meaning that as time grows, probability of death increases to infinity. This implies that the GomRD could be used to model random events whose failure rate increases per time or age.

Maximum Likelihood Estimation of the unknown Parameters of the GomRD
Let n X X X ., ,......... , 2 1 be a sample of size 'n' independently and identically distributed random variables from the GomRD with unknown parameters,  ,  and  defined previously.
The likelihood function of the GomRD using the pdf in equation (7) Differentiating   l  partially with respect to  ,  and  respectively gives the following results: Making equation (21), (22) and (23)

Application to a Real Life Dataset
In this section, an application to real life data is provided to illustrate the flexibility of the GomRD introduced in this study. We have compared the fitness of the GomRD to those of five other extensions of the Rayleigh model such as transmuted Weibull-Rayleigh distribution (TWRD), Weibull-Rayleigh distribution (WRD), transmuted Rayleigh distribution (TRD), Gompertz distribution (GomD) and the conventional Rayleigh distribution (RD). The MLEs of the model parameters are determined and some goodness-of-fit statistics for this distribution are compared with other competitive models.
The above listed distributions were fitted to the dataset and their performance was selected based on the following: Akaike Information Criterion, AIC, Consistent Akaike Information Criterion, CAIC, Bayesian Information Criterion, BIC, Hannan Quin Information Criterion, HQIC, Anderson-Darling (A*), Cramѐr-Von Mises (W*) and Kolmogorov-smirnov (K-S) statistics. More information about these information criteria can be found in Chen and Balakrishnan [24]. Note that the smaller these statistics are, the better the fit of the distribution.
Data set: This dataset has been used by Afify and Aryal [25], Barreto-Souza et al. [26], Bourguignon et al. [27], Oguntunde et al. [12], Ieren et al. [15], Ieren and Yahaya [28], Yahaya and Ieren [29], Koleoso et al. [16] and Smith and Naylor [30]. It is summarized as follows: Based on the statistics from Table 1, it is observed that the real life dataset is negatively skewed, and could be good for skewed models.     Tables 2 presents the values of the Maximum Likelihood Estimates of the model parameters for the real life  dataset used, while table 3 presents the values of AIC, CAIC, BIC and HQIC for all the distributions fitted to  the real life dataset and the values

Summary and Conclusion
This paper has presented a three-parameter probability distribution called "a Gompertz-Rayleigh distribution". The paper has provided a comprehensive study of some statistical properties of the proposed distribution (Gompertz-Rayleigh) which include the derivation of explicit expressions for its survival function, hazard function and the quantile function which is useful for obtaining the median, skewness and kurtosis and simulation of random numbers from the proposed distribution. The estimation of the unknown model parameters using the method of maximum likelihood estimation has also being considered and treated in this study. The graph of the pdf of the proposed distribution has shown that its shape is skewed model and flexible for modeling real life data. Also, the graph of the survival and hazard functions are monotonically decreasing and increasing respectively. This means that the proposed model could be used for analyzing random events failure rate increase per age or time. It is also an indication that the proposed distribution could be a good choice for modeling cured fraction of surviving patients in a population. The performance of the proposed distribution (GomRD) has also been evaluated in this present paper using a real life dataset and the results revealed that the proposed distribution is much flexible compared to the other fitted distributions due to its ability to fit the real life dataset better than the other distributions as considered in this study. Based on this performance therefore we recommend that the proposed model be used for statistical quality control of variables whose distribution is skewed.