Bayesian Analysis of Record Statistics Based on Generalized Inverted Exponential Model

In some situations, only observations that are more extreme than the current extreme value are recorded. This kind of data is called record values which have many applications in a lot of fields. In this paper, the Bayesian estimators using squared error and LINEX loss functions for the generalized inverted exponential distribution parameters are considered depending on upper record values and upper record ranked set sampling. The Bayes estimates and credible intervals are derived by considering the independent gamma priors for the parameters. The Markov Chain Monte Carlo (MCMC) method is developed due to the lack of explicit forms for the Bayes estimates. A Simulation study is implemented to compute and compare the performance of estimators in both sampling schemes with respect to relative absolute biases, estimated risks and the width of credible intervals. Keywords— upper record ranked set sample; Bayesian estimator; squared error (SE) loss function; linear exponential (LINEX) loss function; Markov Chain Monte Carlo.


I. INTRODUCTION
Record data are very important in many situations when the observations are difficult to obtain or are destroyed in experimental tests. Record data arise in a wide variety of practical situations including industrial stress testing, meteorology, sports, hydrology and economics. Record values can be viewed as order statistics from a sample that's its size is determined by the values and the order of occurrence of the observations. A record value of some phenomenon is the largest (smallest) observation anyone has ever made. The theoretical contributions and inference for record values have been studied extensively in the literature. The reader may refer to [1], [2], [3] and [4].
According to [3], record values can be classified into lower record values (LRV) and upper record values (URV).
≥ be a sequence of independent and identically distributed (iid) random variables, an observation j X is called URV (LRV) if its value exceeds (lower than) all of the previous observations, i.e., ( ) Reference [5] presented a new sampling scheme, called record ranked set sampling (RRSS), for generating record data. The new scheme helps the scientists in situations where the only observations that are going to be used are the last record data as in athletic, weather and Olympic data.
The upper record ranked set sampling (URRSS) can be described as follows: Suppose that there exist n independent sequential sequences of continuous random variables, the th i sequence sampling is ceased when the th i record value is observed. The only observations that are used for analysis are the last record value in each sequence. The last record value of the th i sequence in this plane is denoted by , i i U s are independent random variables but not ordered. Bayesian estimation based on record values has been considered by several researchers. Among them, [6]- [19]. Reference [20] discussed the Bayesian estimation problem for the shape parameter of the weighted exponential distribution based on URRSS.
The applicability of the one parameter exponential distribution is the simplest and the most widely discussed distribution for lifetime data but it is restricted to a constant hazard rate. Most generalizations of the exponential distributions possess the constant, non-increasing, nondecreasing and bathtub hazard rates. But in many practical situations, the data shows the inverted bathtub hazard rate (initially increase and then decrease, i.e., unimodal). For such data types, another extension of the exponential distribution known as the one parameter inverted exponential distribution is provided, which have inverted bathtub hazard rate [21]. Reference [22] introduced the twoparameter generalized inverted exponential distribution (GIED) by adding a shape parameter to the inverted exponential distribution.
The probability density function (pdf) of the GIED with the shape parameter and the scale parameter takes the following form 0, , 0.
The cumulative distribution function (cdf) is as follows Recently, there has been a growing interest in the study of inference problems associated with record values and record ranked set samples via the Bayesian approach. The Bayesian estimation for the GIED based on URRSS hasn't been studied in the literature yet. The goal of this paper is to obtain and to compare the Bayes estimates and the average width of the posterior 95% credible intervals for the unknown parameters of GIED on the basis of URV and URRSS. These Bayes estimates and credible intervals width are obtained using independent gamma priors under symmetric (squared error (SE)) and asymmetric (linear exponential (LINEX)) loss functions through MCMC method. The procedures are illustrated through analysing a simulated data. The rest of the paper is organized as follows. Section (II) gives the Bayes estimates based on URV, the Bayes estimates under URRSS, MCMC approach and a simulation. Discussion and results of the simulation study appear in Section (III). Concluding remarks appear in Section (IV).

A. Bayesian Estimators Based on URV
In this section, Bayesian estimators of the unknown parameters of the GIED under the assumption of independent gamma priors on both the shape and scale parameters are considered. Based on URV, the Bayes estimators cannot be obtained in explicit forms. Hence the MCMC technique is carried out to generate samples from the posterior distributions and consequently computing the Bayes estimators and construct the corresponding credible intervals. Here, two types of loss functions are considered for Bayesian computation; symmetric one (SE) and asymmetric one (LINEX). Let 1 m r = (r , , r ) … be a set of URV from GIED ( , ) α λ , the likelihood function according to [3], is given by where, ( ) ., f θ and, ( ) ., F θ are respectively the pdf and the cdf of GIED ( , ) α λ . The likelihood function of the observed URV is obtained, as follows Further, assuming that the prior of parameters and has a gamma distribution with parameters ( , ) and ( , ) respectively. Hence, assuming independence of parameters, the joint prior distribution of parameters, denoted by ( , ), is as follows ; , , , , , 0 The expression for the joint posterior can be written as r d v Generally, as observed, the analytical solution of integrations given by (7) and (8) is very difficult to obtain due to the complicated mathematical form. Therefore, the MCMC technique is employed to approximate these integrations. Therefore Metropolis-Hastings (M-H) method will be implemented which is a powerful MCMC technique to compute the Bayes estimates and credible intervals width.

B. Bayesian Estimators Based on URRSS
This section discusses the Bayes estimates of the unknown shape and scale parameters of the GIED under the assumption of independent gamma priors defined in (5) based on URRSS using SE and LINEX loss functions. Let Hence, the marginal posterior distributions of α and λ , based on URRSS, can be expressed as follows and where, v is a real number.
Again, the integrals (10) and (11) cannot be reduced to a closed form due to its difficult mathematical form. So, M-H algorithm is used to compute the Bayes estimator under the SE and LINEX loss functions.

C. Simulation Study
In this section, a numerical study is performed in order to examine and compare the behaviour of the Bayes estimators for the two parameters of the GIED ( , ) The simulation results are summarized in Tables (I-VI) and represented through Fig. (1-6). From these tables and figures, the following observations can be made

IV. CONCLUSION
In this paper, we presented how to develop Bayes estimates in the context of upper record values and upper record ranked set sampling from generalized inverted exponential distribution under symmetric and asymmetric loss functions.
Based on the URV and URRSS, it is observed that the Bayes estimators cannot be obtained in explicit forms. Therefore, the MCMC technique has been used to generate posterior samples.
We observe from the numerical study that the relative absolute biases, estimated risks and widths of confidence intervals are very small based on the two sampling schemes for both SE and LINEX loss functions. Generally