Bayesian Prediction Based on Type-I Hybrid Censored Data from a General Class of Distributions

One and two-sample Bayesian prediction intervals based on Type-I hybrid censored for a general class of distribution 1-F(x)=[ah(x)+b] c are obtained. For the illustration of the developed results, the inverse Weibull distribution with two unknown parameters and the inverted exponential distribution are used as examples. Using the importance sampling technique and Markov Chain Monte Carlo (MCMC) to compute the approximation predictive survival functions. Finally, a real life data set and a generated data set are used to illustrate the results derived here.


Introduction
Prediction plays an important role in different areas of applied statistics such as medical sciences and reliability analysis. Bayesian prediction have more attention among other issues of prediction. Discussion of the prediction intervals (one-sample and two-sample prediction) for a future sample is valuable in lifetime studies. Bayesian prediction intervals for future observations have been discussed by several authors, including Howlader [1], Geisser [2], Raqab and Nagaraja [3], Al-Hussaini and Jaheen ( [4]; [5]), Abdel-Aty et al. [6], Kundu and Howlader [7], Mohie El-Din et al. ([8]; [9]), Shafay and Balakrishnan [10], Mohie El-Din and Shafay [11] and Shafay et al. [12]. In this article, we use a general class of distribution (see; Khan and Abu-Salih [13], Athar and Islam [14]) to derive general procedure for determining the one-and two-sample Bayesian prediction intervals based on Type-I hybrid censored data. In the rest of this section, we derive the likelihood function and the conditional density functions of Xs:n given the Type-I hybrid censored data. In Section 2, we derive the one-sample Bayesian predictive survival function and the one-sample Bayesian predictions bounds for the ( ) s th r s n − < ≤ ordered lifetime from Type-I hybrid censored sample. Furthermore, we derive the two-sample Bayesian predictive survival function and the two sample Bayesian predictions bounds for the s th − ordered lifetime from a future independent sample. In Section 3, special cases of this general class such as the inverse Weibull distribution when the two parameters are unknown and the inverted exponential distribution are considered as illustrative examples, wherein we adopt the importance sampling technique to compute the approximation predictive survival function in the one-sample case and the Markov Chain Monte Carlo (MCMC) method to compute the approximation predictive survival function in the two-sample case. Finally, some numerical examples are conducted to illustrate the prediction procedures.
Let the general form of distributions be [ ]  The following table gives The likelihood function of a Type-I hybrid censored sample is as follows: Case I.
, r s n ≤ the conditional density function of : , s n X given the Type-I hybrid censored data, is obtained as follows: Case I. 11 1 12 ( | ), < , and Case II:

Bayesian Analysis
Bayesian approach has received a lot of attention for estimating the parameters of statistical distributions and for predicting samples. It makes use of ones prior knowledge about the parameters and also takes into consideration the data available. If ones prior knowledge about the parameter is available, it is suitable to make use of an informative prior but in a situation where one does not have any prior knowledge about the parameter and cannot obtain vital information from experts to this regard, then a non-informative prior will be a suitable alternative to use, Guure et al. [15].
Let the prior distribution denoted by ( ; ), π θ δ where θ ∈ Θ is the vector of parameters of the distribution under consideration and δ is the vector of prior parameters. Then the posterior density function of , θ can be written as: Case I.

One-Sample Bayesian Prediction Intervals
We simply obtain the predictive survival function of : s n X as follows: Case I.
and : X s n L and : X s n U denote the lower and upper bounds, respectively.

Two-Sample Bayesian Prediction Intervals
Let us consider a future sample 1 2 independent of the informative sample 1 2 { , , , } n X X X ⋯ and let 1: (see, Arnold et al. [16]). From (1) and (2), we simply obtain the probability density function of the s -th order statistic from a general class as follows: From (17) and (18) Case II. * * 2 2 : :

Examples
In this section, we discuss the Bayesian prediction of observations from the inverse Weibull distribution when both parameters are unknown and from the inverted exponential distribution. To our knowledge, no one study these distributions for determining the Bayesian prediction intervals for future lifetimes based on an observed Type-I hybrid censored data.

Inverse Weibull Distribution
In this subsection, we take a special case from this general class, the inverse Weibull distribution, when we provide the posterior density function depend on the maximum likelihood distribution given in (4) and (5). Here, we assumed that the model parameters θ and p follow the independent gamma prior density of the following forms: v are the hyper-parameters. Then, the joint posterior density function of θ and , p given the Type-I hybrid censored data, can be written as: Case I.
Similarly as above, we can write the posterior density function of θ and p given k X as is a gamma density function with the shape and scale parameters as is proper density function given by and 2 ( , | ) = [1 ] .

One-Sample Bayesian Prediction
The conditional density function of : s n X given the Type-I hybrid censored data, (from Eqs (7), (8), (9)), can be written as: Case I.
Case II. The conditional density function of : s n X given the Type-I hybrid censored data, in this case, can be written as: e e e w s r q w s r q t w s r q r e e p X d dp and similarly * * 12

Inverted Exponential Distribution
The inverted exponential distribution is a special case from inverse Weibull distribution when the shape parameter is known ( = 1 p ). we provide the posterior density function depend on the maximum likelihood distribution given in (4) and (5), when the shape parameter is = 1 p . It is assumed that the scale parameter has a gamma prior distribution with the shape and scale parameters as u and v , respectively and it has the probability density function The posterior density function of , θ given the Type-I hybrid censored data, can be written as:  (1,7,3,1), (1,7,3,2), (1,8,3,1), (1,8,3,2).  Table 3. Now we generate another data set to illustrate the predictions results for the inverted exponential distribution, we follow the steps 1. given the set of prior parameters, generate the parameter θ , 2. using the generated population parameter, generate an inverted exponential random sample of size n, 3. follow the procedures presented in Section 2.2 to construct one-sample and two-sample Bayesian prediction intervals based on Type-I hybrid censored data.
Given the set of prior parameters (let = 30, = 11, u v ), we generated the parameter θ from prior distribution, = 2.7 θ then generated the inverted exponential random sample of size = 30, n the generated sample is listed as the following: The corresponding results for the one-sample and two-sample prediction intervals are represented in Tables 4  and 5, respectively.

Concluding Remarks
In this paper, we obtained one and two sample prediction bounds based on Type-I hybrid censored data under the general class of distributions. We introduced two examples, the inverse Weibull distribution with unknown two parameters and the inverted exponential distribution, to illustrate the developed results. Bayesian predictive survival function can not be obtained in closed form and so importance sampling technique and Markov Chain Monte Carlo samples, which are then used to compute the approximate predictive survival function. Finally, some numerical results are presented to illustrate the results and we observe the following remarks: 1. From Tables 2-5, we notice that the lengths of the Bayesian prediction intervals are short when there are a large number of observed values. It is clear that when we use the same value of r but larger value of , T the Bayesian prediction intervals become tighter.
2. It is observed that the prediction intervals tend be wider when s increase. This is a natural, since the prediction of the future order statistic that is far a way from the last observed value has less accuracy than that of other future order statistics.
3. It is evident from Tables 2 and 3 that the lower bounds of Bayesian prediction are relatively insensitive to the specification of the hyperparameters 1 1 2 2 ( , , , ) u v u v while the upper bounds are somewhat sensitive.