American Journal of Theoretical and Applied Statistics
Volume 5, Issue 4, July 2016, Pages: 192-201

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

Amr Sadek

Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt

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To cite this article:

Amr Sadek. Bayesian Prediction Based on Type-I Hybrid Censored Data from a General Class of Distributions. American Journal of Theoretical and Applied Statistics. Vol. 5, No. 4, 2016, pp. 192-201. doi: 10.11648/j.ajtas.20160504.15

Received: May 4, 2016; Accepted: May 12, 2016; Published: June 14, 2016


Abstract: 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.

Keywords: Bayesian Prediction, Type-I Hybrid Censored, General Class, Markov Chain Monte Carlo, Importance Sampling Technique


1. 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  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  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

(1)

where a, b and c are constants (a; c  0) s.t  and  is a monotonic and differentiable function of x in the interval  and the parameter may be a real vector, then

(2)

where . The following table gives some distributions with proper choice of a; b; c and h(x) as examples of the general class.

Table 1. Some distributions derived from the general class.

Let  be the order statistics from a random sample of size  from a distribution function  given in (1) with density function  given in (2).

Let  denote the number of ’s that are at most  Then  is a discrete random variable with support  and probability density function as

(3)

where  and

We have one of the two following types of observations:

Case I.  if  with

Case II:  if  with

The likelihood function of a Type-I hybrid censored sample is as follows:

Case I.

(4)

where  and

Case II:

(5)

where  and

When  the conditional density function of  given the Type-I hybrid censored data, is obtained as follows:

Case I.

(6)

where

(7)

and

(8)

with

 

and

.

Case II:

(9)

where

2. 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  where  is the vector of parameters of the distribution under consideration and  is the vector of prior parameters. Then the posterior density function of  can be written as:

Case I.

(10)

where  and  and

Case II.

(11)

where  and  and

2.1. One-Sample Bayesian Prediction Intervals

We simply obtain the predictive survival function of  as follows:

Case I.

(12)

where

(13)

and

(14)

Case II.

(15)

The Bayesian predictive  interval for  can be obtained by solving the following two equations:

(16)

where

and  and  denote the lower and upper bounds, respectively.

2.2. Two-Sample Bayesian Prediction Intervals

Let us consider a future sample  of size , independent of the informative sample  and let  be the order statistics of the future sample. Suppose we are interested in the predictive density of the order statistic  of the future sample, given the informative data set . The probability density function of the -th order statistic of the future sample of size  from a continuous distribution with the distribution function  and the probability density function  is given by

where  (see, Arnold et al. [16]).

From (1) and (2), we simply obtain the probability density function of the -th order statistic from a general class as follows:

where  and we simply obtain the Bayesian predictive density function of  as follows:

Case I.

(17)

Case II.

(18)

From (17) and (18), we simply obtain the predictive survival function of  as follows:

Case I.

(19)

Case II.

(20)

Then, the Bayesian predictive  interval for  can be obtained by solving the following two equations:

(21)

where

and  and  denote the lower and upper bounds, respectively.

3. 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.

3.1. Inverse Weibull Distribution

In this subsection, we take a special case from this general class, the inverse Weibull distribution, when  and  we provide the posterior density function depend on the maximum likelihood distribution given in (4) and (5). Here, we assumed that the model parameters  and  follow the independent gamma prior density of the following forms:

where  and  are the hyper-parameters. Then, the joint posterior density function of  and  given the Type-I hybrid censored data, can be written as:

Case I.

(22)

where  and

Therefore, the posterior density function of  and  given  can be written as

(23)

where  is a gamma density function with the shape and scale parameters as  and , respectively,  is a proper density function given by

(24)

and

(25)

Case II.

(26)

where  and

Similarly as above, we can write the posterior density function of  and  given  as

(27)

where  is a gamma density function with the shape and scale parameters as  and , respectively,  is proper density function given by

(28)

and

(29)

3.1.1. One-Sample Bayesian Prediction

The conditional density function of  given the Type-I hybrid censored data, (from Eqs (7), (8), (9)), can be written as:

Case I.

(30)

where

Similarly, From Eq. (8), we can obtain

(31)

Case II.

The conditional density function of  given the Type-I hybrid censored data, in this case, can be written as:

(32)

where

Then, we obtain the predictive survival function of  as follows:

(33)

and similarly

(34)

(35)

It does not seem to be possible to compute the probabilities in Eqs (33), (34), (35) analytically. Hence, we use the importance sampling technique, (see, Geweke [17]; Chen and Shao ([18]; [19])) to construct the Bayesian prediction interval. The details are explained below.

Importance Sampling technique

Firstly, we need to prove that the  as given in (24) and (28) has a log-concave density function: From (24), the  without the additive constant is

it is easy to show that , which implies that  has a log-concave density function.

Since  has a log-concave density, using the idea of Devroye [20], it is possible to generate sample from . Moreover, since  follows gamma, it is quite simple to generate sample from . Now we would like to provide the importance sampling procedure to compute the probabilities in Eqs (33), (34), (35).

Algorithm:

• Step1: Generate  from  using the method developed by Devroye [20].

• Step2: Generate  from.

• Step3: Repeat Step 1 and Step 2 and obtain

• Step4: The approximate value of  can be obtained as

Similary, we can use the above algorithm to compute

The Bayesian predictive  interval for  can be obtained by solving the two equations given in (16).

3.1.2. Two-Sample Bayesian Prediction

The predictive survival function of  in this special case is obtained as follows:

Case I.

(36)

Case II.

(37)

where

By the same importance sample technique, the Bayesian predictive  interval for  can be obtained by solving the two equations in (21).

3.2. Inverted Exponential Distribution

The inverted exponential distribution is a special case from inverse Weibull distribution when the shape parameter is known (). we provide the posterior density function depend on the maximum likelihood distribution given in (4) and (5), when the shape parameter is . It is assumed that the scale parameter has a gamma prior distribution with the shape and scale parameters as  and , respectively and it has the probability density function

The posterior density function of  given the Type-I hybrid censored data, can be written as:

Case I.

(38)

where  and

Case II.

(39)

where  and

In this case the predictive survival function of  is the same as above when we put

3.2.1. One-Sample Bayesian Prediction

To compute  by using the MCMC technique, we use the following procedure:

• Step 1. Generate  from

• Step 2. Repeat Step 1 and obtain ;

• Step 3. The approximate value of  is then obtained as

Similarly, we can use the above algorithm to compute

The Bayesian predictive  interval for  can be obtained by solving the two equations given in (16).

3.2.2. Two-Sample Bayesian Prediction

The predictive survival function of  in this special case is obtained as follows:

Case I.

(40)

Case II.

(41)

Then, the Bayesian predictive interval for  can be obtained by the same manner.

4. Numerical Results

In this section we consider a real life data set which the inverse Weibull distribution fits it well and another generated data set from the inverted exponential distribution to illustrate the methods proposed in the previous sections.

The real data set is given by Dumonceaux and Antle [21], and it represents the maximum flood levels (in millions of cubic feet per second) of the Susquehenna River at Harrisburg, Pennsylvania over 20 four-year periods (1890-1969) as: 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379,0.324, 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265. Maswadah [22] and Singh et al. [23] checked the suitability of the inverse Weibull distribution to this real data set and concluded that the inverse Weibull distribution fits the data very well.

We consider two different Type-I hybrid censoring schemes:

1. When  and . In this case, the life-test would have terminated at , and we have obtained the folloing data: 0.265, 0.269, 0.297, 0.315, 0.324, 0.338, 0.379, 0.379, 0.392, 0.402, 0.412, 0.416, 0.418 and 0.423;

2. When  and . In this case, the life-test would have terminated at , and we have obtained the following data: 0.265, 0.269, 0.297, 0.315, 0.324, 0.338, 0.379, 0.379, 0.392, 0.402, 0.412, 0.416, 0.418, 0.423 and 0.449.

By using the procedures presented earlier, we construct  one-sample Bayesian prediction intervals for order statistics  as well as  two-sample Bayesian prediction intervals for order statistics  from a future sample of size  To explore the sensitivity of the predictors with respect to the hyperparameters , we have considered the following four hyperparameters:  Table 2 presents the lower and upper  one-sample Bayesian prediction bounds for  for these four choices of the hyperparameters. Similarly, the lower and upper  two-sample Bayesian prediction bounds for for the different choices of the hyperparameters are presented in 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 ,

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 ), we generated the parameter  from prior distribution,  then generated the inverted exponential random sample of size  the generated sample is listed as the following:

0.45175 0.84893 0.95595 1.33698 1.38886 1.42612 1.89285 2.0541 .18509 2.60002 2.64192 2.8265
2.91384 3.34748 3.39207 3.49797 .41589 4.56534 4.67023 5.94493 8.93186 9.01355 9.69029 10.5754
.0716 11.7103 13.3331 78.4651 135.999 848.432            

The corresponding results for the one-sample and two-sample prediction intervals are represented in Tables 4 and 5, respectively.

5. 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  but larger value of  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  while the upper bounds are somewhat sensitive.

Table 2. one-sample Bayesian prediction bounds for  from inverse Weibull distribution.

Table 3.  two-sample Bayesian prediction bounds for  from inverse Weibull distribution.

Table 4. one-sample Bayesian prediction bounds for . from inverted exponential distribution.

Table 5.  two-sample Bayesian prediction bounds for  from inverted exponential distribution.


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