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.
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.
References