Journal of Probability and Statistics
Volume 2011, Article ID 457472, 17 pages
http://dx.doi.org/10.1155/2011/457472
Research Article

## Bayesian Inference on the Shape Parameter and Future Observation of Exponentiated Family of Distributions

1Department of Statistics, St. Anthony's College, Shillong 793 001, India
2Department of Statistics, Visva-Bharati University, Santiniketan 731 235, India

Received 17 May 2011; Accepted 5 September 2011

Copyright © 2011 Sanku Dey and Sudhansu S. Maiti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Bayes estimators of the shape parameter of exponentiated family of distributions have been derived by considering extension of Jeffreys' noninformative as well as conjugate priors under different scale-invariant loss functions, namely, weighted quadratic loss function, squared-log error loss function and general entropy loss function. The risk functions of these estimators have been studied. We have also considered the highest posterior density (HPD) intervals for the parameter and the equal-tail and HPD prediction intervals for future observation. Finally, we analyze one data set for illustration.

#### 1. Introduction

Let be a random variable whose cumulative distribution function (cdf) and probability density function (pdf) are given by respectively. Here is the continuous baseline distribution function with the corresponding probability density function , and may be vector valued, and is the positive shape parameter. Then, is said to be belonging to the exponentiated family of distributions (abbreviated as EFD) or the proportional reversed hazard family.

If the baseline distribution is exponential, then it is called the generalized exponential (GE) distribution in the literature. In recent years, an impressive array of papers has been devoted to study the behavioral patterns of the parameters of the generalized exponential distribution using both classical and Bayesian framework, and a very good summary of this work can be found in Gupta and Kundu [14], Raqab [5], Raqab and Ahsanullah [6], Zheng [7], Raqab and Madi [8], Alamm et al. [9], Singh et al. [10], Dey [11], and the references cited there for some recent developments on GE distribution. If the baseline distribution is Weibull, then it is called the exponentiated Weibull distribution. Mudholkar and Srivastava [12], Nassar and Eissa [13], and Singh et al. [14] have studied this distribution.

In this paper, we assume that is known, but the shape parameter is unknown. Then, the cdf and pdf become respectively. If is symmetric, then will be skewed distribution for different values of . Hence can be considered as a skewness parameter. Gupta and Gupta [15] have shown that positively skewed data can be analyzed very well for normal baseline distribution. Again is the parameter of the proportional reversed hazard model in lifetime data analysis. For its various important roles, we are interested to find out the Bayes estimators and their performances under different loss functions using different priors. In Figure 1, the shape of (i) exponentiated distribution with , (ii) exponentiated Rayleigh, distribution with , and (iii) exponentiated lognormal distribution with has been shown for and , respectively.

Figure 1: Exponentiated distributions for and .

The paper is categorized into the following sections. Section 2 has a brief description of the prior distributions and loss functions. The Bayes estimators and associated risk functions are provided in Sections 3 and 4, respectively. Section 5 presents the highest posterior density (HPD) interval for . Section 6 is devoted to finding out the predictive distributions and equal-tail Bayesian predictive interval for the future observation. Section 7 deals with the Bayes predictive estimator and HPD prediction interval for a future observation. Section 8 presents the data application based on a real life data set. The paper ends with a concluding remark in Section 9.

#### 2. Prior and Loss Functions

The Bayesian inference requires appropriate choice of prior(s) for the parameter(s). From the Bayesian viewpoint, there is no clear cut way from which one can conclude that one prior is better than the other. Nevertheless, very often priors are chosen according to one’s subjective knowledge and beliefs. However, if one has adequate information about the parameter(s), it is better to choose informative prior(s); otherwise, it is preferable to use noninformative prior(s). In this paper we consider both type of priors: the extended Jeffreys’ prior and the natural conjugate prior.

The extended Jeffreys’ prior proposed by Al-Kutubi [16] is given as The conjugate prior in this case will be the gamma prior, and the probability density function is taken as With the above priors, we use three different loss functions for the model (1.1).(1)The first loss function considered is called weighted quadratic loss function and is given by where is a decision rule to estimate . is to be chosen such that is minimum. This can be equivalently written as being minimum. Hence (2)The second one is the squared-log error loss function proposed by Brown [17] and is defined as This loss function is balanced with as or . A balanced loss function takes both error of estimation and goodness of fit into account, but the unbalanced loss function only considers error of estimation. This loss function is convex for and concave otherwise, but its risk function has a unique minimum with respect to . The Bayes estimator for the parameter under squared-log error loss function may be given as where denotes the posterior expectation. (3)The third loss function is a particular type of asymmetric loss functions called the general entropy loss function proposed by Calabria and Pulcini [18] (Podder and Roy [19] called it the modified linear exponential (MLINEX) loss function) and is given by If we replace in place of , that is, , we get the linear exponential (LINEX) loss function, . Without loss of generality, we assume that . If , it is the entropy loss function. Under this general entropy loss function, the Bayes estimator of is obtained as follows:

#### 3. Estimation of Parameter

Let us consider a random sample of size from the exponentiated family of distributions. Then the likelihood function of for the given sample observation is Here, the maximum likelihood estimator (MLE) of is , with .

##### 3.1. Estimation under the Assumption of Extended Jeffreys’ Prior

Combining the prior distribution in (2.1) and the likelihood function, the posterior density of is derived as follows: which is a gamma distribution .

For differnt derivations in this section and subsequent sections, we use the expressions , , , , the digamma function, and , the trigamma function.

Using extended Jeffreys’ prior of the form (2.1), the Bayes estimators of under weighted quadratic, squared-log error and general entropy loss functions are derived as follows: with , respectively.

Remark 3.1. We get the Jeffreys’ noninformative prior for and the Hartigan’s noninformative prior for .

##### 3.2. Estimation under the Assumption of Conjugate Prior

Combining the prior distribution in (2.2) and the likelihood function, the posterior density of is derived as follows: which is a gamma distribution with .

Using a conjugate prior of the form (2.2), the Bayes estimators under weighted quadratic, squared-log error and general entropy loss functions are derived as follows: respectively. It is to be noted that Bayes’ estimators given in (3.5), (3.6), and (3.7) depend on and which are the parameters of the prior distribution of . These parameters, that is, and could be estimated by means of empirical Bayes’ procedure (see Lindley [20] and Awad and Gharraf [21]). Given the random sample , the likelihood function of is gamma density with parameter . Hence it is proposed to estimate the prior parameters and from the sample by and , respectively. Therefore, (3.5), (3.6), and (3.7) will become respectively.

#### 4. Risks of the Bayes Estimators

Since follows the exponentiated family of distributions with parameter , then is distributed as . Then the probability density function of is Therefore, The risk function of is Similarly, the risk functions of and under squared-log error loss and general entropy loss functions are respectively. The risk functions of , , and , assuming conjugate prior are respectively.

The risk functions of under weighted quadratic, squared-log error and general entropy loss functions are respectively.

The estimators, developed in Section 3, are studied here on the basis of their risks obtained under three different loss functions, namely, (a) weighted quadratic loss, (b) squared-log error loss, (c) general entropy loss for , (d) general entropy loss for , and (e) general entropy loss for . Risk functions of the proposed estimators are shown in Figures 25. The thick lines in each figure show the risk of the Bayes estimators under extended Jeffreys’ prior and conjugate prior, and dotted lines show the risk of MLE under different loss functions. In Figure 2, risk functions have been plotted for all the loss functions under extended Jeffreys’ prior for different values of and for . It is observed that risks are increasing with the increase in . Risks under general entropy loss for are ordinarily less than those of weighted quadratic and squared-log error losses. For small values of , risks of Bayes estimators are lower than those of maximum likelihood estimators for each loss function considered. The Bayes estimators perform better for some values of and loss functions under consideration; for example, in the figure, risk of is less for (approximately) whereas that of is less for (approximately). Figures 3 and 4 show the risks for different values of for and 2, respectively. We find that the risks are decreasing with the increase in for all values of .

Figure 2: Risk function of estimators under the extended Jeffreys’ prior for different values of with .
Figure 3: Risk function of estimators under the extended Jeffreys’ prior for different values of with .
Figure 4: Risk function of estimators under the extended Jeffreys’ prior for different with .
Figure 5: Risk function of estimators under conjugate prior for different .

When we consider conjugate prior, we see that risks are less for the squared-log error and weighted quadratic losses than the general entropy loss, and only for these two losses, the Bayes estimators are less than the MLE for small (Figure 5). The risks under conjugate prior are generally higher than those under Jeffreys’ prior.

#### 5. Highest Posterior Density Intervals for 𝛼

In this section our objective is to provide a highest posterior density (HPD) interval for the unknown parameter of the model (1.2). HPD interval is one of the most useful tools to measure posterior uncertainty. This interval is such that it includes more probable values of the parameter and excludes the less probable ones. Since the posterior density (3.2) is unimodal, the HPD interval for must satisfy that is that is simultaneously. The HPD interval is the simultaneous solution of (5.2) and (5.4).

Similarly, the posterior density (3.4) is unimodal, and the HPD interval for must satisfy that is that is simultaneously. Therefore, the HPD interval is the simultaneous solution of (5.6) and (5.8). If and are not known, then substituting the empirical Bayes estimate of and , we get the equations as follows: respectively.

#### 6. Predictive Distribution

In this section our objective is to obtain the posterior predictive density of future observation, based on current observations. Another objective is to attain equal-tail Bayesian prediction interval for the future observation and then compare this interval with frequentist predictive interval. The posterior predictive distribution for given under (3.2) is defined by A equal-tail prediction interval is the solution of Using (6.1), we get (after simplification) The posterior predictive distribution for given under (3.4) is defined by A equal-tail prediction interval is the solution of Using (6.4), we get (after simplification) If and are not known, then substituting the empirical Bayes estimate of and , we get the prediction limits as follows: For deriving classical intervals, we notice that is distributed as a beta-variate of the second kind with parameters 1 and . The pdf of has the following form: Solving for in and using (6.8), we get (after simplification) It is to be noted that if we take in (6.3), we get classical equal-tail prediction interval.

#### 7. Bayes Predictive Estimator and HPD Prediction Interval for a Future Observation

In this section, we introduce the Bayes predictive estimator for different priors under the above-mentioned loss functions, and later we obtain HPD predictive intervals for the future observation. The Bayes predictive estimators of under a weighted quadratic error loss function assuming the extended Jeffreys’ prior and the conjugate prior are respectively. The Bayes predictive estimators of under the squared-log error loss function assuming the extended Jeffreys’ prior and the conjugate prior are respectively. The Bayes predictive estimators of under the general entropy loss function assuming the extended Jeffreys’ prior and the conjugate prior are respectively. The closed form expressions of (7.1)–(7.6) seem to be intractable, and calculations are to be made using numerical method.

For the unimodal predictive density (6.1), the HPD-predictive interval with probability for is the simultaneous solution of the following: that is that is Similarly, for the unimodal predictive density (6.4), the HPD-predictive interval with probability for is the simultaneous solution of the following: that is that is If and are not known, then substituting the empirical Bayes estimate of and , we get the HPD-prediction limits for future observation as follows:

#### 8. Data Analysis

Consider the following data which arose in tests on endurance of deep grove ball bearings (Lawless [22, page 228]). The data are the number of million revolutions before failure for each of the 23 ball bearings in the life test: To study the goodness of fit of the exponentiated exponential model, Gupta and Kundu [2] computed the statistic as 0.783 with the corresponding value being 0.376. The estimate of , the shape parameter, is 5.2589 and that of , the rate of the exponential distribution, is 0.0314. Here our aim is to obtain the Bayes estimates of for this data set under three different loss functions and for two priors by assuming that the base line distribution is exponential with . At the same time, we are interested in studying the HPD intervals for the parameter . Further, our intention is to obtain the future observation based on a given set of observations and the HPD predictive intervals for the future observation based on the current observations. Figure 6 shows the estimated predictive distribution.

Figure 6: Estimate of the predictive distribution for the given data set for extended Jeffreys’ prior and conjugate prior.

Tables 14 summarize the result from the data analysis. Tables 1(a) and 1(b) represents the Bayes estimates of and corresponding risks under the extended Jeffreys’ prior the conjugate prior, respectively. It is evident from Table 1(a) that Bayes estimates under general entropy loss ( and ) give better estimates than all other estimates. It is also evident from Table 1(a) that the estimates decrease and the corresponding risks increase with the increase in . In case of conjugate prior (Table 1(b)), the estimates under weighted quadratic and squared-log error loss functions seem to be better. In Table 2, the HPD intervals under conjugate prior appear to be slightly better than the extended Jeffreys’ with respect to minimum length. Table 3 presents the estimates of future observation based on the data set. It is observed from the last column of the table that the general entropy loss function at gives a quite reasonable estimate. Table 4 shows the Bayesian predictive and HPD predictive intervals for future observation of the data set. The first row in each cell represents the Bayesian predictive intervals, and the second row represents the HPD intervals for future observation. It is apparent that the HPD intervals for future observation with respect to conjugate prior are reasonably good.

Table 1
Table 2: HPD intervals of the shape parameter for the data set.
Table 3: Estimated future observation for the data set under three different loss functions.
Table 4: Bayesian predictive and HPD predictive intervals for future observation of the data set.

#### 9. Concluding Remark

In this paper, we have derived the Bayes estimators of the shape parameter of the exponentiated family of distributions under the extended Jeffreys’ prior as well as conjugate prior using three different loss functions. Though the extended Jeffreys’ prior gives the opportunity of covering wide spectrum of priors, yet at times the conjugate prior also gives better Bayes estimates and HPD intervals of the parameter and of future observations.

#### Acknowledgment

The authors would like to thank the referee for a very careful reading of the manuscript and making a number of nice suggestions which improved the earlier version of the manuscript.

#### References

1. R. D. Gupta and D. Kundu, “Generalized exponential distributions,” Australian and New Zealand Journal of Statistics, vol. 41, no. 2, pp. 173–188, 1999.
2. R. D. Gupta and D. Kundu, “Exponentiated exponential family: an alternative to gamma and Weibull distributions,” Biometrical Journal, vol. 43, no. 1, pp. 117–130, 2001.
3. R. D. Gupta and D. Kundu, “Generalized exponential distribution: different method of estimations,” Journal of Statistical Computation and Simulation, vol. 69, no. 4, pp. 315–337, 2001.
4. R. D. Gupta and D. Kundu, “Discriminating between Weibull and generalized exponential distributions,” Computational Statistics & Data Analysis, vol. 43, no. 2, pp. 179–196, 2003.
5. M. Z. Raqab, “Inferences for generalized exponential distribution based on record statistics,” Journal of Statistical Planning and Inference, vol. 104, no. 2, pp. 339–350, 2002.
6. M. Z. Raqab and M. Ahsanullah, “Estimation of the location and scale parameters of generalized exponential distribution based on order statistics,” Journal of Statistical Computation and Simulation, vol. 69, no. 2, pp. 109–123, 2001.
7. G. Zheng, “On the Fisher information matrix in type II censored data from the exponentiated exponential family,” Biometrical Journal, vol. 44, no. 3, pp. 353–357, 2002.
8. M. Z. Raqab and M. T. Madi, “Bayesian inference for the generalized exponential distribution,” Journal of Statistical Computation and Simulation, vol. 75, no. 10, pp. 841–852, 2005.
9. A. A. Alamm, M. Z. Raqab, and M. T. Madi, “Bayesian prediction intervals for future order statistics from the generalized exponential distribution,” Journal of the Iranian Statistical Society, vol. 6, no. 1, pp. 17–30, 2007.
10. R. Singh, S. K. Singh, U. Singh, and G. P. Singh, “Bayes estimator of generalized-exponential parameters under Linex loss function using Lindley's approximation,” Data Science Journal, vol. 7, pp. 65–75, 2008.
11. S. Dey, “Bayesian estimation of the shape parameter of the generalised exponential distribution under different loss functions,” Pakistan Journal of Statistics and Operations Research, vol. 6, no. 2, pp. 163–174, 2010.
12. G. S. Mudholkar and D. K. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Transactions on Reliability, vol. 42, no. 2, pp. 299–302, 1993.
13. M. M. Nassar and F. H. Eissa, “Bayesian estimation for the exponentiated Weibull model,” Communications in Statistics. Theory and Methods, vol. 33, no. 10, pp. 2343–2362, 2004.
14. U. Singh, P. K. Gupta, and S. K. Upadhyay, “Estimation of parameters for exponentiated-Weibull family under type-II censoring scheme,” Computational Statistics and Data Analysis, vol. 15, pp. 2065–2077, 2005.
15. R. D. Gupta and R. C. Gupta, “Analyzing skewed data by power normal model,” TEST, vol. 17, no. 1, pp. 197–210, 2008.
16. H. S. Al-Kutubi, On comparison estimation procedures for parameter and survival function exponential distribution using simulation, Ph.D. thesis, Baghdad University, Baghdad, Iraq, 2005.
17. L. Brown, “Inadmissiblity of the usual estimators of scale parameters in problems with unknown location and scale parameters,” Annals of Mathematical Statistics, vol. 39, pp. 29–48, 1968.
18. R. Calabria and G. Pulcini, “Point estimation under asymmetric loss functions for left-truncated exponential samples,” Communications in Statistics. Theory and Methods, vol. 25, no. 3, pp. 585–600, 1996.
19. C. K. Podder and M. K. Roy, “Bayesian estimation of the parameter of Maxwell distribution under MLINEX loss function,” Journal of Statistical Studies, vol. 23, pp. 11–16, 2003.
20. D. V. Lindley, Introduction to Probability and Statistics from a Bayesian View Point, Cambridge University Press, Cambridge, Mass, USA, 1969.
21. A. M. Awad and M. K. Gharraf, “Estimation of $P\left[Y in the Burr case: a comparative study,” Communications in Statistics—Simulation and Computation, vol. 15, no. 2, pp. 389–403, 1986.
22. J. F. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley and Sons, New York, NY, USA, 1982.