Estimating the Reliability Function for a Family of Exponentiated Distributions

A family of exponentiated distributions is proposed. The problems of estimating the reliability function are considered. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are derived. A comparative study of the two methods of estimation is done. Simulation study is preformed.


Introduction
The reliability function ( ) is defined as the probability of failure-free operation until time . Thus, if the random variable (rv) denotes the lifetime of an item, then ( ) = ( > ). Another measure of reliability under stressstrength setup is the probability = ( > ), which represents the reliability of an item of random strength subject to random stress .
A lot of work has been done in the literature to deal with inferential problems related to various exponentiated distributions. In particular, Mudholkar and Srivastava [1] considered exponentiated Weibull family for analyzing bathtub failure-real data. The exponentiated Weibull family was also applied to the bus-motor-failure data in Davis [2], to head-and-neck cancer clinical trial data in Efron [3] [see [4]], and in analyzing flood data [see [5]]. Jiang and Murthy [6] considered exponentiated Weibull family and illustrated some of its properties by using graphical approach. Nassar and Eissa [7,8] studied a two-parameter exponentiated Weibull distribution and they gave some of its properties and estimated the parameters by using the maximum likelihood and Bayes methods based on type II censored data. They used the squared-error and linear in exponential (LINEX) loss functions and an informative prior to obtain the Bayes estimates. Pal et al. [9] showed that the failure rate of exponentiated Weibull behaves more like the failure rate of the Weibull distribution than that of the Gamma distribution. They also obtained maximum likelihood estimators, Fisher's information matrix, and confidence intervals for related parameters. Gupta et al. [10] introduced exponentiated exponential distribution in a series of papers. Gupta and Kundu [11][12][13][14][15][16][17], Gupta et al. [18], and Kundu et al. [19] concentrated on the study of the exponentiated exponential distribution. Kundu and Gupta [20] stated that exponentiated exponential distribution is an alternative to the well-known two-parameter Gamma, two-parameter Weibull or two-parameter lognormal distributions. Kundu and Gupta [20] estimated the reliability of the stress-strength model ( < ), when and are independent exponentiated exponential random variables. Raqab and Ahsanullah [21] estimated the parameters of the exponentiated exponential based on ordered statistics. Inference for exponentiated Weibull distribution based on record values was made by Raqab [22]. Abdel-Hamid and AL-Hussaini [23] obtained the maximum likelihood estimates of the parameters when step-stress accelerated life testing is applied for exponentiated exponential distribution. Al-Hussaini and Hussein [24,25] fitted the exponentiated Burr model to the data of breaking strength of single carbon fibers of particular length (data in [26]) in the complete sample case and when type II censoring is imposed on data. SELF and LINEX loss functions are used in the Bayes estimation. Gupta et al. [10] introduced a new distribution, called the exponentiated Pareto distribution. Shawky and Abu-Zinadah [27] obtained maximum likelihood estimators of the different parameters of the exponentiated Pareto distribution. They also considered five other estimation procedures and compared their performances through numerical simulations.
The purpose of the present paper is manyfold. We propose a family of exponentiated distributions, which covers as many as ten exponentiated distributions as specific cases. We consider the problems of estimating ( ) and " . " Uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES) are derived. In order to obtain these estimators, the major role is played by the estimator of the probability density function (pdf) at a specified point, and the functional forms of the parametric functions to be estimated are not needed. It is worth mentioning here that, in order to estimate " , " in the literature, the authors have considered the case when and follow the same distributions, may be with different parameters. We have generalized the result to the case when and may follow any distribution from the proposed exponentiated family of distributions. Simulation study is carried out to investigate the performance of the estimators.
In Section 2, we introduce the family of exponentiated distributions. In Section 3, we derive the UMVUES of ( ) and " . " In Section 4, we obtain the MLES of ( ) and " . " In Section 5, analysis of a simulated data has been presented for illustrative purposes. Finally, in Section 6, conclusions have been provided.

The Family of Exponentiated Distributions
Let the rv follow the distribution having the pdf Here, ( ; , ) is a function of and may also depend on (may be vector-valued) parameter . Moreover, ( ; , ) is a real-valued, monotonically increasing function of with ( ; , ) = 0, (∞; , ) = ∞, and ( ; , ) denotes the derivative of ( ; , ) with respect to . Our case is a special model of the beta extended Weibull family proposed by Cordeiro et al. [28] and can be obtained from equation (7) when = 0. We note that (1) represents a family of exponentiated distributions, as it covers the following exponentiated distributions as special cases.
The following lemma provides the UMVUES of the powers of .
Using Lemma 1 of Chaturvedi and Tomer [37] and Lemma 2, the UMVUE of ( ; , , , ) at a specified point " " iŝ or ( ; , , , ) which coincides with Lemma 3. Thus, the UMVUE of the power of can be used to derive the UMVUE of ( ; , , , ) at a specified point.
In the following theorem, we obtain the UMVUE of ( ).
Theorem 5. The UMVUE of ( ) is given bŷ Proof. Let us consider the expected value of the integral ∫ ∞̂( ; , , , ) with respect to ; that is, 4

Journal of Probability and Statistics
We conclude from (14) that the UMVUE of ( ) can be obtained simply integratinĝ( ; , , , ) from to ∞. Thus, denoting by −1 (⋅), the inverse function of ( ; , ), from Lemma 3, we obtain and the theorem follows.
be a random sample of size from 1 ( ; 1 , 1 , 1 , 1 ) and let 1 , 2 , . . . , be a random sample of size from 2 ( ; 2 , 2 , 2 , 2 ). Let us denote by In what follows, we obtain the UMVUE of " ". Theorem 6. The UMVUE of " " is given bŷ Proof. From the arguments similar to those adopted in proving Theorem 5, it can be shown that the UMVUE of " " is given bŷ= Thus, using Lemma 3, Let us first consider the case when In this case, from (18), Now we consider the case when In this case, from (18), The theorem now follows on combining (19) and (20).
(i) In the literature, the researchers have dealt with the estimation of ( ) and " , " separately. If we look at the proofs of Theorem 5 and Theorem 6, we observe that the UMVUE of the sampled pdf is used to obtain the UMVUES of ( ) and " . " Thus, we have established interrelationship between the two estimation problems.
(ii) In the literature, the researchers have derived the UMVUES of " " for the case when and follow the same distribution (may be with different parameters). We have obtained UMVUES of " " for all the three situations, when and follow the same distribution having all the parameters same other than 's, when and have the same distribution with different parameters and when and have different distributions.

MLES of ( ) and '' '' When All the Parameters Are Unknown
Looking at the distributions belonging to the family (1), we notice that, except for exponentiated Pareto and generalized exponentiated Pareto distributions [( ) and ( )], = 0; that is, " " is known. For these two distributions contains " . " First of all, we replace " " by (1) in (22); then we differentiate (22) with respect to all the unknown parameters and equate these differential equations to zero. The MLES of unknown parameters are obtained on solving these equations simultaneously. Let̃,̃, and̃be the maximum likelihood estimators of , , and , respectively.
The following lemma provides The MLE of ( ; , , , ) at a specified point " ".
Proof. The proof follows from (1) and the one-to-one property of the MLE.
In the following theorem, we derive the MLE of ( ).

Theorem 9. The MLE of ( ) is given bỹ
Proof. Using Lemma 8 and invariance property of the MLES, and the theorem follows.
In the following theorem, we obtain the maximum likelihood estimator of " . " 6 Journal of Probability and Statistics Theorem 10. The MLE of " " is given bỹ Proof. Using Lemma 8 and the one-to-one property of the MLES̃= and the result follows. Hence, Thus, the UMVUE of is more efficient than its MLE. Similarly, we can compare the performances of these estimators for other powers of .

Simulation Studies
In order to verify the consistency of estimators, we have drawn samples of sizes = 5, 20 and 30 from (1) with ( ; , ) = , = 0, = 3, and = 1. In Figure 1, we have plotted̂( ; , ) corresponding to these samples. We conclude that as the sample size increases, the curves of̂( ; , ) come closer to the curve of ( ; , ), which is plotted in the same figure only for = 30. For = 30, the curves overlap. A similar pattern follows for the curves of ( ; , ). For the case when is unknown but other parameters are known, we have conducted simulation experiments using bootstrap resampling technique for sample sizes = 5, 10, 20, and 50. The samples are generated from (1), with, ( ; , ) = , = 0, = 3, and = 1. For different values of , we have computed̂( ),̃( ), their corresponding bias, variance, 95% confidence length, and corresponding coverage percentage. All the computations are based on 500 bootstrap replications and results are reported in Table 1.
In order to demonstrate the application of the theory developed in Section 4, we consider the following example. Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage.
Let the rv follow exponentiated exponential distribution with pdf where , , and are unknown. Denoting, 1 ( , , | ), the likelihood, the log-likelihood is given by log 1 ( , , ) = log + log + log Considering negative log-likelihood, then differentiating it with respect to all unknown parameters, and equating these differential coefficients to zero, from (32), Journal of Probability and Statistics Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage.

Conclusions
We propose a class of distributions, which covers as many as ten exponentiated distributions as special cases. The problems of estimating ( ) and " " are considered. UMVUES and MLES are derived. A comparative study of the two methods of estimation is done. The estimators of " " are derived, which cover the cases when and may follow the same, as well as, different distributions. By estimating the sampled pdf to obtain the estimators of ( ) and " , " an interrelationship between the two estimation problems is established. Simulation study is performed, and a real-data example is presented.