Estimation of the reliability function for two-parameter exponentiated Rayleigh or Burr type X distribution

Problem Statement: The two-parameter exponentiated Rayleigh distribution has been widely used especially in the modelling of life time event data. It provides a statistical model which has a wide variety of application in many areas and the main advantage is its ability in the context of life time event among other distributions. The uniformly minimum variance unbiased and maximum likelihood estimation methods are the ways to estimate the parameters of the distribution. In this study, we explore and compare the performance of the uniformly minimum variance unbiased and maximum likelihood estimators of the reliability functions R(t) = P (X > t) and P = P (X > Y ) for the two-parameter exponentiated Rayleigh distribution. Approach: A new technique of obtaining the estimators of these parametric functions is introduced in which major role is played by the estimators of powers of the parameter(s) and the functional forms of the parametric functions to be estimated are not needed. We explore the performance of these estimators numerically under varying conditions. Through the simulation study a comparison are made on the performance of these estimators with respect to the bias, mean square error (MSE), 95% confidence length and corresponding coverage percentage. Conclusion: Based on the results of simulation study, the uniformly minimum variance unbiased estimators of R(t) and ‘P ’ for the twoparameter exponentiated Rayleigh distribution are found to be superior than maximum likelihood estimators of R(t) and ‘P ’.


Introduction
Reliability theory is mainly concerned with the determination of the probability that a system, consisting possibly of several components, will operate adequately for a given period of time in its intended application.The reliability function R(t) is defined as the probability of failure-free operation until time t.Thus, if the random variable (rv) X denotes the lifetime of an item, then R(t) = P (X > t).Another measure of reliability under stress-strength set-up is the probability P = P (X > Y ), which represents the reliability of an item of random strength X subject to random stress Y .Many researchers have considered the problems of estimation of R(t) and 'P ' for various lifetime distributions and for a brief review, one may refer to [3,4,5,13,16,24,2,28,10,18,29,8,9],etc.
In [7], Burr introduced twelve different forms of cumulative distribution functions for modelling lifetime data.Among those distributions, Burr Type X and Burr Type XII are the most popular ones.Several authors considered different aspects of the Burr Type X and Burr Type XII distributions, see, for example, [21,23,30,11,12,1,19,25].For an excellent review for the two distributions the readers are refereed to [15].[26] (see also [27]) introduced two-parameter Burr Type X distribution and named as the two-parameter exponentiated Rayleigh distribution.The twoparameter exponentiated Rayleigh distribution has the following probability density function (pdf) f (x; α, λ) = 2αλ 2 xe −(λx) 2  (1 − e −(λx) 2 ) α−1 ; x, α, λ > 0 (1) and the distribution function Here, α and λ are the shape and scale parameters, respectively.In [20], the authors observed that for α ≤ 1/2, the pdf of two-parameter exponentiated Rayleigh distribution is a decreasing function and it is a right skewed unimodal function for α > 1/2.They found different forms of the density function.It is also observed that the hazard function of a two-parameter exponentiated Rayleigh distribution can be either bathtub type or an increasing function, depending on the shape parameter α.For α ≤ 1/2, the hazard function of two-parameter exponentiated Rayleigh distribution is bathtub type and, for α > 1/2, it has an increasing hazard function.Surles and Padgett [26] showed that the two-parameter exponentiated Rayleigh distribution can be used quite effectively in modelling strength data and also modelling general lifetime data.Kundu and Raqab [17] proposed different methods of estimation for generalized Rayleigh distribution.
The rest of the study is arranged as follows.In Section 2, we derive the UMVUES of the reliability function R(t) and 'P ' assuming α to be unknown but λ known.In Section 3, we obtain the MLES of the reliability function R(t) and 'P ', when all the parameters are unknown.In Section 4, simulation study is carried out to investigate the performance of estimators.Finally, In Section 5, discussion is made and followed by conclusion.
2. UMVUES of the powers of α, R(t) and 'P ' when λ is known Let X 1 , X 2 , . . ., X n be a random sample of size n from (1).

Lemma 1
Then, S is complete and sufficient for the distribution given at (1).Moreover, the pdf of S is (3) It follows from (3) and Fisher-Neymann factorization theorem [see [22], p. 341] that S is sufficient for the distribution given in (1).In (1), if we make the transformation which is χ 2 (2) .Thus, from the additive property of gamma distribution [see [14], p. 170] Hence, the distribution of S follows from (4).Since the distribution of S belongs to exponential family, it is also complete [see [22], p. 170].
The following lemma provides the UMVUES of the powers of α.
In the following lemma, we provide the UMVUE of the sampled pdf (1) at specified point 'x'.

Lemma 2
The UMVUE of f (x; α, λ) at a specified point 'x' is otherwise.

Proof
Since S is complete and sufficient for the distribution f (x; α, λ), any function H(S) of S satisfying E[H(S)] = f (x; α, λ) will be the UMVUE of f (x; α, λ).
From (1) and Lemma 1, we have or, or, or, Let us choose Hence the lemma holds.

Remark 1
We can write (1) as Using (2.4), Theorem 1 and Lemma 1 of Chaturvedi and Tomer (2002 otherwise, which coincide with Lemma 2. Thus, the UMVUES of the powers of α can be used to derive the UMVUE of f (x; α, λ) at a specified point 'x'.
In the following theorem, we obtain UMVUE of R(t).

Theorem 2
The UMVUE of R(t) is given by ], p.207)] are satisfied for the change of order of integration.Let us consider the expected value of the integral We conclude from ( 7) that the UMVUE of R(t) can be obtained simply integrating f (x; α, λ) from t to ∞.Thus, from Lemma 2, and the theorem follows.
Let X and Y be two independent rv's following the distributions f 1 (x; α 1 , λ 1 ) and f 2 (y; α 2 , λ 2 ), respectively, where Here, we assume that α 1 and α 2 are unknown but λ 1 and λ 2 are known.Let X 1 , X 2 , . . ., X n be a random sample of size n from f 1 (x; α 1 , λ 1 ) and In what follows, we obtain the UMVUE of 'P '.

Theorem 3
The UMVUE of 'P ' is given by

Proof
It follows from Lemma 2 that the UMVUES of f 1 (x; α 1 , λ 1 ) and f 2 (y; α 2 , λ 2 ) at specified points 'x' and 'y', respectively, are and From the arguments similar to those adopted in the proof of Theorem 2, it can be shown that the UMVUE of 'P ' is given by

Proof
From Theorem 3, for S < T , ) i and the first assertion follows.From Theorem 3, for S > T , ) j+1 and the second assertion follows.
3. MLES of R(t) and 'P ' when all the parameters are unknown Following the lines of derivations in [17], it can be shown that the MLES of α and λ are solutions of and respectively.
Remarks 1(i) In the literature, the researchers have dealt with the estimation of R(t) and 'P ', separately.If we look at the proof of Theorems 2 and 3, we observe that the UMVUE of the sampled pdf is used to obtain the UMVUES of R(t) and 'P ', respectively, which is also true for MLES.Thus we have established interrelationship between the two estimation problems.Moreover, in the present approach, one does not require the expressions of R(t) and 'P '.
(ii) Since the UMVUES and MLES of powers of α are obtained under same conditions, we compare their performances.For q = −1 the UMVUE and MLE of α are, respectively α = (n − 1)(−T ) −1 and α = (n)(−T ) −1 .For these estimators, Thus, Thus, the UMVUE of α is more efficient than its MLE.Similarly, we can compare the performances of these estimators for other powers of α.

Numerical Findings
In order to compare the efficiency of the estimators α and α, when λ is known, we have calculated variances of α and α, for samples of sizes n = 5, 10, 20, 30 and 50 corresponding to = 0.80(0.60)4.20 and these results are reported in Table 1.
From Table 1, it is clear that α is more efficient than α.In order to verify the consistency of the estimators obtained, we have drawn sample of sizes n = 30 from (1), with α = 4 and λ = 3.In Fig. 1, we have  plotted f (x; α, λ), f (x; α, λ) and f (x; α, λ), respectively, corresponding to this sample.We conclude from Fig. 1 that curves of f (x; α, λ) and f (x; α, λ) overlap to the curve of f (x; α, λ) for n = 30.This justifies the consistency property of the estimators.
In order to demonstrate the application of the theory developed in Section 3, we generated a sample of size n = 30 from (1) for α = 4 and λ = 3. Solving (13) and   In order to obtain the MLE of 'P ', we have generated one more sample of size m = 30 from (1) for α = 2.5 and λ = 3. Solving as above, we get λ = 2.486173, λ = 2.963452 and −2 ln L = −32.0487.Using this population as Y and above population as X, we get P = 0.6153846 and P = 0.6146800.
For the case when α is unknown but λ is known, we have conducted simulation experiments using bootstrap re-sampling technique for sample sizes n = 5, 10, 20 and 50.The samples are generated from (1), with α = 3 and λ = 2.5.For different values of t, we have computed R(t), R(t), 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 2.

Discussion and Conclusion
In Table 1, we compared UMVUE and MLE of α, keeping λ to be constant for two-parameter exponentiated Rayleigh distribution.The table shows that UMVUE of α is more efficient than MLE of α.From table we observe that as we increase the sample size variance of estimators of α decrease (for both of estimators UMVUE as well as for MLE).Table 1 also shows that as we increase values of the parameter α, variance increases corresponding to both of the estimators.
With the help of Fig. 1, we justified the consistency property of the estimators.Through Table 2, we compared the efficiency of R(t) and R(t).Table 2 shows that UMVUE of R(t) is more efficient than MLE of R(t).It is also clear that as we increase sample size Biasness, MSE and Confidence Length decreases but on the other hand corresponding Coverage Percentage increases.These statements are also true for the estimators P and P .

Table II .
Simulation results for R(t).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.

Table III .
Simulation results for 'P '.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.