Exponentiated Gamma Distribution : Different Methods of Estimations

The exponentiated gamma (EG) distribution and Fisher information matrices for complete, Type I, and Type II censored observations are obtained. Asymptotic variances of the different estimators are derived. Also, we consider different estimators and compare their performance through Monte Carlo simulations.


Introduction
Gupta et al. 1 introduced the exponentiated gamma EG distribution.This model is flexible enough to accommodate both monotonic as well as nonmonotonic failure rates.The EG distribution has the distribution function c.d.f.: Therefore, EG distribution has the density function: , θ, λ, x > 0, 1.2 the survival function R x; θ, λ 1 − 1 − e −λx λx 1 matrix for the parameter vector Ω θ, λ based on an observation in terms of the expected values of the first and second derivatives of the log-likelihood function is where for i, j 1, 2, Now, we will derive Fisher information matrix of EG θ, λ under complete sample.It can be shown that ln f x; θ, λ ln θ 2 ln λ − λx ln x θ − 1 ln 1 − e −λx λx 1 .

2.3
Differentiating 2.3 with respect to θ and λ respectively, we have

2.4
Therefore, the second derivatives are

2.5
Thus, the elements of Fisher information matrix for single observation from EG θ, λ are in the forms: where Moreover, Fisher information matrix for a complete sample of size n from EG θ, λ is simply nI θ, λ .

Fisher Information Matrix under Type II Censoring
Let X 1 , X 2 , . . ., X n be a random sample of size n from F x; Ω .In life-time analysis, n items are on test.The test continues until the rth smallest outcome is observed, 1 < r < n.Thus, we observe the smallest r-order statistics from F x; Ω , denoted by X 1:n ≤ X 2:n ≤ • • • ≤ X r:n , which are called the Type II censored data.Denote X r:n by X np :n , where np is the integer part of np, 0 < p < 1.Thus, r/n → p as n → ∞.Denote Fisher information matrix in X 1:n , X 2:n , . . ., X np :n by I 1••• np :n Ω see, Zheng 11 , where Ω θ, λ in the case of the EG θ, λ distribution, and define The estimates based on X 1:n , X 2:n , . . ., X np :n , under suitable conditions, are asymptotically normal, where the asymptotic covariance matrix is the inverse of I 0, p Ω .
Assuming the regularity conditions hold, the following expression for I 0, p Ω , where Ω is any finite-dimensional vector, can be expressed as where v p is the pth percentile of F x; Ω , T denotes the transpose, and h x; Ω is the hazard function.
If there is no censoring p 1 , then 2.11 becomes the usual Fisher information in a single variable 2.1 .
It can be shown that

2.13
Differentiating 2.13 with respect to θ and λ, respectively, we have

2.14
Thus, it is easily to see, for p 1−e −λv p λv p 1 θ , that the elements of I 0, p Ω from EG θ, λ are

2.17
It follows, by 2.15 , that the percentage of Fisher information about θ, I 0, p θ /a 11 , is independent of θ, and thus I 0, p θ is a decreasing function of θ.

Fisher Information Matrix under Type I Censoring
If the observation of X is right censored at a fixed time point t, that is, one observe min X, t , Fisher information for the parameter vector Ω based on a censored observation is thus where, for i, j 1, 2,

2.19
The Fisher information matrix of EG θ, λ under Type I censoring can be similarly derived as shown in Type II censoring.For p 2.20

Maximum Likelihood Estimators for Complete Sample
In this section, the maximum likelihood estimators MLE's of EG θ, λ are considered.First, we consider the case when both θ and λ are unknown.Let x 1 , x 2 , . . ., x n be a random sample of size n from EG θ, λ , then the log−likelihood function is

3.1
The normal equations become x 2 i e −λx i 0.

3.3
It follows, by 3.2 , that the MLE of θ as a function of λ, say θ λ , where

3.4
Substituting θ λ in 3.1 , we obtain the profile log-likelihood of λ as

3.5
Therefore, MLE of λ, say λ MLE , can be obtained by maximizing 3.5 with respect to λ as follows:

3.6
Once λ MLE is obtained, the MLE of θ, say θ MLE , can be obtained from 3.4 as θ MLE θ λ MLE .Now, we state the asymptotic normality results to obtain the asymptotic variances of the different parameters.It can be stated as follows: where I θ, λ is the information matrix 2.1 whose elements are given by 2.6 , 2.7 , and 2.8 .Now, consider the MLE of θ, when the scale parameter λ is known.Without loss of generality, we can take λ 1.If λ is known, the MLE of θ, say θ MLESCK , is It follows, by the asymptotic properties of the MLE, that where a 11 is the single information about θ which is defined in 2.6 .Now, note that if X i 's are independently and identically distributed EG θ, 1 , then

3.10
Using 3.8 , an unbiased estimate of θ can be obtained by where Let us consider the MLE of λ when the shape parameter θ is known.For known θ the MLE of λ, say λ MLESHK , can be obtained by numerical solving of the following equation: x 2 i e −λx i 0.

3.13
It follows, by the asymptotic properties of the MLE, that where a 22 is the single information about λ which is defined in 2.8 .

Maximum Likelihood Estimators under Censored Samples
Let X 1:n ≤ X 2:n ≤ • • • ≤ X r:n be the Type II censored data, then the likelihood function is given Lawless 20 by Next, let X i , t , i 1, 2, . . ., n, be independent, then x i min X i , t are the Type I censored data, thus the likelihood function is given Lawless 20 by

3.16
We now turn to the computationally more complicated case of censored data.Type I and Type II censorings will be considered simultaneously, since they give the same form of likelihood function above.We will deal with the MLE under Type II censoring from EG θ, λ and it is the same for Type I censoring.
In life testing, under the Type II censoring from EG θ, λ , the log likelihood function is 3.17

Journal of Applied Mathematics
The normal equations become where

3.20
The MLE of θ and λ, say θ MLETII and λ MLETII , can be obtained by solving numerically the two nonlinear equations 3.18 and 3.19 .
The MLE θ MLETII and λ MLETII , based on Type II censored data are strongly consistent and asymptotically normal see, Zheng 11 , that is, where Ω θ, λ and I 0, p Ω is the Fisher information matrix 2.12 whose elements are given by 2.15 , 2.16 , and 2.17 .Now, consider the MLE of θ, based on the Type II censored data when the scale parameter λ is known.Without loss of generality, we can take λ 1.If λ is known, the MLE of θ, say θ MLETIISCK , could be obtained by solving numerically the nonlinear 3.18 .It follows, by the asymptotic properties of the MLE, that where I 0, p θ is defined in 2.15 .
Let us consider the MLE of λ when the shape parameter θ is known.For known θ the MLE of λ, say λ MLETIISHK , can be obtained by solving numerically the nonlinear equation 3.19 .It follows, by the asymptotic properties of the MLE, that where I 0, p λ is defined in 2.16 .

Estimators Based on Percentiles
If the data come from a distribution function which has a closed form, then it is quite natural to estimate the unknown parameters by fitting straight line to the theoretical points obtained by the distribution function and the sample percentile points.
with respect to θ and λ.We call these estimators as percentile estimators PCEs and could be obtained by solving numerically the following two nonlinear equations: Several estimators of p i can be used here see, Murthy et al. 21 .In this section, we mainly consider p i i/ n 1 , which is the expected value of F x i:n .Now, let us consider the case when one parameter is known.If the shape parameter θ is known, then the PCE of λ, say λ PCESHK , can be obtained from 4.5 .Now let us consider the case when the scale parameter λ is known.Without loss of generality, we can assume that λ 1.If we denote F x; θ F x; θ, 1 , then ln F x; θ θ ln 1 − e −x x 1 .

4.6
Therefore, the PCE of θ, say θ PCESCK , can be obtained by minimizing Journal of Applied Mathematics with respect to θ and hence

4.8
Interestingly, θ PCESCK is also in a closed form like θ MLESCK when λ is known.

Least-Squares and Weighted Least-Squares Estimators
The least-squares estimators and weighted least-squares estimators were originally proposed by Swain Using the expectations and the variances, two variants of the least-squares methods can be used.
Method 1.The least-squares estimators of the unknown parameters can be obtained by minimizing with respect to the unknown parameters.Therefore, in case of EG distribution, the leastsquares estimators of θ and λ, say θ LSE and λ LSE , respectively, can be obtained by minimizing 1 − e −λx j:n λx j:n 1 θ − j n 1 θλx 2 j:n e −λx j:n 1 − e −λx j:n λ x j:n 1 θ−1 0.

5.5
Method 2. The weighted least-squares estimators of the unknown parameters can be obtained by minimizing with respect to the unknown parameters.Therefore, in case of an EG distribution, the weighted least-squares estimators of θ and λ, say θ WLSE and λ WLSE , respectively, can be obtained by minimizing with respect to θ and λ.
It could be found by solving the following nonlinear equations j:n e −λx j:n 1 − e −λx j:n λx j:n 1 θ−1 0. 5.9

Method of Moment Estimators
In this section, we provide the method of moment estimators MMEs of the parameters of an EG distribution.If X follows EG θ, λ , then where A 1 θ and A 2 θ are defined in 2.9 .
It is well known that the principle of the moment's method is to equate the sample moments with the corresponding population.From 6.1 and 6.2 , we obtain the coefficient of variation C.V. as The C.V. is independent of the scale parameter λ.Therefore, equating the sample C.V. with the population C.V., we obtain where We need to solve 6.4 to obtain the MME of θ, say θ MME .Once we estimate θ, we can use 6.1 to obtain the MME of λ.
If the scale parameter is known without loss of generality, we assume λ 1 , then the MME of θ, say θ MMESCK , can be obtained by solving the nonlinear equation: Now consider the case when the shape parameter θ is known, then the MME of λ, say λ MMESHK , is Note that 6.5 follows easily from 6.1 .Although λ MMESHK is not an unbiased estimator of λ, 1/ λ MMESHK is unbiased estimator of 1/λ and, therefore, 6.8

L-Moment Estimator
In this section, we propose a method of estimating the unknown parameter of an EG distribution based on the linear combination of order statistics see, 24, 25 .The estimators obtained by this method are popularly known as L-moment estimators LMEs .It is observed The standard method to compute the L-moment estimators is to equate the sample L-moments with the population L-moments.
First, we discuss the case of obtaining the LMEs when both the parameters of an EG distribution are unknown.If x 1:n ≤ x 2:n ≤ • • • ≤ x n:n denote the ordered sample, then using the same notation as in 15, 25 , we obtain the first and second sample L-moments as Similarly, the first two population L-moments see David and Nagaraja 24 are respectively, where μ i:n E X i:n .Then, for EG θ, λ , we obtain where A 1 θ is defined by 2.9 and Therefore, LMEs can be obtained by solving the following two equations: First, we obtain the LME of θ, say θ LME , as the solution of the following nonlinear equation: Once θ LME is obtained, the LME of λ, say λ LME , can be obtained from 7.5 as Note that if θ or λ is known, then the LME of λ or θ is the same as the corresponding MME obtained in Section 6.

Numerical Experiments and Discussions
In this section, we present the results of some numerical experiments to compare the performance of the different estimators proposed in the previous sections.We apply Monte Carlo simulations to compare the performance of different estimators, mainly with respect to their biases and mean squared errors MSEs for different sample sizes and for different parameter values.Since λ is the scale parameter and all the estimators are scale invariant, we take λ 1 in all our computations.We set θ 0.25, 0.5, 1, 2, 2.5 and n 10, 20, 50, 100.We compute the bias estimates and MSEs over 1000 replications for different cases.
First of all, we consider the estimation of θ when λ is known.In this case, the MLE, unbiased estimator UBE , and PCE of θ can be obtained from 3.4 , 3.11 , and 4.8 respectively.The least-squares and weighted least-squares estimators of θ can be obtained by solving numerically the nonlinear equations 5.4 and 5.8 , respectively.The MME similarly LME of θ can be obtained by solving numerically the nonlinear equation 6.5 as well.The results are reported in Table 1.
It is observed in Table 1 that for each method the MSEs decrease as sample size increases at θ ≥ 1.It indicates that all the methods deduce asymptotically unbiased and consistent estimators of the shape parameter θ for known λ.Moreover, all methods except PCE and MME usually overestimate θ whereas PCE underestimate θ in all cases that are considered and for some values of θ in case UBE.Therefore, the estimates of all methods are underestimate for most values of θ except MLE that forms overestimate for all θ.Also, all estimators are unbiased except LSE and WLSE that are the worst in biasness.The estimates of all methods are consistent except for some values at θ ≤ 0.5 because of the shape of the curve reversed J shaped .The first entry is the simulated bias.
The second entry is simulated MSE.
In the context of computational complexities, UBE, MLE, and PCE are the easiest to compute.They do not involve solvable nonlinear equation, whereas the LSE, WLSE, and MME involve solvable nonlinear equations and they need to be calculated by some iterative processes.Comparing all the methods, we conclude that for known scale parameter, the UBE should be used for estimating θ.
The negative sign in some results of the first entry only, in cells in all tables because of calculating of bias see, Abouammoh and Alshingiti 26 .Now consider the estimation of λ when θ is known.In this case, the MLE, PCE, LSE, and WLSE of λ can be obtained by solving the nonlinear equations 3.13 , 4.5 , 5.5 , and 5.9 , respectively, but the MME LME is exactly the same of λ can be obtained directly from 6.7 .The results are reported in Table 2.
In this case, it is observed, at the same sample size, that the value of θ increases for all methods the MSEs decrease.Comparing the computational complexities of the different estimators, it is observed that when the shape parameter is known, PCE and MME can be computed directly, while some iterative techniques are needed to compute MLE, LSE, and WLSE.We apply Newton-Raphson method using Mathematica 6 to solve the nonlinear equations required.Comparing all methods, we conclude that all the estimates are consistent except WLSE and LSE for some θ.
Also, for most estimates, the MSEs decrease as the values of θ decrease.We recommend to use PCE for estimate at θ ≤ 0.5, n 10, 20, and to use LSE at θ ≤ 0.5, n 100, while use WLSE at θ ≤ 0.5, n 50.All the estimates are consistent and unbiased at θ ≥ 1 for all values of n.
Finally, consider the estimation of θ and λ when both of them are unknown.The λ MLE can be obtained by solving the nonlinear equation 3.6 , once λ MLE is obtained, θ MLE can be obtained from 3.4 .The PCE of θ and λ can be obtained by solving the nonlinear equations 4.4 and 4.5 .Similarly, LSE of θ and λ can be obtained by solving the nonlinear equations 5.4 and 5.5 .Also, WLSE of θ and λ can be obtained by solving the nonlinear equations 5.8 and 5.9 .The θ MME or θ LME can be obtained by solving the nonlinear equation 6.4 or 7.7 , and then λ MME or λ LME can be obtained from 6.7 or 7.8 .The results for θ and λ are presented in Tables 3 and 4, respectively.
It is observed in Tables 3 and 4 that for each method, the MSEs decrease as sample size increases.It indicates that all the methods deduce asymptotically unbiased and consistent estimators of θ and λ when both are unknown.
Comparing the performance of all the estimators, it is observed that as far as the minimum biases are concerned, the MLE performs.Considering the MSEs, the MLE and PCE perform better than the rest in most cases considered.The performances of the LSE's and WLSE's for θ ≤ 1 are the worst as far as the bias or MSE's are concerned.Moreover, it is observed from Table 4 for PCE method that the MSE's of λ depend on θ, that is, for θ < 2.5, and n > 10 as θ, increases the MSEs of λ decrease.Most of the estimators are consistent and most of the estimators PCE are underestimate for all n.Now if we consider the computational complexities, it is observed that MLEs, MMEs, and LMEs involve one-dimensional optimization, whereas PCEs, LSEs, and WLSEs involve two-dimensional optimization.Considering all the points above, we recommend to use MLE's for estimating θ and λ when both are unknown.

n j 1 1 1 1 1 θ − j n 1 1 − 1 θ ln 1
− e −λx j:n λx j:n 1 θ and λ.It could be obtained by solving the following nonlinear equations: n j − e −λx j:n λx j:n e −λx j:n λx j:n − e −λx j:n λ x j: Murthy et al.21discussed this method for Weibull distribution while Gupta and Kundu 22 studied the generalized exponential distribution.First, let us consider the case when both parameters are unknown.Since ln 1 − F x; θ, λ 1/θ −λx ln λx 1 .4.2 Let X 1:n < X 2:n < • • • < X n:n be the order statistics obtained by EG θ, λ .If p i denotes some estimate of F x i:n ; θ, λ , then the estimate of θ and λ can be obtained by minimizing n i 1 et al. 23 to estimate the parameters of Beta distributions.It can be described as follows.Suppose Y 1 , Y 2 , . . ., Y n is a random sample of size n from a distribution function G • and Y 1:n < Y 2:n < • • • < Y n:n denotes the order statistics of the observed sample.It is well known that

Table 1 :
Bias estimates and MSEs of θ are presented, when λ is known.

Table 2 :
Bias estimates and MSEs of λ are presented, when θ is known.

Table 3 :
Bias estimates and MSEs of θ are presented, when λ is unknown.

Table 4 :
Bias estimates and MSEs of λ are presented, when θ is unknown.