Exponentiated Gamma Distribution: Different Methods of Estimations

Abstract

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.

1. 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.):𝐹(𝑥;𝜃,𝜆)=1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃,𝜃,𝜆,𝑥>0. (1.1) Therefore, EG distribution has the density function:𝑓(𝑥;𝜃,𝜆)=𝜃𝜆²𝑥𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃−1,𝜃,𝜆,𝑥>0, (1.2) the survival function𝑅(𝑥;𝜃,𝜆)=1−1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃,𝜃,𝜆,𝑥>0, (1.3) and the hazard functionℎ(𝑥;𝜃,𝜆)=𝜃𝜆²𝑥𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃−11−1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃,𝜃,𝜆,𝑥>0, (1.4) here 𝜃, and 𝜆 are the shape and scale parameters, respectively. The two-parameter EG distribution will be denoted by EG(𝜃,𝜆). For details, see Bakoban [2], Coronel-Brizio et al. [3] and Shawky and Bakoban [4–10].

Computation of Fisher information for any particular distribution is quite important, see, for example, Zheng [11]. Fisher information matrix can be used to compute the asymptotic variances of the different functions of the estimators, for example, maximum likelihood estimators (MLEs). The problem is quite important when the data are censored. We compute Fisher information matrices of EG distribution for complete and censored samples. We then study the properties of the MLEs of EG distribution under complete and censored samples in great details. Also, we consider different estimators of EG distribution and study how the estimators of the different unknown parameter(s) behave for different sample sizes and for different parameter values. We mainly compare the MLEs, estimators based on percentiles (PCEs), least squares estimators (LSEs), weighted least squares estimators (WLSEs), method of moment estimators (MMEs) and the estimators based on the linear combinations of order statistics (LMEs) by using extensive simulation techniques. It is important to mention that many authors interested with estimating parameters of such distributions, for example, Wingo [12] derived MLEs of Burr XII distribution parameters under type II censoring. Estimations based on order statistics under complete and censored samples compared with MLE which in turn based on a complete sample were studied by Raqab [13] for the Burr X distribution. Also, Gupta and Kundu [14] presented the properties of MLE’s for generalized exponential (GE) distribution, and they discussed other methods for GE in [15]. Kundu and Raqab [16] discussed the generalized Rayleigh distribution too. Hossain and Zimmer [17] compared several methods for estimating Weibull parameters with complete and censored samples. Surles and Padgett [18] considered the MLEs and discussed the asymptotic properties of these estimators for complete and censored samples from Burr X distribution.

The rest of the paper is organized as follows. In Section 2, we obtain the Fisher information matrices of EG distribution. In Section 3, we derive MLEs of EG distribution and study its properties. In Sections 4 to 7, we describe other methods of estimations. Simulation results and discussions are provided in Section 8.

2. Fisher Information Matrix

2.1. Fisher Information Matrix for Complete Sample

Let 𝑋 be a continuous random variable with the cumulative distribution function (c.d.f.) 𝐹(𝑥;Ω) and the probability density function (p.d.f.) 𝑓(𝑥;Ω). For the simplicity, we consider only two parameters 𝜃 and 𝜆, although the results are true for any finite-dimensional vector. Under the standard regularity conditions (Gupta and Kundu [19]), the Fisher information 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 is𝐈()=1𝜃,𝜆𝑛𝐸−𝜕²ln𝐿(𝜃,𝜆)𝜕𝜃²𝐸−𝜕²ln𝐿(𝜃,𝜆)𝐸−𝜕𝜕𝜃𝜕𝜆₂ln𝐿(𝜃,𝜆)𝐸−𝜕𝜕𝜆𝜕𝜃²ln𝐿(𝜃,𝜆)𝜕𝜆²=𝑎₁₁𝑎₁₂𝑎₂₁𝑎₂₂, (2.1) where for 𝑖,𝑗=1,2,𝑎_𝑖𝑗=^∞_−∞𝜕𝜕Ω_𝑖()𝜕ln𝑓𝑥,𝛀𝜕Ω_𝑗ln𝑓(𝑥,𝛀)𝑓(𝑥,𝛀)𝑑𝑥. (2.2) Now, we will derive Fisher information matrix of EG(𝜃,𝜆) under complete sample. It can be shown thatln𝑓(𝑥;𝜃,𝜆)=ln𝜃+2ln𝜆−𝜆𝑥+ln𝑥+(𝜃−1)ln1−𝑒^−𝜆𝑥(𝜆𝑥+1). (2.3) Differentiating (2.3) with respect to 𝜃 and 𝜆 respectively, we have𝜕ln𝑓(𝑥;𝜃,𝜆)=1𝜕𝜃𝜃+ln1−𝑒^−𝜆𝑥(,()𝜆𝑥+1)𝜕ln𝑓𝑥;𝜃,𝜆=2𝜕𝜆𝜆−𝑥+(𝜃−1)𝜆𝑥²𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)⁻¹. (2.4) Therefore, the second derivatives are𝜕²ln𝑓(𝑥;𝜃,𝜆)𝜕𝜃²1=−𝜃²,𝜕²ln𝑓(𝑥;𝜃,𝜆)𝜕𝜆²2=−𝜆²−(𝜃−1)𝜆𝑥³𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)⁻¹+(𝜃−1)𝑥²𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)⁻¹−(𝜃−1)𝜆²𝑥⁴𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)⁻²,𝜕²ln𝑓(𝑥;𝜃,𝜆)𝜕𝜆𝜕𝜃=𝜆𝑥²𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)⁻¹. (2.5) Thus, the elements of Fisher information matrix for single observation from EG(𝜃,𝜆) are in the forms:𝑎₁₁−𝜕=𝐸²ln𝑓(𝑥;𝜃,𝜆)𝜕𝜃²=1𝜃², (2.6)𝑎₁₂=𝑎₂₁−𝜕=𝐸²ln𝑓(𝑥;𝜃,𝜆)=𝜕𝜃𝜕𝜆2−𝜃2+𝐴₁(𝜃)𝜆(𝜃−1), (2.7)𝑎₂₂−𝜕=𝐸²ln𝑓(𝑥;𝜃,𝜆)𝜕𝜆²=𝜃²𝐴₂(𝜃)−(4+2/𝜃)𝐴₁(𝜃)−2𝜆²(𝜃−2), (2.8) where𝐴_𝑟(𝜃)=^∞_𝑗=1^𝑗_𝑘=0(−1)^𝑗𝑗𝑗𝑘𝜃−1Γ(𝑟+𝑘+2)1+𝑗^𝑟+𝑘+2,𝑟=1,2,…. (2.9) Moreover, Fisher information matrix for a complete sample of size 𝑛 from EG(𝜃,𝜆) is simply 𝑛𝐈(𝜃,𝜆).

2.2. Fisher Information Matrix under Type II Censoring

Let 𝑋₁,𝑋₂,…,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 𝐼_[0,𝑝](𝛀)=lim_𝑛→∞1𝑛𝐼_{1⋯[𝑛𝑝]∶𝑛}(𝛀). (2.10)

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 as𝐼_[0,𝑝](𝛀)=^𝑣𝑝_−∞𝜕(𝜕𝜕Ωlnℎ𝑥;𝛀)(𝜕Ωlnℎ𝑥;𝛀)^𝑇𝑓(𝑥;𝛀)𝑑𝑥, (2.11) 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).

In the following, we use (2.11) to obtain Fisher information matrix under Type II censoring for the EG(𝜃,𝜆) distribution. For Ω=(𝜃,𝜆), denote Fisher information matrix 𝐼_[0,𝑝](Ω) as𝐼_[0,𝑝](𝛀)=𝐼_[0,𝑝](𝜃)𝐼_[0,𝑝]()𝐼𝜃,𝜆_[0,𝑝](𝜃,𝜆)𝐼_[0,𝑝](𝜆), (2.12) where 𝐼_[0,𝑝](𝜃),𝐼_[0,𝑝](𝜆), and 𝐼_[0,𝑝](𝜃,𝜆) can be obtained by (2.11).

It can be shown thatlnℎ(𝑥;𝜃,𝜆)=ln𝜃+2ln𝜆+ln𝑥−𝜆𝑥+(𝜃−1)ln1−𝑒^−𝜆𝑥(𝜆𝑥+1)−ln1−1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃. (2.13) Differentiating (2.13) with respect to 𝜃 and 𝜆, respectively, we have𝜕lnℎ(𝑥;𝜃,𝜆)=1𝜕𝜃𝜃+ln1−𝑒^−𝜆𝑥(𝜆𝑥+1)1−1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃,𝜕lnℎ(𝑥;𝜃,𝜆)=2𝜕𝜆𝜆−𝑥1−𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥()+𝜆𝑥+1𝜃𝜆𝑥²𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)1−1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃. (2.14) Thus, it is easily to see, for 𝑝=[1−𝑒^{−𝜆𝑣𝑝}(𝜆𝑣_𝑝+1)]^𝜃, that the elements of 𝐼_[0,𝑝](Ω) from EG(𝜃,𝜆) are𝐼_[0,𝑝](𝜃)=𝜃𝜆^2𝑣𝑝₀𝜕(𝜕𝜃lnℎ𝑥;𝜃,𝜆)²𝑥𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃−1𝑑𝑥=𝜃^𝜆𝑣𝑝₀1𝜃+ln1−𝑒^−𝑦𝑦+11−1−𝑒^−𝑦𝑦+1^𝜃2𝑦𝑒^−𝑦1−𝑒^−𝑦𝑦+1^𝜃−1=1𝑑𝑦𝜃^2𝑝₀1+ln𝑥1−𝑥²1𝑑𝑥=𝜃²𝑝𝑝+1−𝑝(ln𝑝)²,𝐼_[0,𝑝](𝜆)=𝜃𝜆^2𝑣𝑝₀𝜕(𝜕𝜆lnℎ𝑥;𝜃,𝜆)²𝑥𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃−1=𝜃𝑑𝑥𝜆^{2𝜆𝑣𝑝}₀𝑦2−1−𝑒^−𝑦1−𝑒^−𝑦+𝑦+1𝜃𝑦²𝑒^−𝑦1−𝑒^−𝑦𝑦+11−1−𝑒^−𝑦𝑦+1^𝜃2×𝑦𝑒^−𝑦1−𝑒^−𝑦𝑦+1^𝜃−1𝐼𝑑𝑦,_[0,𝑝](𝜃,𝜆)=𝜃𝜆^2𝑣𝑝₀𝜕(𝜕𝜕𝜃lnℎ𝑥;𝜃,𝜆)(𝜕𝜆lnℎ𝑥;𝜃,𝜆)𝑥𝑒^−𝜆𝑥1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃−1=𝜃𝑑𝑥𝜆^𝜆𝑣𝑝₀𝑦2−1−𝑒^−𝑦1−𝑒^−𝑦+𝑦+1𝜃𝑦²𝑒^−𝑦1−𝑒^−𝑦𝑦+11−1−𝑒^−𝑦𝑦+1^𝜃×1𝜃+ln1−𝑒^−𝑦𝑦+11−1−𝑒^−𝑦𝑦+1^𝜃𝑦𝑒^−𝑦1−𝑒^−𝑦𝑦+1^𝜃−1𝑑𝑦. (2.15)(2.16)(2.17) It follows, by (2.15), that the percentage of Fisher information about 𝜃,𝐼_[0,𝑝](𝜃)/𝑎₁₁, is independent of 𝜃, and thus 𝐼_[0,𝑝](𝜃) is a decreasing function of 𝜃.

2.3. 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 thus𝐼_𝑅^(𝑐)()=𝑏𝑡,𝜃¹¹𝑏₁₂𝑏₂₁𝑏₂₂, (2.18) where, for 𝑖,𝑗=1,2,𝑏_𝑖𝑗=^𝑡₀𝜕𝜕Ω_𝑖()𝜕lnℎ𝑥,𝛀𝜕Ω_𝑗lnℎ(𝑥,𝛀)𝑓(𝑥,𝛀)𝑑𝑥. (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=[1−𝑒^−𝜆𝑡(𝜆𝑡+1)]^𝜃,𝑏₁₁=1𝜃²𝑝𝑝+𝑝𝑝1−ln²,𝑏₂₂=𝜃𝜆²₀^𝜆𝑡𝑦2−1−𝑒^−𝑦1−𝑒^−𝑦+𝑦+1𝜃𝑦²𝑒^−𝑦1−𝑒^−𝑦𝑦+1{1−1−𝑒^−𝑦𝑦+1^𝜃}²×𝑦𝑒^−𝑦1−𝑒^−𝑦𝑦+1^𝜃−1𝑏𝑑𝑦,₁₂=𝑏₂₁=𝜃𝜆₀^𝜆𝑡𝑦2−1−𝑒^−𝑦1−𝑒^−𝑦+𝑦+1𝜃𝑦²𝑒^−𝑦1−𝑒^−𝑦𝑦+11−1−𝑒^−𝑦𝑦+1^𝜃×1𝜃+ln1−𝑒^−𝑦𝑦+11−1−𝑒^−𝑦𝑦+1^𝜃𝑦𝑒^−𝑦1−𝑒^−𝑦𝑦+1^𝜃−1𝑑𝑦. (2.20)

3. Maximum Likelihood Estimators

3.1. 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 𝑥₁,𝑥₂,…,𝑥_𝑛 be a random sample of size 𝑛 from EG(𝜃,𝜆), then the log−likelihood function isln𝐿(𝜃,𝜆)=𝑛ln𝜃+2𝑛ln𝜆−𝜆^𝑛_𝑖=1𝑥_𝑖+^𝑛_𝑖=1ln𝑥_𝑖+(𝜃−1)^𝑛_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1. (3.1) The normal equations become𝜕ln𝐿(𝜃,𝜆)=𝑛𝜕𝜃𝜃+^𝑛_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1=0, (3.2)𝜕ln𝐿(𝜃,𝜆)=𝜕𝜆2𝑛𝜆−^𝑛_𝑖=1𝑥_𝑖+(𝜃−1)𝜆^𝑛_𝑖=11−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1⁻¹𝑥²_𝑖𝑒^{−𝜆𝑥𝑖}=0. (3.3) It follows, by (3.2), that the MLE of 𝜃 as a function of 𝜆, say 𝜃(𝜆), where𝜃(𝜆)=−𝑛∑^𝑛_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1. (3.4) Substituting 𝜃(𝜆) in (3.1), we obtain the profile log-likelihood of 𝜆 as𝑔(𝜆)=ln𝐿𝜃(𝜆),𝜆=𝑛ln𝑛−𝑛ln−^𝑛_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1−𝑛+2𝑛ln𝜆−𝜆^𝑛_𝑖=1𝑥_𝑖+^𝑛_𝑖=1ln𝑥_𝑖−^𝑛_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1. (3.5) Therefore, MLE of 𝜆, say 𝜆_MLE, can be obtained by maximizing (3.5) with respect to 𝜆 as follows:𝜕𝑔(𝜆)𝜕𝜆=−^𝑛_𝑖=1𝜆𝑥²_𝑖𝑒^{−𝜆𝑥𝑖}1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖++12𝑛𝜆−^𝑛_𝑖=1𝑥_𝑖−𝑛∑^𝑛_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1^𝑛𝑖=1𝜆𝑥²_𝑖𝑒^{−𝜆𝑥𝑖}1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1=0. (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:√𝑛𝜃_MLE−𝜃,√𝑛𝜆_MLE−𝜆⟶𝑁₂𝟎,𝐈⁻¹(𝜃,𝜆), (3.7) 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, is𝜃_MLESCK=−𝑛∑^𝑛_𝑖=1ln1−𝑒^−𝑥𝑖𝑥_𝑖+1. (3.8) It follows, by the asymptotic properties of the MLE, that𝜃_MLESCK1≈𝑁𝜃,𝑛𝑎₁₁, (3.9) where 𝑎₁₁ 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 −𝜃∑^𝑛_𝑖=1ln[1−𝑒^−𝑥𝑖(𝑥_𝑖+1)] follows 𝐺(𝑛,1). Therefore, for 𝑛>2,𝐸𝜃_MLESCK=𝑛𝜃𝑛−1𝜃,Var_MLESCK=𝑛²(𝑛−1)²()𝜃𝑛−2². (3.10) Using (3.8), an unbiased estimate of 𝜃 can be obtained by𝜃_USCK=𝑛−1𝑛𝜃_MLESCK=−𝑛−1∑^𝑛_𝑖=1ln1−𝑒^−𝑥𝑖𝑥_𝑖+1, (3.11) whereVar𝜃_USCK=𝜃²𝑛−2. (3.12) 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:2𝑛𝜆−^𝑛_𝑖=1𝑥_𝑖+(𝜃−1)𝜆^𝑛_𝑖=11−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1⁻¹𝑥²_𝑖𝑒^{−𝜆𝑥𝑖}=0. (3.13) It follows, by the asymptotic properties of the MLE, that𝜆_MLESHK1≈𝑁𝜆,𝑛𝑎₂₂, (3.14) where 𝑎₂₂ is the single information about 𝜆 which is defined in (2.8).

3.2. 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]) by𝐿(𝛀)=𝑛!(𝑛−𝑟)!^𝑟_𝑖=1𝑓𝑥_𝑖;𝛀1−𝐹𝑥_𝑟∶𝑛^𝑛−𝑟. (3.15) 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]) by𝐿(𝛀)=𝑛!(𝑛−𝑟)!^𝑟_𝑖=1𝑓𝑥_𝑖;𝛀[1−𝐹(𝑡)]^𝑛−𝑟. (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 isln𝐿(𝜃,𝜆)=ln𝑛!(𝑛−𝑟)!+𝑟ln𝜃+2𝑟ln𝜆−𝜆^𝑟_𝑖=1𝑥_𝑖+^𝑟_𝑖=1ln𝑥_𝑖+(𝜃−1)∑^𝑟_𝑖=1ln1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1+(𝑛−𝑟)ln1−1−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1^𝜃. (3.17) The normal equations become𝜕ln𝐿(𝜃,𝜆)=𝑟𝜕𝜃𝜃+ln𝑢−(𝑛−𝑟)𝑣^−𝜃−1⁻¹()ln𝑣=0,𝜕ln𝐿𝜃,𝜆=𝜕𝜆2𝑟𝜆−^𝑛_𝑖=1𝑥_𝑖+(𝜃−1)𝜆^𝑛_𝑖=11−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1⁻¹𝑥²_𝑖𝑒^{−𝜆𝑥𝑖}+(𝑛−𝑟)𝜃𝜆𝑥²_𝑟∶𝑛𝑒^{−𝜆𝑥𝑟∶𝑛}𝑣𝑣^−𝜃−1⁻¹=0, (3.18)(3.19) where𝑢=^𝑟_𝑖=11−𝑒^{−𝜆𝑥𝑖}𝜆𝑥_𝑖+1,𝑣=1−𝑒^{−𝜆𝑥𝑟}𝜆𝑥_𝑟+1. (3.20) The MLE of 𝜃 and 𝜆, say 𝜃_MLETIIand 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,√𝑛𝜃_MLETII−𝜃,√𝑛𝜆_MLETII−𝜆⟶𝑁𝟎,𝐼⁻¹_0,𝑝(𝛀), (3.21) 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, that𝜃_MLETIISCK1⟶𝑁𝜃,𝑛𝐼_[0,𝑝](𝜃), (3.22) 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, that𝜆_MLETIISHK1⟶𝑁𝜆,𝑛𝐼_[0,𝑝](𝜆), (3.23) where 𝐼_[0,𝑝](𝜆) is defined in (2.16).

4. 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. Murthy et al. [21] discussed 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. Since𝐹(𝑥;𝜃,𝜆)=1−𝑒^−𝜆𝑥(𝜆𝑥+1)^𝜃, (4.1) therefore,ln1−(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=1𝜆𝑥_𝑖∶𝑛−ln𝜆𝑥_𝑖∶𝑛+1+ln1−𝑝_𝑖^1/𝜃2, (4.3) 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:^𝑛_𝑖=1𝜆𝑥_𝑖∶𝑛−ln𝜆𝑥_𝑖∶𝑛+1+ln1−𝑝_𝑖^1/𝜃1−𝑝_𝑖^1/𝜃−1𝑝_𝑖^1/𝜃ln𝑝_𝑖𝜃²=0, (4.4)^𝑛_𝑖=1𝜆𝑥_𝑖∶𝑛−ln𝜆𝑥_𝑖∶𝑛+1+ln1−𝑝_𝑖^1/𝜃𝑥_𝑖∶𝑛−𝜆𝑥_𝑖∶𝑛+1⁻¹𝑥_𝑖∶𝑛=0. (4.5) 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), thenln𝐹(𝑥;𝜃)=𝜃ln1−𝑒^−𝑥(𝑥+1). (4.6) Therefore, the PCE of 𝜃, say 𝜃_PCESCK, can be obtained by minimizing^𝑛_𝑖=1ln𝑝_𝑖−𝜃ln1−𝑒^{−𝑥𝑖∶𝑛}𝑥_𝑖∶𝑛+1², (4.7) with respect to 𝜃 and hence𝜃_PCESCK=∑^𝑛_𝑖=1ln𝑝_𝑖ln1−𝑒^{−𝑥𝑖∶𝑛}𝑥_𝑖∶𝑛+1∑^𝑛_𝑖=1ln1−𝑒^{−𝑥𝑖∶𝑛}𝑥_𝑖∶𝑛+1². (4.8) Interestingly, 𝜃_PCESCK is also in a closed form like 𝜃_MLESCK when 𝜆 is known.

5. Least-Squares and Weighted Least-Squares Estimators

The least-squares estimators and weighted least-squares estimators were originally proposed by Swain et al. [23] to estimate the parameters of Beta distributions. It can be described as follows. Suppose 𝑌₁,𝑌₂,…,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 that𝐸𝐺𝑌_𝑗∶𝑛=𝑗𝐺𝑌𝑛+1,Var_𝑗∶𝑛=𝑗𝑛−𝑗+1(𝑛+1)²(𝑛+2). (5.1) 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 ^𝑛_𝑗=1𝐺𝑌_𝑗∶𝑛−𝑗𝑛+1², (5.2) with respect to the unknown parameters. Therefore, in case of EG distribution, the least-squares estimators of 𝜃 and 𝜆, say 𝜃_LSE and 𝜆_LSE, respectively, can be obtained by minimizing ^𝑛_𝑗=11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−𝑗𝑛+1², (5.3) with respect to 𝜃 and 𝜆.
It could be obtained by solving the following nonlinear equations: ^𝑛_𝑗=11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−𝑗𝑛+11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃ln1−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1=0, (5.4)^𝑛_𝑗=11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−𝑗𝑛+1𝜃𝜆𝑥²_𝑗∶𝑛𝑒^{−𝜆𝑥𝑗∶𝑛}1−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−1=0. (5.5)

Method 2. The weighted least-squares estimators of the unknown parameters can be obtained by minimizing _𝑛𝑗=11Var𝐺𝑌_𝑗∶𝑛𝐺𝑌_𝑗∶𝑛−𝑗𝑛+1², (5.6) 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 _𝑛𝑗=1(𝑛+1)²(𝑛+2)𝑗𝑛−𝑗+11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−𝑗𝑛+1², (5.7) with respect to 𝜃 and 𝜆.
It could be found by solving the following nonlinear equations _𝑛𝑗=1(𝑛+1)²(𝑛+2)𝑗𝑛−𝑗+11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−𝑗×𝑛+11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃ln1−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1=0,_𝑛𝑗=1(𝑛+1)²(𝑛+2)𝑗𝑛−𝑗+11−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−𝑗𝑛+1×𝜃𝜆𝑥²_𝑗∶𝑛𝑒^{−𝜆𝑥𝑗∶𝑛}1−𝑒^{−𝜆𝑥𝑗∶𝑛}𝜆𝑥_𝑗∶𝑛+1^𝜃−1=0. (5.8)(5.9)

6. 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(𝑋)=𝜃𝜇=𝐸𝜆2+𝐴₁(𝜃), (6.1)𝜎²(𝑋)=𝜃=𝑉𝜆²6+𝐴₂(𝜃)−𝜃2+𝐴₁(𝜃)², (6.2) where 𝐴₁(𝜃) and 𝐴₂(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.) as𝜎C.V.=𝜇=𝜃6+𝐴₂(𝜃)−𝜃²2+𝐴₁(𝜃)²𝜃2+𝐴₁(𝜃). (6.3) The C.V. is independent of the scale parameter 𝜆. Therefore, equating the sample C.V. with the population C.V., we obtain𝑆𝑋=𝜃6+𝐴₂(𝜃)−𝜃²2+𝐴₁(𝜃)²𝜃2+𝐴₁(𝜃), (6.4) where 𝑆²=(∑^𝑛_𝑖=1(𝑋_𝑖−𝑋)²)/(𝑛−1) and 𝑋=(1/𝑛)∑^𝑛_𝑖=1𝑋_𝑖. 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:𝑋=𝜇, (6.5) that is,𝑋=𝜃2+𝐴₁(𝜃). (6.6)

Now consider the case when the shape parameter 𝜃 is known, then the MME of 𝜆, say 𝜆_MMESHK, is𝜆_MMESHK=𝜃2+𝐴₁(𝜃)𝑋. (6.7) 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,𝜆Var𝜆_MMESHK=1𝑛6+𝐴₂(𝜃)𝜃2+𝐴₁(𝜃)²−1. (6.8)

7. 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 (see, Gupta and Kundu [15]) that the LMEs have certain advantages over the conventional moment estimators.

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 as𝑙₁=1𝑛^𝑛𝑖=1𝑥_𝑖∶𝑛,𝑙₂=2𝑛(𝑛−1)^𝑛𝑖=1(𝑖−1)𝑥_𝑖∶𝑛−𝑙₁. (7.1) Similarly, the first two population L-moments (see David and Nagaraja [24]) are𝜆₁=𝜇=𝐸(𝑋),𝜆₂=12𝜇_2∶2−𝜇_1∶2=^∞_−∞𝑥[2𝐹(𝑥)−1]𝑓(𝑥)𝑑𝑥, (7.2) respectively, where 𝜇_𝑖∶𝑛=𝐸(𝑋_𝑖∶𝑛).

Then, for EG(𝜃,𝜆), we obtain𝜆₁=𝜃𝜆2+𝐴₁(𝜃),𝜆₂=2𝜃𝜆2+𝐴₁(−𝜃2𝜃)𝜆2+𝐴₁(𝜃), (7.3) where 𝐴₁(𝜃) is defined by (2.9) and𝐴_𝑟(2𝜃)=^∞_𝑗=1^𝑗_𝑘=0(−1)^𝑗𝑗𝑗𝑘2𝜃−1Γ(𝑟+𝑘+2)1+𝑗^𝑟+𝑘+2,𝑟=1,2,…. (7.4) Therefore, LMEs can be obtained by solving the following two equations:𝑙₁=𝜃𝜆2+𝐴₁(𝜃), (7.5)𝑙₂=2𝜃𝜆2+𝐴₁(−𝜃2𝜃)𝜆2+𝐴₁(𝜃). (7.6) First, we obtain the LME of 𝜃, say 𝜃_LME, as the solution of the following nonlinear equation:2+2𝐴₁(2𝜃)−𝐴₁(𝜃)2+𝐴₁(𝜃)=𝑙₂𝑙₁. (7.7) Once 𝜃_LME is obtained, the LME of 𝜆, say 𝜆_LME, can be obtained from (7.5) as𝜆_LME=𝜃_LME2+𝐴₁𝜃_LME𝑙₁. (7.8) 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.

8. 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.

Table 1 

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

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).

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.

Table 2 

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

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.

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.

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.

Acknowledgment

The authors thank the Editor and the Referees for their helpful remarks that improved the original manuscript.