Improved maximum likelihood estimators for the parameters of the two parameter lindley distribution

Abstract

Keywords: Bootstrap Bias-Correction; Cox Snell Bias-Correction; Maximum Likelihood Estimators; Monte Carlo Simulation; Two Parameter Lindley Distribution

Introduction

The Lindley distribution with one parameter was proposed by Lindley [1] as an alternative to the exponential distribution in the context of fiducial distributions and the Bayes theorem. It is also known that the one-parameter Lindley distribution is a mixture of exponential (θ) and gamma (2, θ) distributions. Its probability density function (pdf) and cumulative distribution function (CDF) are given by (1)

The Lindley distribution, which was overshadowed by the exponential distribution for many years, has been studied by many authors as a lifetime model in recent years. Ghitany et al. [2] discussed various statistical properties such as moments, failure rate function, entropies, stochastic ordering, maximum likelihood (ML), and method of moments (MoM) estimations. Using a real data set, they also showed that the Lindley distribution can be a better model than the exponential distribution. Mazucheli

*Corresponding author. *E-mail address: hasan.huseyin@giresun.edu.tr This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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and Achcar [3] used the Lindley distribution within the simple competing risks distribution as a possible alternative to the exponential or Weibull distributions. Krishna and Kumar [4] derived and studied model properties and reliability measures of Lindley distribution. They used progressively Type-II censored sample data in the estimation process. Al-Mutairi et al. [5] studied the estimation of stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables. Some researchers have studied parameter estimation of Lindley distribution hybrid censored data. See, for example, Gupta and Singh [6], Al-Zahrani and Ali [7], and Jia and Song [8]. The pdf and cdf of the two-parameter Lindley (TPL) distribution proposed by Shanker et al. [9] are given by (3)

(4) At α = 1, the TPL distribution is the one-parameter Lindley distribution, and at α = 0, the TPL distribution is the exponential distribution. This article deals with deriving modified maximum likelihood estimators (MLEs) that are analytic second-order biases for the parameters of TPL distribution. The choice of estimation method for estimating the parameters of any probability distribution is a very important issue. Among the estimation methods, the MLE is the most widely used method due to its important appealing. For instance, they are asymptotically unbiased and normally distributed, efficient, consistent, etc. However, it is to be noted that most of these properties depend on the large sample size condition. Therefore, especially the unbiasedness property may not be applicable for small and moderate sample sizes. For this reason, it is important to develop nearly unbiased estimators for TPL distribution. In this article, two different methods are used to reduce the bias of the MLE from order 𝒪(n-1) to order 𝒪(n-2) for the TPL distribution. The first is the analytical methodology proposed by Cox and Snell [10] which is called the “corrective” approach to derive “bias-corrected” MLEs of second order. This analytical method means that bias correction is done by subtracting the bias (estimated at the MLE of the parameter) from the original MLE. The second is Efron’s [11] bootstrap resampling method which is called “parametric bootstrap”. This method, unlike the “bias-correction” method introduced by [10], is performed numerically without deriving an analytical expression for the bias function. In the literature, several authors have studied to develop nearly unbiased estimators for the parameters of several distributions. Readers may refer to Corderio et al. [12], Cribari-Neto and Vasconcellos [13], Saha and Paul [14], Lemonte et al. [15], Lemonte [16], Giles [17], Giles et al. [18], Ling and Giles [19], Giles et

al. [20], Schwartz and Giles [21], Wang and Wang [22], Reath et al. [23], Mazucheli and Dey [24], Mazuheli et al. [25], Mazucheli et al. [26], Menezes and Mazucheli [27], Menezes et al. [28], Tsai et al. [29], Dey and Wang [30]. The main objective of this study is to obtain almost unbiased estimators for the parameters of the TPL distribution. In the literature, bias-corrected estimators for the TPL distribution have not yet been investigated and no comprehensive study has been conducted. This paper addresses this issue using both analytical and simulation-based methods. For this purpose, two different methods are considered, one analytical method suggested by [10] and the other bootstrap-based bias-corrected. Then, a Monte-Carlo simulation study is performed to compare the performance of the proposed estimators with respect to the bias and mean square error (MSE) criteria. In addition, a real data application is presented to demonstrate the applicability of the methods. In the next sections 2 and 3, point estimation by the maximum likelihood method and its two different bias-corrected versions for the TPL distribution is discussed. Section 4 reports a Monte-Carlo simulation experiment that evaluates the performance among the improved modified Cox-Snell bias-corrected estimator (BCE) and bootstrap-based bias-corrected estimator (PBE). For illustrative purposes, a real data set is presented in Section 5. Finally, some remarks in Section 6 closes the paper. Maximum Likelihood Estimation Suppose that y = (y1,…,yn) be a random sample of size n from the TPL distribution with pdf (1). The log-likelihood function is: (5) where Θ = (θ, α). The MLEs of θ̂ and α̂ of the unknown parameters θ and α are obtained by solving the non-linear equations: (6)

(7) The expected Fisher information matrix is given in the Appendix. Equations (6) and (7) do not seem to be solved directly. Therefore, a suitable numerical algorithm must be used. The Nelder-Mead optimization method (Nelder and Mead, [31]) in MATLAB software is used to obtain the estimated parameters.

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Bias-Corrected MLES

Cox-Snell Method For a p-dimensional parameter vector Θ, [10] demonstrated that when the sample data are independent, but not necessarily identically distributed, the bias of the s-th element of the MLE Θ, Θ̂, is calculated as (8)

where s = 1,…, p, K = [-κij] is the (i,j)-th element of the inverse of the expected Fisher information,

The analytic expressions are given in the Appendix. The elements of A(1) are obtained as follows:

Eq. (8) can be rewritten in the following form: (9) Now, let define the following matrices,

The elements of A(2) are: They also showed that the 𝒪(n-1) the bias of the MLE of Θ in Eq. (9) can be re-expressed as:

where vec (.) is an operator that creates a column vector from a matrix by stacking the column vectors below one another. Then, a “bias-corrected” MLE for Θ can be obtained as: (11) and , and it can be shown that where the bias of will be 𝒪(n-2). For the TPL distribution, after extensive algebraic manipulation, see the Appendix, the following equations are obtained:

Using Corderio and Klein’s [32] modification of [10]’s method, the bias of Θ̂ can be written as follows: (12)

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Parametric Bootstrap Method An alternative approach considered to obtain modified nearly unbiased estimators for the TPL distribution is based on the parametric bootstrap resampling method ([11], Efron and Tibshirani [33], Davison and Hinkley [34]). In this method bias correction is performed numerically without deriving an analytical expression for the bias function. It uses the empirical distribution of the sample as an approximation of the population distribution. The estimated bias of Θ̂ s for a parameter vector Θ is given by (13)

where Θ̂ j is the MLE of Θ obtained from the j-th Bootstrap sample. Thus, the bootstrap bias-corrected estimator (PBE) is (14) Although it does not involve any analytical derivatives, it provides a second-order unbiased estimator, which is a very important and remarkable feature. For more details, see [33].

Numerical Evaluations

In this section, a Monte Carlo simulation study is carried out to compare the performance of the finite-sample behavior of the MLEs, BCEs, and PBEs of the TPL

distribution. The Monte Carlo experiments are evaluated by selecting sample sizes n=10, 30, and 50, θ=1.0, 1.5, 2.0, 3.0, and 5.0, α=0.5, 1.0, 1.5, and 3.0. The pseudo-random samples are simulated using the inverse transform method from TPL distribution, that is x = (x1,…, xn) is generated from:

where ui are random numbers from a uniform distribution and W(.) is the Lambert W function. The Monte-Carlo and Bootstrap replication numbers are taken as M=10,000 and B=1000, respectively. The simulation results are reported in Tables 1-4. It is observed from Table 1 that, the biases of αMLE, αBCE and αPBE is smaller than the biases of θMLE, θBCE and θPBE. In most cases, the biases and MSEs of all estimators of θ and α approach zero as n increases. This shows that all estimators are consistent. The bias-corrected estimators αBCE and θBCE clearly outperform the other estimators according to both the bias and MSE criteria. Similar results are observed in Tables 2-4. These Monte-Carlo simulation results reveal that the bias-corrected estimators perform favorably in bringing the estimates closer to their true values. In the case of α = 0.5, the plots for the θ and α parameters are given in Figures 1-4 for the estimated bias and MSE.

Table 1. Estimated bias (mean square errors) for θ and α, (α = 0.5) Estimator of θ θ 1.0

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Table 2. Estimated bias (mean square errors) for θ and α, (α = 1.0) Estimator of θ θ 1.0

Table 3. Estimated bias (mean square errors) for θ and α, (α = 1.5) Estimator of θ θ 1.0

Illustrative Example In this section, to demonstrate the applicability of the improved bias-corrected estimators for the TPL distribution, two different data sets are used. Data Set 1: The following data set represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test by Lawless [35]. The data are given in Table 5.

Descriptive statistics of the failure times of electronic components data are given in Table 6. Data Set 2: The following data set represents the number of cycles to failure for 25 100-cm specimens of yarn, tested at a particular strain level, by [35]. Both data sets were recently used as an illustrative example for the TPL distribution by [9].

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Table 4. Estimated bias (mean square errors) for θ and α, (α = 3) Estimator of θ

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Table 6. Summary of the descriptive statistics for the failure times of electronic components data Mean

Table 7. Number of cycles to failure for 25 100-cm specimens of yarn 15

Table 8. Summary of the descriptive statistics for the Number of cycles to failure for 25 100-cm specimens of yarn data. Mean

Table 9. MLEs, BCEs, and PBEs (Bootstrap standard errors) Data Set 1

The point estimates of θ and α along with standard errors with MLEs, BCEs, and PBEs of the TPL distribution for both data sets are presented in Table 9. In comparing the three considered estimation methods for the TPL distribution, the smallest standard error is taken

as a criterion. It is observed that bias-corrected estimates have smaller standard errors for both parameters θ and α for both data sets. Moreover, the BCEs and PBEs gave similar bootstrap standard errors. In addition, Table 9 shows that the MLEs overestimates θ and α for both data sets.

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Conclusion

In the literature, there are no studies on the bias reduction of MLE estimates of the parameters of the two-parameter Lindley distribution. This gap in the literature is very important for improving the parameter estimation of this distribution. In this paper, a “corrective” approach to derive analytical expressions for bias-corrected maximum likelihood estimator suggested by [10] for the parameters of the two-parameter Lindley distribution has been adopted. Besides, Efron’s bootstrap resampling technique, which is not an analytical method, is considered an alternative bias-correction technique. In addition, to demonstrate the applicability of the above-mentioned techniques, a real data application is performed. The results show that the proposed bias-corrected estimators are preferred in terms of bias and mean square error for the two-parameter Lindley distribution over the maximum likelihood and bootstrap bias-corrected estimators.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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