UNIVERSITY OF CALGARY. A New Hybrid Estimation Method for the. Generalized Exponential Distribution. Shan Zhu A THESIS

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1 UNIVERSITY OF CALGARY A New Hybrid Estimation Method for the Generalized Exponential Distribution by Shan Zhu A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS CALGARY, ALBERTA June, 2013 c Shan Zhu 2013

2 Abstract The generalized exponential distribution (GED) is a popular distribution for analyzing lifetime data and has been introduced and studied quite extensively. It can be used as an alternative to gamma or Weibull distribution in many situations. As one of the most important research areas of GED, the estimation of the parameters has been discussed by many authors. Among the existing methods, the maximum likelihood method and Bayesian method are the methods that many authors recommend to use since they can provide balanced and good performances for different sample sizes and parameter values. In order to improve the estimation in terms of bias and mean squared error (MSE), and to simplify the computation for the Bayesian method, in this thesis, we introduce a new hybrid estimation method for the GED, which is based on the idea of minimizing a goodness-of-fit measure and incorporating useful maximum likelihood information. Through simulation, we compare the new hybrid method with the MLE method and the Bayesian methods as well as other existing methods, and show that this new hybrid method can not only reduce the estimation bias but also improve the MSE especially for the small sample case. Key Words: Maximum likelihood estimators; Bayesian estimators; Method of moments; Least squares and weighted least squares estimators, Percentile estimators; L-moment estimators; EDF statistics; Estimation bias and mean squared error. ii

3 Acknowledgements I wish to express my sincere gratitude to my supervisor Dr. Gemai Chen for the constant support and encouragement over the past two years. The excellent research methods and correct learning attitude that you teach me will benefit me the whole life. I feel so lucky to be one of your students. Also, I would like to thanks all the statistics faculty members at the department for the full help in my study and life. You make me better integrate into the statistical world. Especially, I benefited a lot from the courses I took by Dr. John Collins, Dr. Xuewen Lu, Dr. Murray Burke and Dr. Jingjing Wu. Finally, I am grateful to all my friends for the insightful suggestions from you. There are still so many people that I am grateful to. I will never forget the two years of life. iii

4 Table of Contents Abstract ii Acknowledgements iii Table of Contents iv List of Tables v List of Figures vi 1 Introduction Generalized Exponential Distribution Framework of Thesis Estimation Methods for GED Maximum Likelihood Estimation Method of Moments Estimation based on Percentiles Least Squares and Weighted Least Squares Estimations L-Moment Estimation Bayesian Estimation A New Hybrid Estimation Empirical Distribution Function Statistics Computation of the New Hybrid Estimates Simulation Study Comparisons of Finite-Sample Performances An Example Final Conclusions and Future Research A Computation for the Lindley s Approximation B An R Function for Computing the GED Estimates Bibliography iv

5 List of Tables 3.1 The average relative mean squared error of α The average relative mean squared error of λ The estimated GED parameters for the lifetime of deep groove ball bearings 32 v

6 List of Figures and Illustrations 2.1 Bias and MSE for estimating GED parameters α and λ using the MGF estiamtors with replications with n = 50 and λ = Average relative bias comparisons for estimating the GED parameters α and λ with replications with sample size n = 15 and λ = Average relative bias comparisons for estimating the GED parameters α and λ with replications with sample size n = 50 and λ = Average relative bias comparisons for estimating the GED parameters α and λ with replications with sample size n = 100 and λ = Average relative mean squared error comparisons for estimating the GED parameters α and λ with replications with sample size n = 20 and λ = Average relative mean squared error comparisons for estimating the GED parameters α and λ with replications with sample size n = 100 and λ = The estimated GED cumulative distribution functions using different methods (MLE, MME, PCE, LME) versus the EDF for the deep groove ball bearings The estimated GED cumulative distribution functions using different methods (HYB, BAYE, LSE, WLSE) versus the EDF for the deep groove ball bearings. 34 vi

7 Chapter 1 Introduction In the mid-20th century, the two-parameter (scale parameter and shape parameter) gamma and Weibull distributions are the most popular distributions for analyzing lifetime data. Both distributions are flexible for analyzing any positive real data sets. However, both of them also have drawbacks. For example, the gamma distribution function or its survival function cannot be expressed in a closed form if the parameter is not an integer; and the maximum likelihood estimators of the Weibull distribution may not behave properly for all parameter values, as indicated by Bain and Engelhardt (1978). Then, a new distribution named generalized exponential distribution (GED) was introduced and was extensively studied by Gupta and Kundu (1999, 2001), Raqab and Ahsanullah (2001), which can be used as an alternative to the gamma and Weibull distributions in situations where a skewed distribution for a non-negative random variable is needed. 1.1 Generalized Exponential Distribution The Generalized Exponential Distribution (GED) is a two-parameter distribution which was introduced by Gupta and Kundu (1999) with the distribution function (cdf) F (x; α, λ) = (1 e λx ) α ; α, λ, x > 0, (1.1) and corresponding density function (pdf) f(x; α, λ) = αλ(1 e λx ) α 1 e λx, (1.2) and survival function S(x; α, λ) = 1 (1 e λx ) α. (1.3) 1

8 Here α is the shape parameter and λ is the scale parameter. A GED with a shape parameter α and a scale parameter λ will be denoted by GED(α, λ). The density functions of the generalized exponential distribution can have different shapes. For α 1, it is a decreasing function and for α > 1, the pdf is unimodal, skewed, right tailed and similar to the Weibull or gamma density function. Also it is observed that when the shape parameter is very large, it is not symmetric. For λ = 1, the mode is at log α for α > 1, and is at 0 for α 1. The median is ln(1 (0.5) 1 α ). The mean, median and mode are all non-linear functions of α. When α goes to infinity, all of them also go to infinity. The hazard function of the generalized exponential distribution is given by h(x; α, λ) = f(x; α, λ) 1 F (x; α, λ) = αλ(1 e λx ) α 1 e λx 1 (1 e λx ) α. (1.4) The shape of the hazard function does not depend on λ since it is the scale parameter. When λ is fixed, the generalized exponential distribution has an increasing hazard function for α > 1 which increases from 0 to λ; and a decreasing hazard function for α < 1, which decreases from to λ, same as the gamma distribution. For α = 1, the hazard function is constant. The two-parameter GED is a particular member of the three-parameter exponentiated Weibull distribution, introduced by Mudholkar and Srivastava (1993) and it can be used quite effectively to analyze positive lifetime data. When the shape parameter α = 1, it becomes the one-parameter exponential distribution. Therefore, the generalized exponential, Weibull and gamma distributions are all extensions of the one-parameter exponential distribution in different ways. 1.2 Framework of Thesis The main aim of this thesis is to compare the different estimation methods of the unknown parameters of the GED for different sample sizes and different parameter values. We will first review the maximum likelihood (ML) method, the method of moments (MM), the method 2

9 based on percentiles (PC), the least square (LS) and weight least square (WLS) method, the method based on the linear combinations of order statistics (LM), and the Bayesian method (Bay). Then a new hybrid estimation method (HYB) for the GED will be studied at the end of Chapter 2. Chapter 3 contains the simulation studies. We will compare all of these estimation methods with the new hybrid estimation method with respect to their bias and mean square errors (MSE). In Chapter 4, we will analyze a deep groove ball bearings data set by using different estimation methods. Finally in Chapter 5, the final conclusion and some advantages of the new hybrid estimations will be given, and the future work will be summarized. 3

10 Chapter 2 Estimation Methods for GED In this chapter, we will consider eight different estimation methods. Most of them have already been studied by Gupta and Kundu (1999). Among these methods, the MLE and the Bayesian are the most popular methods which received great attention in the literature. In order to make improvement over the existing estimation methods, we will introduce a new hybrid method at the end of this chapter, which is a combination of the maximum likelihood method and the maximum goodness-of-fit method. This method had been used for generalized pareto distribution by Chunlin Wang (2011), and it is the first time that it is used for the generalized exponential distribution. 2.1 Maximum Likelihood Estimation The maximum likelihood method is one of the most well-known and classical methods of estimation. For the generalized exponential distribution, the MLE is not very difficult to get. Given that x 1,...x n is a random sample from GE(α, λ) with the cdf given in (1.1), the log-likelihood function, l(α, λ), is where α, λ > 0. l(α, λ) = n ln(α) + n ln(λ) + (α 1) ln(1 e λx i ) λ x i, (2.1) Consider the first derivatives of the GED log-likelihood with respect to α and λ and set them to be zero, we get the following likelihood equations: l α = n n α + ln(1 e λx i ) = 0, (2.2) l λ = n n λ + (α 1) x i e λx i x (1 e λx i = 0. (2.3) i ) 4

11 From (2.2), it is very easy to obtain the MLE of α as a function of λ n ˆα(λ) =. (2.4) ln(1 e λx i ) Substituting α in (2.1) with ˆα(λ), we have the profile log-likelihood function only with λ given by g(λ) = C n ln ( ln(1 e λx i )) + n ln(λ) ln(1 e λx i ) λ x i, (2.5) here C is a constant independent of λ. Therefore, by maximizing (2.5) with respect to λ we can obtain the MLE of λ, say ˆλ MLE, and then to get the MLE of α from (2.4). To maximize (2.5) and obtain ˆλ MLE, Gupta and Kundu (1999b) gives a method. They observed that L(ˆα(λ), λ) is an unimodal function for all x 1,, x n > 0 and then they differentiate g(λ) with respect to λ and equate it 0, g (λ) = n x i e λx i (1 e λx i ) + n ln(1 e λx i λ n ) therefore, ˆλ is the fixed point solution of h(λ) = λ, where h(λ) = x i e λx i (1 e λx i ) + 1 ln(1 e λx i n ) x i e λx i (1 e λx i ) x i e λx i (1 e λx i ) + x i = 0, x i Also, Gupta and Kundu (1999b) provided the consistency and the asymptotic normality results of MLE of the GE parameters. Consider the expectation of the second derivative of l(α, λ), they got that for α > 0, the maximum likelihood estimators (ˆα, ˆλ) are consistent and n(ˆα α, ˆλ λ) is asymptotically normal with mean vector 0 and dispersion matrix I 1, 1. where I = E( 2 l α 2 ) E( 2 l λ α ) E( 2 l α λ ) E( 2 l λ 2 ). 5

12 The elements for the above Fisher Information matrix are as follows: For α > 2, ( 2 ) l E = n α 2 α, 2 E ( 2 ) ( l 2 ) l = E = n α λ λ α λ [ α (ψ(α) ψ(1)) (ψ(α + 1) ψ(1))], α 1 ( 2 ) l E λ 2 = n α(α 1) [1 + λ2 α 2 (ψ (1) ψ (α 1) +(ψ(α + 1) ψ(1)) 2 )]. For 0 < α 2, ( 2 ) l E = n α 2 α 2 E ( 2 ) ( l 2 ) l = E = nα α λ λ α λ 0 xe 2x (1 e x ) α2 dx <, E ( 2 ) l λ 2 = n nα(α 1) x 2 e 2x (1 e x ) α 3 dx <, λ2 λ 2 0 where ψ( ) denotes the digamma function and ψ ( ) denotes the derivative of ψ( ). 2.2 Method of Moments The method of moments is an estimation method that by equating the sample moments to the population moments and then solving the obtained equations to get the estimators. Firstly, we need to find the population mean and variance of GED. Gupta and Kundu (1999), see also Narisi (2010) derived the moment generating function M(t) of the GED(α, λ) as M(t) = Ee tx = Γ(α + 1)Γ(1 t λ ) Γ(α t λ + 1), t < λ, 6

13 where Γ( ) is the gamma function Γ(z) = 0 t z 1 e t dt. Differentiating log M(t) and evaluating at t = 0, we get the mean and variance of GED as µ = E(X) = 1 (ψ(α + 1) ψ(1)), (2.6) λ σ 2 = V (X) = 1 λ 2 (ψ (α + 1) ψ (1)). (2.7) For a fixed λ, the mean of a generalized exponential distribution is increasing to and the variance increases to π2 6λ variation (C.V.) as as α increases. From (2.6) and (2.7), we obtain the coefficient of σ µ = C.V. = ψ (1) ψ (α + 1) ψ(α + 1) ψ(1), which is independent of the scale parameter λ. Therefore, we can equate the sample C.V. with the population C.V and obtain the following equation: 1 S (xi X = x) n 1 2 = xi 1 n ψ (1) ψ (α + 1) ψ(α + 1) ψ(1). (2.8) Using statistical software R, we can solve this equation (2.8) to obtain the MME of α, say ˆα MME, and then use (2.6) to get the MME of λ, which is ˆλ = ψ(ˆα+1) ψ(1). Also, Gupta and X Kundu (2000) provided a iterative method to solve the equation (2.8). The MME of α and λ have the following asymptotic property: where and C = [ n(ˆα MME α), n(ˆλ MME λ)] N 2 [0, DA 1 CA 1 D], A = D = λ, ψ (α + 1) ψ (α + 1) ψ(α + 1) ψ(1) ψ (1) ψ (α + 1) 2( ψ (α + 1) + ψ (1)) ψ (α + 1) ψ (1) ψ (α + 1) ψ (1) ψ 3 ψ 3 (α + 1) (ψ (1) ψ (α + 1)) 2 Here ψ ( ) and ψ 3 are the second and the third derivative of the digamma function. 7.

14 2.3 Estimation based on Percentiles As we mentioned before, the generalized exponential distribution is quite similar to the Weibull and the gamma distributions. Therefore, some estimation methods are interchangeable for these distributions. The graphical approximation to the best linear unbiased estimator is one of the most easily obtained estimators of the Weibull distribution. The main idea is fitting a straight linear to the theoretical points obtained from the distribution function by minimizing the sum of squared distance between observed data and the line, and making it have the lowest variance. This method was originally introduced by Kao (1958,1959), see also Mann, Schafer and Singpurwalla (1974). Johnson, Kotz and Balakrishnan (1994) used this method to estimate the three-parameter Weibull distribution with distribution F (x) = 1 exp[ ( x ξ α )c ]. Then, they found the estimators of ξ, α and c by solving the equation x ps = ξ + α[ log(1 p s )] 1/c. Here c is the shape parameter and ξ can be assumed to be 0. For the Generalized exponential distribution, this method also can be used since this distribution has the similar distribution structure as the Weibull distribution. We can use the same concept to find the estimators of α and λ based on the percentiles. The cdf of GED is F (x; α, λ) = (1 e λx ) α, therefore, the equation we need can be written as x = 1 λ ln(1 [(1 e λx ) α ] 1/α ). Suppose that p i denotes some estimate of F (x, α, λ) and x i are the corresponding estimators of x, then the estimate of α and λ can be obtained by minimizing [xi + λ 1 ln(1 p 1/α i )] 2 (2.9) with respect to α and λ. (2.9) is a non-linear function of α and λ, but using R, it is easy to obtain the estimators of α and λ. We call them percentile estimators (PCE), ˆα P CE and ˆλ P CE. 8

15 Gupta and Kundu (2001) mentioned that approximating the true least squares estimators as tacitly and incorrectly assuming that the covariance matrix of the vector of order statistics is some constant times the identity matrix, which is not a correct assumption. p i, as some estimators of F (x i ), can have many different expressions. Mann, Schafer, and Singpurwalla (1974) find p i = (i 1 )/n can do the best in terms of mean-squared error for the Weibull 2 distribution. The convention based on p i equal to the median rank of the ıth reduced order statistic is used extensively. Another choice for p i = (i 3)/(n + 1 ) is also good. Here we 8 4 follow Gupta and Kundu to use p i = i/(n 1), possibly because i/(n 1) is the expect value of F (x i ). 2.4 Least Squares and Weighted Least Squares Estimations In this section we study the regression based estimators of the unknown parameters. This method was originally introduced by Swain, Venkatraman and Wilson (1988). They described a least-squares procedure that applies to any distribution fitting problem involving a continuous target distribution function F x ( ) from which we have taken a random sample x j : 1 j n with the corresponding order statistics x (j), 1 j n. Let x (1) x (2)... x (n) denote the order statistics based on a random sample of size n from F x ( ). Then for 1 j n, the transformed variate R j = F x (x (j) ) has the distribution of the ith uniform order statistic, where R j has mean ρ j and the covariance between R (j) and R (k) is ρ j(1 ρ k ) n+2. In 1995, Johnson, Kotz and Balakrishnan also used this method in their book. Suppose Y 1,... Y n is a random sample from a distribution function G( ), and Y (j), j = 1,... n is the ordered sample. We have the following equations: E(G(Y (j) )) = i n + 1, V (G(Y (j))) = j(n j + 1) (n + 1) 2 (n + 2) and Cov(G(Y (j) ), G(y (k) )) = j(n k + 1) ; for j < k. (n + 1) 2 (n + 2) 9

16 Using the expectations and variances, two variants of the least-squares method are used ( in minimizing ω j G(Y (j) ) j ) 2 with respect to the parameters, where ω j are some n + 1 weights. Case 1 (Least Squares estimators) When ω j = 1 for all j, we can obtain the estimators by minimizing ( G(Y (j) ) j ) 2 n + 1 with respect to the unknown parameters. For the generalized exponential distribution, we can obtain the least squares estimators of α and λ, say ˆα LSE, ˆλ LSE by minimizing with respect to α and λ. Case 2 (Weighted Least Squares estimators) ( (1 e λx (j) ) α j ) 2 (2.10) n + 1 Same as Case 1, the weighted least squares estimators can be obtained by minimizing ( ω j G(Y (j) ) j ) 2 n + 1 with respect to the unknown parameters, with ω j = (1/V (G(Y j ))) = ((n+1) 2 (n+2))/(j(n j+1)). For the generalized exponential distribution, we can obtain the weighted least squares estimators of α and λ, say ˆα W LSE, ˆλ W LSE, by minimizing with respect to α and λ. ( ω j (1 e λx (j) ) α j ) 2 (2.11) n L-Moment Estimation L-moment estimation (LME) uses expectations of certain linear combinations of order statistics, see David (1981) and Hosking (1990). L-moment estimators are analogous to the conventional moments estimators through the use of linear combinations of order statistics, 10

17 namely, L-statistics. Compared with the conventional moments, the theoretical advantages of the L-moments are that they are less affected by the effects of sampling variability and they can characterize a wider range of distributions. In particular, L-moments are more robust than the conventional moments to outliers in the data and they can enable more secured inferences to be made from small samples about an underlying probability distribution. Also, L-moments are less subject to bias in estimation and the estimators obtained from the L-moments are sometimes more accurate than even the maximum likelihood estimates in small samples. To estimate the GED parameters, Hosking (1990) let x 1, x 2,, x n be the sample and x 1:n x 2:n x n:n the ordered sample, and define the rth sample L-moment to be l r = ( ( ) ) n 1 r 1 r 1 i 1 <i 2 <,,<i r n r 1 r 1 ( 1) k x ir k :n, for r = 1,, n. Using this k=0 k definition, Gupta and Kundu (2001) obtain the first and second sample L-moments as l 1 = 1 x (i), l 2 = n 2 n(n 1) (i 1)x (i) l 1 and the first two population L-moments, which are from the distribution function of the ith order statistic of the GED random variable, as λ 1 = 1 λ [ψ(α + 1) ψ(1)], λ 2 = 1 [ψ(2α + 1) ψ(α + 1)]. λ Same as the method of moments, we equate the sample L-moments with the population L-Moments, then we obtain the following two equations l 1 = 1 n x (i) = 1 [ψ(α + 1) ψ(1)], (2.12) λ l 2 = 2 n(n 1) (i 1)x (i) l 1 = 1 [ψ(2α + 1) ψ(α + 1)]. (2.13) λ Now we can obtain the LME of α, say ˆα LME, by solving the nonlinear equation l 2 l 1 = ψ(2α + 1) ψ(α + 1). (2.14) ψ(α + 1) ψ(1) 11

18 Then the LME of λ can be obtained from (2.12), as ˆλ LME = ψ(ˆα LME+1) ψ(1) l 1. Compared with the asymptotically optimal method of maximum likelihood, the method of L-moments usually needs less frequent recourse to iterative procedures. The asymptotic standard errors of the L-moment estimators are also usually reasonably more efficient. 2.6 Bayesian Estimation Bayesian estimation of the GPD parameters is also one of the most important estimation methods and has been studied by several authors. Jaheen (2004) considered the empirical Bayesian estimate of the shape parameter when the scale parameter is known, to be 1; Raqab and Madi (2005, 2009) considered the Bayesian estimation based on progressively censored data; Gupta and Kundu (2008) used the idea of Lindley and the Markov Chain Monte Carlo method to compute the approximate Bayesian estimators of the unknown parameters; Kim and Song (2010) considered doubly censored samples; and Kumar, Hakkak and Khan (2012) estimated α and λ by using Markov Chain Monte Carlo method for informative priors. Two of the above papers are related to the topic of this thesis. First, Raqab and Madi (2005) considered two situations where both the parameters are unknown: one is that the items are kept under observation until they all fail; the second, the experiment is terminated at the rth failure. They assumed the prior of the parameters α and λ follows the gamma distribution with means a 0 /b 0 and a 1 /b 1. By combining the prior and the MLE function, they obtained the joint posterior density of α and λ. Next, they got the marginal posterior density of α and λ. Finally, they obtained the expression of the bayesian estimators by using the marginal and the fact E(α λ, x) = (a 0 + n)/(d λ + b 0 ), where D λ = ln(1 e λx i ). The final result is: E(λ x) = E [λw (λ)] E [W (λ)], 12

19 E(α x) = E [W (λ)(a 0 + n)/(d λ + b 0 )], E [W (λ)] where E denote the expectation with respect to the gamma distribution G(a 1 + n, b 1 + n x) and W (λ) = (D λ + b 0 ) (a 0+n). This is just a brief description of this method without the simulation here, because Gupta and Kundu (2008) used the idea of Lindley to compute the approximate Bayesian estimators of the unknown parameters and had observed that Lindley s method can be extended to a more general class of distributions, for example, proportional reversed hazard models or for exponentiated Weibull distribution. Since both α and λ are non-negative, as in Raqab and Madi (2005), Gupta and Kundu (2008) also assumed the gamma priors on α and λ, although they were not the conjugate priors. In many practical situations, the information about the shape and scale of the sampling distribution is available in an independent manner, see Basu, Basu and Mukhopadhyay (1999), where the authors assumed that the parameters α and λ are independent a priori. Here we mainly consider the squared error loss function. It is observed that the Bayesian estimators can not be expressed in explicit forms and they can be only obtained by two dimensional numerical integrations. For computing the Bayesian estimators of α and λ, we follow Gupta and Kundu (2008), assuming that α and λ have the following gamma prior distributions: π 1 (λ) λ b 1 e aλ ; λ > 0, π 2 (α) α d 1 e cα ; α > 0. Here all the hyper parameters a, b, c, d are assumed to be known and non-negative. If x 1, x 2,, x n are random sample from GED(α, λ), the likelihood function is L(x i α, λ) = α n λ n e λ n x i n (1 e λx i ) α 1. Based on the likelihood of the observed data, the joint posterior density function of α and λ 13

20 can be written as: l(α, λ data) = 0 L(data α, λ)π 1 (λ)π 2 (α) 0 L(data α, λ)π 1 (λ)π 2 (α)dαdλ. Therefore, the Bayesian estimator of any function of α and λ say g(α, λ) under the squared error loss function is g(ˆα, ˆλ) = g(α, λ)l(data α, λ)π 1 (λ)π 2 (α)dαdλ. L(data α, λ)π 1 (λ)π 2 (α)dαdλ 0 It is not easy to compute this equation to get the estimators of α and λ, so Gupta and Kundu(2008) use the Lindley s method to find the approximate estimators. For the two parameters λ 1 and λ 2, Lindley s approximation can be written as ĝ = g( ˆλ 1, ˆλ 2 ) [A + l 30B 12 + l 03 B 21 + l 21 C 12 + l 12 C 21 ] + p 1 A 12 + p 2 A 21 where A = 2 2 j=1 ω ij τ ij, l ij = i+j l(λ 1, λ 2 ) λ i 1 λ j, i, j = 0, 1, 2, 3, i + j = 3, 2 p i = p λ i, ω i = g λ i, ω ij = 2 g λ i λ j, p = ln π(λ 1, λ 2 ), A ij = ω i τ ii + ω j τ ji, B ij = (ω i τ ii + ω j τ ij )τ ii, C ij = 3ω i τ ii τ ij + ω j (τ ii τ jj + 2τ 2 ij). Here π(λ 1, λ 2 ) is the joint prior density function of (λ 1, λ 2 ), τ ij is the (i, j) th element of the inverse of the Fisher information matrix. The computing process is given in the Appendix A. Finally, we obtain the approximate Bayesian estimators of α and λ under the squared error loss function are ˆα Bay = ˆα [2n ˆα 3 τ n ˆλ 3 + (ˆα 1) n x 3 i e ˆλx i (1 + e ˆλx i ) τ (1 e ˆλx i) 21 τ 22 3 x 2 i e ˆλx ( ) ( ) i d 1 b 1 (τ (1 e ˆλx i) 22 τ τ )] + ˆα c τ 11 + ˆλ a τ 12, (2.15) 14

21 ˆλ Bay = ˆλ [2n ˆα 3 τ 12τ n ˆλ 3 + (ˆα 1) n x 3 i e ˆλx i (1 + e ˆλx i ) τ 2 (1 e ˆλx i) x 2 i e ˆλx ( ) ( ) i d 1 b 1 (τ (1 e ˆλx i) 22 τ 21 )] + 2 ˆα c τ 21 + ˆλ a τ 22, (2.16) respectively, where the ˆα and ˆλ denote the MLE of α and λ, and τ 11 = W UW V, τ V 2 12 = UW V, τ 2 22 = U UW V 2, U = ṋ α 2, V = n x i e ˆλx i (1 e ˆλx i), W = ṋ λ 2 + (ˆα 1) n x 2 i e ˆλx i (1 e ˆλx i) 2. For computing the approximate Bayesian estimators, we assume a = b = c = d = 0, since we do not have any prior. Although it implies improper priors on ˆα, but the corresponding posteriors are proper. Finally, by solving the equations (2.15) and (2.16), we can obtain the approximate Bayesian estimators of α and λ. 2.7 A New Hybrid Estimation Wang (2011) used a hybrid estimation method to estimate the two-parameter generalized Pareto distribution (GPD) with the distribution function F (x; σ, k) = 1 (1 kx/σ) 1/k, k 0. For GPD, the maximum likelihood estimator should be performed over the parameter space R = (k < 0, σ > 0) (k > 0, σ/k > X (n) ). Wang (2011) observed that when k > 0 and n < 25, the maximum likelihood method might have convergence problem, and the situation of no maximum could occur when k approaches and goes beyond 0.5. Compared with the ML method, the Hybrid estimation method does not suffer from convergence problems and can always provide valid estimates for the entire parameter space. It makes a big improvement over some existing estimation methods. For the generalized exponential distribution, the ML method and other existing method do not have convergence problem, but we also can use the hybrid method to estimate GED parameters to see if it can still make improvements. 15

22 2.7.1 Empirical Distribution Function Statistics Before introducing the hybrid estimation method, we first introduced the empirical distribution function (EDF) statistic. EDF is the cumulative distribution function associated with the empirical measure of the sample. The essential idea to assess the goodness-of-fit of fitting a continuous probability distribution to data is to measure the distance between the EDF and the underlying distribution function. The original idea of an estimation method using the EDF statistics can be dated back to Wolfowitz (1953, 1957) under the name of minimum distance estimation method. The corresponding estimator can be obtained by minimizing any of the EDF statistics. Let x = (x 1, x 2,, x n ) be a given simple random sample from a continuous distribution function F(x), then the empirical distribution function F n (x) is defined by: F n (x) = numbers of elements in the sample x n = 1 I xi (x), n where I xi (x) = 1 if X i x, and I xi (x) = 0 if X i > x. Then any statistics that measures the discrepancy between F n (x) and F (x) is called an EDF statistic. For every fixed x, F n (x) converges to F (x) almost surely by the Strong Law of Large Numbers, that is, F n (x) converges to F pointwise. Glivenko and Cantelli strengthened this result by proving that F n (x) converges uniformly to F (x), namely, as n, sup F n (x) F (x) 0 almost surely. x There are mainly two classes of EDF statistics: the supremum EDF statistics which include the Kolmogorov-Smirnov (KS) statistic, the Kuiper statistic; and the integral EDF statistics which include the Craḿer von Mises (CM) statistic, the Anderson-Darling (AD) statistic, etc. The following are the definitions of some of the EDF statistics: Kolmogorov-Smirnov (KS) statistic: D n = sup F n (x) F (x; α, λ). x R 16

23 Craḿer von Mises (CM) statistic: Anderson-Darling (AD) statistic: W 2 (x; α, λ) = n {F n (x) F (x; α, λ)} 2 df (x; α, λ). A 2 (x; α, λ) = n {F n (x) F (x; α, λ)} 2 {F (x; α, λ)(1 F (x; α, λ))} 1 df (x; α, λ). Previous research, (Stephens, 1986), showed that in many cases, CM statistic and AD statistic are preferred statistics in terms of better performance than the KS statistic. Due to the non-differentiability of the KS statistic and the poor performances of the higher degree of AD statistics, (Luceño, 2006), we will only discuss the CM statistic and the AD statistic. Since the F n (x) is a step function with a jump at each order statistic, the above EDF statistics can be easily expressed in alternative forms for computational purposes. Denoting the ith order statistic by X (i) and applying the probability integral transformation to the ordered sample to get Z i = F (X (i) ; α, λ), i = 1,, n, we can rewrite the EDF statistics for GED as functions of α and λ as follows: ( W 2 (α, λ) = Z i i 1/2 ) n 12n, (2.17) A 2 (α, λ) = n 1 {(2i 1) ln Z i + (2n + 1 2i) ln(1 Z i )}. n (2.18) By minimizing the functions (2.17) and (2.18) with respect to α and λ, we can obtain the maximum goodness of fit (MGF) estimators. To evaluate the performance of the MGF estimators based on W 2 and A 2, we conduct a simulation study with the shape parameter α in the range 0 < α 2.5. Without loss of generality, the scale λ = 1 is fixed. The results of estimation bias and mean squared error (MSE) are displayed in Figure 2.1 based on 10,000 simulation samples of size n = 50. Comparing the estimators based on W 2 and the estimators based on A 2 in terms of bias and MSE for α and λ, it appears that the MGF estimators based on W 2 generally have the worse performance than that based on A 2 for both α and λ. These results confirm again the previous research, (Stephens, 1986), and the MGF method when the AD statistic is used is preferred in the rest of this thesis. 17

24 Figure 2.1: Bias and MSE for estimating GED parameters α and λ using the MGF estiamtors with replications with n = 50 and λ =1. 18

25 2.7.2 Computation of the New Hybrid Estimates Among the EDF statistics we mentioned, we only focus on the Anderson-Darling statistic A 2 (α, λ), which can give the best and the most balanced performance compared with the W 2. Specifically, we consider minimizing the target function G, which is the univariate minimization problem min G(α, λ; X) = min A 2 (α, λ; X) over α and λ. From the maximum likelihood estimation, we can replace α by ˆα(λ), see (2.4). Our hybrid estimator ˆλ HY B of λ is defined to be the value of λ at which G(ˆα(λ), λ; X) is minimized. Given a sample x = (x 1, x 2,, x n ) from the GED with the distribution function defined in (1.1). The target function G based on A 2 can be written in a simple computational form G(λ; x) = n 1 {(2i 1) ln Z i + (2n + 1 2i) ln(1 Z i )} n = n 1 ln(1 e λx i ) { n(2i 1) n n ln(1 e λx i ) +(2n + 1 2i) ln[1 (1 e λx i ) n/ n ln(1 e λx i) ]}. Using u(λ) = n ln(1 e λx i ), finally, we can have our target function G as G = n 1 { n(2i 1) ln(1 e λx i ) + (2n + 1 2i) ln [ 1 (1 e λx i ) n/u(λ)]}.(2.19) n u(λ) Consider the continuous function G in (2.19) and we can find its first derivative G = 1 n { n(2i 1)[ 1 u(λ) 2 ( xe λxi 1 e λx i u(λ) ln(1 e λx i )u (λ))] +(2n + 1 2i)[1 (1 e λx i ) n/u(λ) ] 1 [ nx ie λx i (1 e λx i ) n/u(λ) 1 u(λ) (1 e λx i ) n/u(λ) ln(1 e λx i ) nu (λ) u(λ) 2 ]} 19

26 Then the numerical search for the value of λ that minimizes G(λ : X) can be performed using the standard Newton-type algorithm or the bisection search algorithm. Hence by minimizing the function (2.19) with respect to λ, the estimate ˆλ HY B is obtained. Finally, we can obtain the estimator of α, say ˆα HY B, using n ˆα HY B =. ln(1 e ˆλ HY B x i ) 20

27 Chapter 3 Simulation Study It is difficult to compare the theoretical performances of the different estimators which are reviewed in the previous chapter. So in this chapter, a finite-sample Monte Carlo simulation study is conducted to compare the performances of the new hybrid estimators proposed in Section 2.7 with other estimators. As the widely accepted criteria for evaluating the quality of an estimator,the estimation bias and mean squared error (MSE) are calculated for different sample sizes and parameter values. 3.1 Comparisons of Finite-Sample Performances The generation of the GED(α, λ) is very simple. If U follows the uniform distribution in [0, 1], then X = ( ln(1 U 1/α )/λ) follows GED(α, λ). This process is quite easy to carry out in R. Our finite-sample simulation comparisons are based on 10,000 replications where each random sample is generated from the GED(α, λ) with sizes n = 15, 20, 30, 50 and 100. Since λ is the scale parameter and all the estimators are scale invariant, we take λ = 1 in all our computations and we consider different values of α. To compare the performance of the HYB, MLE, MME, PCE, LSE, WLSE, LME and BAYE, we take α = 0.5, 1.0, 2.0, 2.5, 4.0 or 5.5, and λ = 1. For each combination of n and α we generate a sample of size n from GE(α, 1) and estimate α and λ by different methods. We report the bias and the average relative mean squared error, which is the MSE of ˆα/α. Tables 3.1 and 3.2 give the average relative mean squared error of the shape parameter α and the average relative mean squared error of the scale parameter λ. The average relative biases and MSE for different estimators of α and λ are plotted against α in Figure 3.1 to Figure 3.5 for sample sizes n = 15, 20, 50 and

28 Table 3.1: The average relative mean squared error of α n Method α = 0.5 α = 1.0 α = 2.0 α = 2.5 α = 4.0 α = 5.5 MLE MME PCE n = 15 LSE WLSE LME BAYE HYB MLE MME PCE n = 20 LSE WLSE LME BAYE HYB MLE MME PCE n = 30 LSE WLSE LME BAYE HYB MLE MME PCE n = 50 LSE WLSE LME BAYE HYB MLE MME PCE n = 100 LSE WLSE LME BAYE HYB

29 Table 3.2: The average relative mean squared error of λ n Method α = 0.5 α = 1.0 α = 2.0 α = 2.5 α = 4.0 α = 5.5 MLE MME PCE n = 15 LSE WLSE LME BAYE HYB MLE MME PCE n = 20 LSE WLSE LME BAYE HYB MLE MME PCE n = 30 LSE WLSE LME BAYE HYB MLE MME PCE n = 50 LSE WLSE LME BAYE HYB MLE MME PCE n = 100 LSE WLSE LME BAYE HYB

30 From the Tables 3.1 and 3.2 and Figures 3,1, 3.2 and 3.3, it is clear that the biases and the average MSE decrease as sample size increases. We can observe that the average relative bias, average relative MSE of α and λ depend on α. When the value of α increases, the relative MSE and the relative bias of λ decrease. Consider the MSE only, we can say that the estimation of α is more accurate for smaller α, whereas the estimation of λ is more accurate for larger α. As in Gupta and Kundu (2006), we also find that most of the estimators overestimate both α and λ except PCE, which usually underestimates the corresponding parameters, particularly for moderate and large sample sizes. In measuring the accuracy of different estimators, unbiasness is always desired. From Figures 3.1, 3.2 and 3.3, we can see that the biases of ˆα and ˆλ from the methods of HYB, LSE, WLSE, LME, MLE and BAYE are all very close and all of them are always positive. Among these methods, our new hybrid estimation method always give better performance than the ML method and the BAYE method. The MME also has positive biases which are much bigger than the others. The PCE works best only for small sample size, however, when the sample size increases, its performance becomes worse. We also need to consider the MSE of all these estimation methods. From Tables 3.1 and 3.2, and Figures 3.4 and 3.5, it is seen that the performance of the HYB, BAYE and MLE are all good in almost all the cases considered for estimating both α and λ. The MLE and BAYE are quite close to each other. Compared with these two methods, we see that for a small sample size, the HYB works better than the MLE and BAYE. The performance of the LME is also quite close to that of the MLE. The MSE of PCE works fine but not that good compared with the MSE of other estimators. Also, the MSE of WLSE is usually smaller than that of the LSE, whereas the MME usually has the largest MSE in most of the cases. Now if we consider the computational complexities, we see that the HYB, MLE, MME and LME involve one dimensional optimization, whereas PCE, LSE and WLSE involve two dimensional minimization. For the Bayesian estimation using Lindley s method, we need 24

31 Figure 3.1: Average relative bias comparisons for estimating the GED parameters α and λ with replications with sample size n = 15 and λ = 1. 25

32 Figure 3.2: Average relative bias comparisons for estimating the GED parameters α and λ with replications with sample size n = 50 and λ = 1. 26

33 Figure 3.3: Average relative bias comparisons for estimating the GED parameters α and λ with replications with sample size n = 100 and λ = 1. 27

34 Figure 3.4: Average relative mean squared error comparisons for estimating the GED parameters α and λ with replications with sample size n = 20 and λ = 1. 28

35 Figure 3.5: Average relative mean squared error comparisons for estimating the GED parameters α and λ with replications with sample size n = 100 and λ = 1. 29

36 to solve the very complex equations. Also, we need to evaluate ψ( ) function at different points in order to get MME and LME. Therefore, considering all the aspects, namely, bias, MSE and computation, we can see that the new hybrid estimation method has the best performance for estimating α and λ. It makes improvements over the MLE and BAYE although the improvements are not very big. 30

37 Chapter 4 An Example In this chapter, a real-world example will be presented. The data set originally discussed in Lawless (1982) arose in tests on the endurance of deep groove ball bearings. The data are the numbers of million revolutions before failure for each of the 23 ball bearings using the same test. Since the generalized exponential distribution is always used in analyzing lifetime data, we consider an analysis of this data set using the GED here. The data set is: 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, , , , , Gupta and Kundu (1999b) analyzed this data set and found that the data has the sample mean , the sample variance and the sample coefficient of variation Since the C.V. is independent of the scale parameter, by using the C.V., they guessed α should be between 5.00 and From the maximum likelihood estimation, they got ˆλ = From a Newton-Raphson type iterative procedure, finally, they had ˆα = The 95% confidence interval of α is (3.0706, ) and the 95% confidence interval of λ is (0.0306, ). They obtained the confidence interval using the following asymptotic normality result: ( ) n(ˆα α, ˆλ λ) N2 0, Now we estimate this data set using all of the estimation methods and the estimates of the parameters α and λ are shown in Table 4.1. It is easy to see from Table 4.1 that the estimates of the HYB, ML, BAYE and MOM are very close to the estimators that Gupta and Kundu (1999) got. For the PCE, LS, WLS and 31

38 Table 4.1: The estimated GED parameters for the lifetime of deep groove ball bearings HYB MLE BAYE MOM PCE LSE WLSE LME ˆα ˆλ LM, the estimates of the shape parameter are a little bit different from those of the first 4 methods. We plot the estimated GED cumulative distribution functions using the parameter estimates from the different methods versus the empirical distribution function for the data set. From Figures 4.1 and 4.2, it seems that our hybrid estimation method gives an overall good fit to the deep groove ball bearings data, with slight improvements over the MLE and BAYE in particular. 32

39 Figure 4.1: The estimated GED cumulative distribution functions using different methods (MLE, MME, PCE, LME) versus the EDF for the deep groove ball bearings. 33

40 Figure 4.2: The estimated GED cumulative distribution functions using different methods (HYB, BAYE, LSE, WLSE) versus the EDF for the deep groove ball bearings. 34

41 Chapter 5 Final Conclusions and Future Research As an alternative to the gamma and Weibull distribution, the generalized exponential distribution was widely used to analyze lifetime data since it was first introduced by Gupta and Kundu in Due to its improvements to the gamma and Weibull distribution, it was extensively considered in statistics. The GED unknown or known parameters estimation is one of the most popular research areas and has been discussed by several authors. In this thesis, we review seven existing estimation methods and introduce a new hybrid estimation method to estimate the unknown parameters α and λ of generalized exponential distribution. As we mentioned before, for the generalized Pareto distribution, this new hybrid method can avoid the convergence problem which the maximum likelihood method may have and can always provide valid estimates. For the generalized exponential distribution, the MLE and other estimation methods do not have any convergence problems. For this situation, we still can use this hybrid estimation method and it also has some advantages. First, same as the MLE, the new hybrid estimation method also do not have any problems and always give valid estimates. Also, the new hybrid method can provide a visible better fit than the MLE method and the Bayesian method, especially when the sample size n is small. Second, the new hybrid method can allow practitioners to conduct reliable and accurate data inference using the generalized exponential distribution. In this thesis, we only focus on the Anderson-Darling statistics among other EDF statistics, this hybrid idea can be similarly carried out for any other EDF statistics. Finally, comparing with some other estimation methods, the new hybrid method is easy to use in practice and do not have a very complex computation, which can give improvements in efficiency and better to avoid the computation mistakes in research. 35

42 A useful R program for computing the GED estimations and the plots is given in the Appendix B. I also use the hybrid method, MLE and Bayesian method to estimate the GED paramaters for the data with outliers. When the outliers are obvious, for example, very large or very small comparing with other data, all of the three methods can not provide good fits. However, this situation is not significant since we can avoid this kind of outliers. The other situation is that the data comes from different generalized exponential distribution but with similar values, as the GED(α, λ) and GED(β, λ), α β. For this kind of data set, all of the three estimation methods can give well performances which show that the hybrid, MLE and Bayesian methods are all robust. For the future research, we can think the following ideas to work on: 1. Try other types of outliers to check if the new hybrid method can provide a better performance than other estimation methods. 2. Develop a large-sample theory for the proposed hybrid estimators. 3. Generalize the hybrid estimation method and apply it to some other distributions suffering from the finite parameter-dependent endpoint problem. 4. Apply the hybrid estimation method to dependent data to see if it still works. 36

43 Appendix A Computation for the Lindley s Approximation From Gupta and Kundu (2006), we had the log-likelihood function l(α, λ) = n ln(α) + n ln(λ) + (α 1) ln(1 e λx i ) λ x i. Therefore, we obtained l 30 = 2n ˆα 3, l 03 = 2n ˆλ 3 + (ˆα 1) n x 3 i e ˆλx i (1 + e ˆλx i ), (1 e ˆλx i) 3 l 12 = When g(α, λ) = α, then x 2 i e ˆλx i (1 e ˆλx i) 2, l 21 = 0. ω 1 = 1, ω 2 = 0, ω i,j = 0, i, j = 1, 2. Therefore, A = 0, B 12 = τ 2 11, B 21 = τ 21 τ 22, C 12 = 3τ 11 τ 12, C 21 = τ 22 τ τ 2 21, A 12 = τ 11, A 21 = τ 12. Then the first part of Lindley s approximation follows by using p 1 = d 1 α c, and p 2 = b 1 λ a. Same as the first part, for the second part, we had g(α, λ) = λ, then ω 1 = 0, ω 2 = 1, ω i,j = 0, i, j = 1, 2. Therefore, A = 0, B 12 = τ 12 τ 11, B 21 = τ 2 22, C 12 = τ 11 τ τ 2 12, C 21 = 3τ 22 τ 21, A 12 = τ 21, A 21 = τ 22, and the following used same p 1 and p 2. 37

44 Appendix B An R Function for Computing the GED Estimates gexp <- function(n, alpha, nrep=10000){ # Estimate alpha and lambda = 1 in a generalized exponential model when the data # are iid GE(alpha, lambda) # bias and mean square error are computed z <- matrix(0,nrep,16) for(i in 1:nrep) { u <- runif(n) x <- -log(1-u^(1/alpha)) z[i,c(1,2)] <- gexpmle(x) z[i,c(3,4)] <- gexpmme(x) z[i,c(5,6)] <- gexppce(x) z[i,c(7,8)] <- gexplse(x) z[i,c(9,10)] <- gexpwlse(x) z[i,c(11,12)] <- gexplm(x) z[i,c(13,14)] <- gexpbaye(x) z[i,c(15,16)] <- gexphybrid(x) } bias <- apply(z,2,mean) - c(alpha, 1, alpha, 1, alpha, 1, alpha, 1, alpha, 1, alpha, 1, alpha, 1, alpha, 1) mse <- c(mean((z[,1]-alpha)^2),mean((z[,2]-1)^2),mean((z[,3]-alpha)^2), mean((z[,4]-1)^2),mean((z[,5]-alpha)^2),mean((z[,6]-1)^2), mean((z[,7]-alpha)^2),mean((z[,8]-1)^2),mean((z[,9]-alpha)^2), 38

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