Parameter and Quantile Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.     

A Procedure for the Sequential Interval Estimation of the Shape Parameter of the Gamma Distribution

A procedure for interval estimation of the shape parameter of the gamma distribution is proposed which enables the experimenter to obtain upper and lower confidence limits after each observation is sampled. An approximate lower bound for the probability that the resulting sequence of confidence intervals will contain the shape parameter is obtained. The proposed procedure may be useful in estimating the coefficient of variationassociated with non-negative variables whose distribution is skewed to the right. Comparisons with a single sample procedure are discussed.     

Model Misspecification of Generalized Gamma Distribution for Accelerated Lifetime-Censored Data

Abstract

The performance of reliability inference strongly depends on the modeling of the product’s lifetime distribution. Many products have complex lifetime distributions whose optimal settings are not easily found. Practitioners prefer to use simpler lifetime distribution to facilitate the data modeling process while knowing the true distribution. Therefore, the effects of model mis-specification on the product’s lifetime prediction is an interesting research area. This article presents some results on the behavior of the relative bias (RB) and relative variability (RV) of pth quantile of the accelerated lifetime (ALT) experiment when the generalized Gamma (GG₃) distribution is incorrectly specified as Lognormal or Weibull distribution. Both complete and censored ALT models are analyzed. At first, the analytical expressions for the expected log-likelihood function of the misspecified model with respect to the true model is derived. Consequently, the best parameter for the incorrect model is obtained directly via a numerical optimization to achieve a higher accuracy model than the wrong one for the end-goal task. The results demonstrate that the tail quantiles are significantly overestimated (underestimated) when data are wrongly fitted by Lognormal (Weibull) distribution. Moreover, the variability of the tail quantiles is significantly enlarged when the model is incorrectly specified as Lognormal or Weibull distribution. Precisely, the effect on the tail quantiles is more significant when the sample size and censoring ratio are not large enough. Supplementary materials for this article are available online.     

Minimum Distance Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) is widely used for extreme values over a threshold. Most existing methods for parameter estimation either perform unsatisfactorily when the shape parameter k is larger than 0.5, or they suffer from heavy computation as the sample size increases. In view of the fact that k > 0.5 is occasionally seen in numerous applications, including two illustrative examples used in this study, we remedy the deficiencies of existing methods by proposing two new estimators for the GPD parameters. The new estimators are inspired by the minimum distance estimation and the M-estimation in the linear regression. Through comprehensive simulation, the estimators are shown to perform well for all values of k under small and moderate sample sizes. They are comparable to the existing methods for k < 0.5 while perform much better for k > 0.5.     

Maximum Likelihood Parameter Estimation for the Quartic Exponential Distribution

The quartic exponential (QE) distribution defined by the probability density function of the type

is examined in detail.

The problem of obtaining maximum likelihood point estimates of the population parameters reduces to that of identifying the α as functions of the population moments μ _r ′, r = 1, 2.3.4.

The invalidity is explained of methods proposed by previous authors to deal with the nonlinear relationships involved, and a new algorithm is developed which overcomes these objections. The new algorithm is applied to practical data, and the resulting distributions fitted to observed frequencies are shown to compare favourably with those obtained by previous Methods.     

Maximum-Likelihood Estimation of the Parameters of a Four-Parameter Generalized Gamma Population from Complete and Censored Samples

Consider the four-parameter generalized Gamma population with location parameter c, scale parameter a, shape/power parameter b, and power parameter p (shape parameter d = bp) and probability density function f(x; c, a, b, p) = p(x — c) ^bp–1 exp {–[(x – c)/a] ^p }/a ^bp Γ(b), where a, b, p > 0 and x ≥ c ≥ 0. The likelihood equations for parameter estimation are obtained by equating to zero the first partial derivatives, with respect to each of the four parameters, of the natural logarithm of the likelihood function for a complete or censored sample. The asymptotic variances and covariances of the maximum-likelihood estimators are found by inverting the information matrix, whose components are the limits, as the sample size n → ∞, of the negatives of the expected values of the second partial derivatives of the likelihood function with respect to the parameters. The likelihood equations cannot be solved explicitly, but an iterative procedure for solving them on an electronic computer is described. The results of applying this procedure to samples from Gamma, Weibull, and half-normal populations are tabulated, as are the asymptotic variances and covariances of the maximum-likelihood estimators.     

Estimation of the Parameters of the Gamma Distribution by Sample Quantiles

This paper deals with large sample estimation of the location parameter (α₁ and the scale parameter α₂ in the gamma distribution with known shape parameter. Best linear unbiased estimates based on k sample quantiles are used. For a given k, the optimum spacings of the sample quantiles can be replaced by simpler “nearly optimum” spacings at virtually no loss of asymptotic efficiency. The theory behind the nearly optimum spacings is briefly reviewed. The major part of the paper concerns estimation of α₂ when α₂ is known. Nearly optimum spacings together with the coefficients to be used in computing the estimates are presented in a number of tables for k = 1(1) 10, and various values of the shape parameter. The paper also contains brief discussions of estimation of α₁, when α₂ is known, and simultaneous estimation of α₁ and α₂.     

Aids for Fitting the Gamma Distribution by Maximum Likelihood

This paper presents a new table and some approximating polynomials especially designed to facilitate maximum likelihood estimation of the parameters of the gamma distribution, and also applicable to the a-parameter Type V; it discusses methods of computing the sampling variances of the likelihood estimators; it illustrates the use of the tables in a numerical example; it mentions applications to the Erlang distribution and difficulties of application to the general Type III. Finally it inquires when the numbers extracted from the tables are maximum likelihood estimates, and what they are estimates of.     

指数分布场合基于竞争失效数据的参数估计

讨论了指数分布场合竞争失效产品在恒定应力加速寿命试验和步进应用加速寿命试验情形加速方程中有关参数的加权最小二乘估计，利用Monte Carlo方法和参数的最大似然估计作了模拟比较。     

Further Remarks on Maximum Likelihood Estimators for the Gamma Distribution

The maximum likelihood estimators , â, for the parameters ρ, a of the gamma density f(x) = k(x/a)^ρ–1 exp(?x/a) are solutions of the equations In – ψ() = ln(A/G), â = A, where Ψ is the logarithmic derivative of the gamma function, A and G being the sample (of size n) arithmetic and geometric means, respectively. The moments of and â are developed in descending powers of ρ. A comparison of the assessments of the moments by the present approach and a method involving series in descending powers of n is made.

Approximate expressions are also given for the first four moments of ? = l/â, which is the maximum likelihood estimator of c = l/a.     

Parametric Estimation

The exponential distribution is often used in reliability work to describe the distribution of time to “chance” failure and is characterized by a constant failure rate. In this paper the small sample powers are compared for four test statistics for the hypothesis of constant failure rate vs. the hypothesis of non-constant failure rate. The tests are compared for samples of size n = 10(5)50 using the Weibull distribution for the alternative distribution. The shape parameter of the Weibull is varied from 0.5 to 2.5. For the two test statistics which involve arbitrary grouping of the data the effect of group size and number was also examined.     

Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias

The numerical technique of the maximum likelihood method to estimate the parameters of Gamma distribution is examined. A convenient table is obtained to facilitate the maximum likelihood estimation of the parameters and the estimates of the variance-covariance matrix. The bias of the estimates is investigated numerically. The empirical result indicates that the bias of both parameter estimates produced by the maximum likelihood method is positive.     

q-对称熵损失函数下Gamma分布的尺度参数的估计

本文在对称熵损失函数的基础上定义了q-对称熵损失函数,并用参数估计的方法研究了在q-对称熵损失函数下Gamma分布的尺度参数的最小风险同变估计(MRE)、贝叶斯(Bayes)估计、最小最大(Mininax)估计等。我们还对这些估计量的可容许性和不可容许性进行了讨论,最后分别对指数分布和Gamma分布在两种损失函数下的估计结果进行了数值比较。     

Sequential Optimum Procedures for Unbiased Estimation of a Binomial Parameter

Let x ₁, x ₂, … x _n, … be a sequence of independent random variables with a common density function P(x = 1) = p, P(x = 0) = 1 – p, 0 < p < 1. This paper considers the non-randomized sequential procedures δ's for estimating p and the following three problems on choice of δ. (i) Choose δ to minimize E _v (Z _δ – P)² subject to E _p N _δ ≤ m where m ≥ 1 and Z _δ is an unbiased estimate of p; (ii) Choose δ to minimize E_p N _δ subject to E _p (Z _δ – p)² ≤ α where α is a real positive number; (iii) choose δ to minimize C E_p N _δ + E _p (Z _δ – p)² where C is the cost of an observation. In each case the minimization is to be done uniformly in p if possible; otherwise the supremum over p of the risk in question is to be minimized. A procedure is constructed for problem (i) when m is not an integer. A fixed sample size procedure is shown to be admissible and minimax for problem (ii). A procedure is constructed which is asymptotically uniformly better than the fixed sample size for problem (ii). Furthermore, for problem (iii) some optimum procedures are constructed.     

多元Marshall～Olkin型指数分布的特征及其参数估计

导出了多元Marshall～Olkin型指数分布的一个特征,利用该特征,获得了多元Marshall～Olkin型指数分布参数的最大似然估计及矩估计.     

一类分块混合效应模型的参数估计

研究一类分块混合效模型中部分固定效应向量的估计问题，为得到其可行的最佳线性无偏估计，提出了一种减约模型，并给出了由此减约模型得出的可行估计等于原模型下相应的最佳线性无偏估计的充分必要条件，并以实例说明文中结果。     

刻度参数的贝叶斯经验贝叶斯估计

我们采用贝叶斯经验贝叶斯方法估计了一个特殊的指数分布族中的刻度参数,证明了刻度参数的贝叶斯经验贝叶斯估计几乎处处收敛到它的贝叶斯估计且获得了它的渐近最优性。最后,提出了一个模拟试验去验证贝叶斯经验贝叶斯估计的渐近最优性。     

Inference on the Gamma Distribution

This study develops inferential procedures for a gamma distribution. Based on the Cornish–Fisher expansion and pivoting the cumulative distribution function, an approximate confidence interval for the gamma shape parameter is derived. The generalized confidence intervals for the rate parameter and other quantities such as mean are explored. The proposed generalized inferential procedures are extended to construct prediction limits for a single future measurement and for at least p of m measurements at each of r locations. The performance of the proposed procedures is evaluated using Monte Carlo simulation. The simulation results show that the proposed procedures are very satisfactory. Finally, three real examples are used to illustrate the proposed procedures. Supplementary materials for this article are available online.     

一种工业过程时变参数估计新算法—修正目标函数法

针对某些工业过程控制参数高度时变的特点和LS拟合算法存在病态估计的现象,提出了一类基于修正目标函数的时变参数估计算法,并给出递推辅助变量算法。此类算法不仅具有较强的实时跟踪能力和较高的估计精度,而且能克服病态估计,摆脱了伪随机信号在工程中带来的麻烦。新算法没有增加计算量,非常适合于工程应用,仿真及实际应用结果都证明新算法是有效的。中给出了有关新算法的定理及其证明。     

The Inverse Yates Algorithm

The choice of an experimental design suitable for fitting a graduating polynomial can be made according to a number of criteria, depending on the problem involved. Difficulties arise when, although the factors are continuous in nature, the number of levels is specified by some external considerations. For example, if some factors can be examined at only two levels, the graduating function cannot include quadratic terms in those variables, but all second order terms for variables to be examined at three or more levels can be permitted. For such cases, a restricted model and special experimental designs are needed. This paper considers the problem when there are some factors at two levels and some factors at three levels, and when there are some factors at two levels and some factors at four levels.