A new hybrid estimation method for the generalized pareto distribution

ABSTRACT

The generalized Pareto distribution (GPD) is important in the analysis of extreme values, especially in modeling exceedances over thresholds. Most of the existing methods for estimating the scale and shape parameters of the GPD suffer from theoretical and/or computational problems. A new hybrid estimation method is proposed in this article, which minimizes a goodness-of-fit measure and incorporates some useful likelihood information. Compared with the maximum likelihood method and other leading methods, our new hybrid estimation method retains high efficiency, reduces the estimation bias, and is computation friendly.     

Parameter estimation of the generalized Pareto distribution—Part II

This is the second part of a paper which focuses on reviewing methods for estimating the parameters of the generalized Pareto distribution (GPD). The GPD is a very important distribution in the extreme value context. It is commonly used for modeling the observations that exceed very high thresholds. The ultimate success of the GPD in applications evidently depends on the parameter estimation process. Quite a few methods exist in the literature for estimating the GPD parameters. Estimation procedures, such as the maximum likelihood (ML), the method of moments (MOM) and the probability weighted moments (PWM) method were described in Part I of the paper. We shall continue to review methods for estimating the GPD parameters, in particular methods that are robust and procedures that use the Bayesian methodology. As in Part I, we shall focus on those that are relatively simple and straightforward to be applied to real world data.     

Parameter estimation of the generalized Pareto distribution—Part I

The generalized Pareto distribution (GPD) has been widely used in the extreme value framework. The success of the GPD when applied to real data sets depends substantially on the parameter estimation process. Several methods exist in the literature for estimating the GPD parameters. Mostly, the estimation is performed by maximum likelihood (ML). Alternatively, the probability weighted moments (PWM) and the method of moments (MOM) are often used, especially when the sample sizes are small. Although these three approaches are the most common and quite useful in many situations, their extensive use is also due to the lack of knowledge about other estimation methods. Actually, many other methods, besides the ones mentioned above, exist in the extreme value and hydrological literatures and as such are not widely known to practitioners in other areas. This paper is the first one of two papers that aim to fill in this gap. We shall extensively review some of the methods used for estimating the GPD parameters, focusing on those that can be applied in practical situations in a quite simple and straightforward manner.     

Comparison of methods of estimation for parameters of generalized Poisson distribution through simulation study

In this article, we take a brief overview of different functional forms of generalized Poisson distribution (GPD) and various methods of its parameter estimation found in the literature. We compare the method of moment estimation (ME) and maximum likelihood estimation (MLE) of parameters of GPD through simulation study in terms of bias, MSE and covariance. To simulate random numbers from GPD, we develop a Matlab function gpoissrnd(). The simulation study leads to the important conclusion that the ME performs better or equally good as compared to MLE when sample size is small.

Further we fit the GPD to various datasets in literature using both estimation methods and observe that the results do not differ significantly even though the sample size is large. Overall, we conclude that for GPD, use of ME in place of MLE will lead to almost similar results. The computational simplicity in calculation of ME as compared to MLE also gives support to the use of ME in case of GPD for practitioners.     

Moving truncations barrier-function methos for estimation in three-parameter lognormal models

A method of centres algorithm for maximum likelihood estimation in the three-parameter lognormal model is presented and discussed, The algorithm is a member of the class of moving truncations algorithms for solving nonlinear programming problems and is able to move the numerical search out of the region of the infinite maximum of the conditional likelihood function, thereby permitting convergence to an interior relative maximum of this function. The algorithm also includes an optimality test to locate the primary relative maximum of the likelihood function.     

Problems of inference for Azzalini's skewnormal distribution

This paper considers various unresolved inference problems for the skewnormal distribution. We give reasons as to why the direct parameterization should not be used as a general basis for estimation, and consider method of moments and maximum likelihood estimation for the distribution's centred parameterization. Large sample theory results are given for the method of moments estimators, and numerical approaches for obtaining maximum likelihood estimates are discussed. Simulation is used to assess the performance of the two types of estimation. We also present procedures for testing for departures from the limiting folded normal distribution. Data on the percentage body fat of elite athletes are used to illustrate some of the issues raised.     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

NEYMAN SMOOTH TESTS FOR THE GENERALIZED PARETO DISTRIBUTION

ABSTRACT

The generalized Pareto distribution (GPD) is commonly used as extreme values's distribution. We present goodness of fit tests for the GPD based on Neyman's smooth tests statistics. The methods of maximum likelihood, moments and probability-weighted moments are used for estimating the GPD's parameters. Simulations are done to study the power of these tests.     

Estimation for the scaled half-logistic distribution under Type-I progressively hybrid censoring scheme

This paper addresses the estimation for the unknown scale parameter of the half-logistic distribution based on a Type-I progressively hybrid censoring scheme. We evaluate the maximum likelihood estimate (MLE) via numerical method, and EM algorithm, and also the approximate maximum likelihood estimate (AMLE). We use a modified acceptance rejection method to obtain the Bayes estimate and corresponding highest posterior confidence intervals. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyze one dataset for illustrative purposes.     

The truncated generalized poisson distribution and its estimation

A discrete probability model always gets truncated during the sampling process and the point of truncation depends upon the sample size. Also, the generalized Poisson distribution cannot be used with full justification when the second parameter is negative. To avoid these problems a truncated generalized Poisson distribution is defined and studied. Estimation of its parameters by moments method, maximum likelihood method and a mixed method are considered. Some examples are given to illustrate the effect on the parameters’ estimates when a non-truncated GPD is used instead of a truncated GPD.     

The factoring likelihood method for non-monotone missing data

We address the problem of parameter estimation in multivariate distributions under ignorable non-monotone missing data. The factoring likelihood method for monotone missing data, termed by Rubin (1974), is applied to a more general case of non-monotone missing data. The proposed method is asymptotically equivalent to the Fisher scoring method from the observed likelihood, but avoids the burden of computing the first and second partial derivatives of the observed likelihood. Instead, the maximum likelihood estimates and their information matrices for each partition of the data set are computed separately and combined naturally using the generalized least squares method. A numerical example is presented to illustrate the method.     

Penalized maximum likelihood estimation in the modified extended Weibull distribution

We address the issue of performing inference on the parameters that index the modified extended Weibull (MEW) distribution. We show that numerical maximization of the MEW log-likelihood function can be problematic. It is even possible to encounter maximum likelihood estimates that are not finite, that is, it is possible to encounter monotonic likelihood functions. We consider different penalization schemes to improve maximum likelihood point estimation. A penalization scheme based on the Jeffreys’ invariant prior is shown to be particularly useful. Simulation results on point estimation, interval estimation, and hypothesis testing inference are presented. Two empirical applications are presented and discussed.     

Unconstrained parametrizations for variance-covariance matrices

The estimation of variance-covariance matrices through optimization of an objective function, such as a log-likelihood function, is usually a difficult numerical problem. Since the estimates should be positive semi-definite matrices, we must use constrained optimization, or employ a parametrization that enforces this condition. We describe here five different parametrizations for variance-covariance matrices that ensure positive definiteness, thus leaving the estimation problem unconstrained. We compare the parametrizations based on their computational efficiency and statistical interpretability. The results described here are particularly useful in maximum likelihood and restricted maximum likelihood estimation in linear and non-linear mixed-effects models, but are also applicable to other areas of statistics.     

Maximum likelihood estimation for the generalized poisson distribution when sample mean is larger than sample variance

The generalized Poisson distribution (GPD), studied by many researchers and containing two parameters θ and λ, has been found to fit very well data sets arising in biological, ecological, social and marketing fields. Consul and Shoukri (1985) have shown that for negative values of λ the GPD gets truncated and the model becomes deficient; however, the truncation error becomes less than 0.0005 if the minimum number of non-zero probability classes ≥ 4 for all values of θ and λ and the GPD model can be safely used in all such cases. The problem of admissible maximum likelihood (ML) estimation when the sample mean is larger than the sample variance is considered in this paper which complements the earlier work of Consul and Shoukri (1984) on the existence of unique ML estimators of θ and λ when the sample mean is smaller than or equal to the sample variance.     

Inference in the Pareto distribution based on progressive Type II censoring with random removals

This study considers the estimation problem for the Pareto distribution based on progressive Type II censoring with random removals. The number of units removed at each failure time has a discrete uniform distribution. We are going to use the maximum likelihood method to obtain the estimator of parameter. The expectation and variance of the maximum likelihood estimator will be derived. The expected time required to complete such an experiment will be computed. Some numerical results of expected test times are carried out for this type of progressive censoring and other sampling schemes.     

Maximum likelihood estimation of the logarithmic series distribution

This paper discusses the maximum likelihood estimation of the parameter of the logarithmic series distribution. The univariate case is treated in Part I, the multivariate case in Part II. A simple numerical estimation procedure is suggested using a fixed point approach. Convergence to the maximum likelihood estimator is shown. In Part III convergence rate is proven to be linear which is also demonstrated through example. In addition, comparisons with Newton’s method and the secant method in the univariate case, and with Newton’s method and the projected gradient method in the multivariate case are provided.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.     

Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring

In this paper, we investigate the estimation problem concerning a progressively type-II censored sample from the two-parameter bathtub-shaped lifetime distribution. We use the maximum likelihood method to obtain the point estimators of the parameters. We also provide a method for constructing an exact confidence interval and an exact joint confidence region for the parameters. Two numerical examples are presented to illustrate the method of inference developed here. Finally, Monte Carlo simulation studies are used to assess the performance of our proposed method.     

Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples

In this paper, the estimation of parameters for a three-parameter Weibull distribution based on progressively Type-II right censored sample is studied. Different estimation procedures for complete sample are generalized to the case with progressively censored data. These methods include the maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators. We also proposed the use of a censored estimation method with one-step bias-correction to obtain reliable initial estimates for iterative procedures. These methods are compared via a Monte Carlo simulation study in terms of their biases, root mean squared errors and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of inference developed here.