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.     

Inference and Optimal Design Based on Step—Partially Accelerated Life Tests for the Generalized Pareto Distribution under Progressive Type-I Censoring

Various types of failure, censored and accelerated life tests, are commonly employed for life testing in some manufacturing industries and products that are highly reliable. In this article, we consider the tampered failure rate model as one of such types that relate the distribution under use condition to the distribution under accelerated condition. It is assumed that the lifetimes of products under use condition have generalized Pareto distribution as a lifetime model. Some estimation methods such as graphical, moments, probability weighted moments, and maximum likelihood estimation methods for the parameters are discussed based on progressively type-I censored data. The determination of optimal stress change time is discussed under two different criteria of optimality. Finally, a Monte Carlo simulation study is carried out to examine the performance of the estimation methods and the optimality criteria.     

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.     

The four-parameter Burr XII distribution: Properties,regression model,and applications

This paper introduces a new four-parameter lifetime model called the Weibull Burr XII distribution. The new model has the advantage of being capable of modeling various shapes of aging and failure criteria. We derive some of its structural properties including ordinary and incomplete moments, quantile and generating functions, probability weighted moments, and order statistics. The new density function can be expressed as a linear mixture of Burr XII densities. We propose a log-linear regression model using a new distribution so-called the log-Weibull Burr XII distribution. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimation are discussed. We prove empirically the importance and flexibility of the new model in modeling various types of data.     

Using jackknife to correct bias of MLE for the truncated Pareto distribution

The Pareto distribution is a well-known probability distribution in statistics, which has been widely used in many fields, such as finance, physics, hydrology, geology and astronomy. However, the parameter estimation for the truncated Pareto distribution is much more complicated than that for the Pareto distribution. In this paper, we demonstrate that the bias of the maximum likelihood estimation for the truncated Pareto distribution can be significantly reduced by its jackknife estimation, which has a very simple form.     

The generalized transmuted-G family of distributions

We introduce a new class of continuous distributions called the generalized transmuted-G family which extends the transmuted-G class. We provide six special models of the new family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of three applications to real data sets.     

Modeling of maximum precipitation using maximal generalized extreme value distribution

Distribution of maximum or minimum values (extreme values) of a dataset is especially used in natural phenomena including sea waves, flow discharge, wind speeds, and precipitation and it is also used in many other applied sciences such as reliability studies and analysis of environmental extreme events. So if we can explain the extremal behavior via statistical formulas, we can estimate how their behavior would be in the future. In this paper, we study extreme values of maximum precipitation in Zahedan using maximal generalized extreme value distribution, which all maxima of a data set are modeled using it. Also, we apply four methods to estimate distribution parameters including maximum likelihood estimation, probability weighted moments, elemental percentile and quantile least squares then compare estimates by average scaled absolute error criterion and obtain quantiles estimates and confidence intervals. In addition, goodness-of-fit tests are described. As a part of result, the return period of maximum precipitation is computed.     

Estimation for the generalized pareto distribution with censored data

We present a methodology for computing the point and interval maximum likelihood parameter estimation for the two-parameter generalized Pareto distribution (GPD) with censored data. The basic idea underlying our method is a reduction of the two-dimensional numerical search for the zeros of the GPD log-likelihood gradient vector to a one-dimensional numerical search. We describe a computationally efficient algorithm which implement this approach. Two illustrative examples are presented. Simulation results indicate that the estimates derived by maximum likelihood estimation are more reliable against those of method of moments. An evaluation of the practical sample size requirements for the asymptotic normality is also included.     

Estimation for the Parameters of Generalized Half-normal Distribution Based on Progressive Type-I Interval Censoring

Based on progressively Type-I interval censored sample, the problem of estimating unknown parameters of a two parameter generalized half-normal(GHN) distribution is considered. Different methods of estimation are discussed. They include the maximum likelihood estimation, midpoint approximation method, approximate maximum likelihood estimation, method of moments, and estimation based on probability plot. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX, and general entropy is calculated. The Lindley’s approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, analysis is also carried out for a real dataset.     

Generalised Smooth Tests of Goodness of Fit Utilising L‐moments

In this paper we present a semiparametric test of goodness of fit which is based on the method of L‐moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L‐moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L‐moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.     

Statistical Inference for a New Class of Multivariate Pareto Distributions

Various solutions to the parameter estimation problem of a recently introduced multivariate Pareto distribution are developed and exemplified numerically. Namely, a density of the aforementioned multivariate Pareto distribution with respect to a dominating measure, rather than the corresponding Lebesgue measure, is specified and then employed to investigate the maximum likelihood estimation (MLE) approach. Also, in an attempt to fully enjoy the common shock origins of the multivariate model of interest, an adapted variant of the expectation-maximization (EM) algorithm is formulated and studied. The method of moments is discussed as a convenient way to obtain starting values for the numerical optimization procedures associated with the MLE and EM methods.     

Large-sample quantile estimation in pareto laws

A large-sample method of estimation for the parameters of Pareto laws is investigatedo The estimates are derived by using a small subset of k sample quantiles out of the original observations. The optimum spacing of the k quantiles is also examined. A Monte Carlo study compares this method with the method of moments and that of maximum likelihood for a selected set of parameter values and sample sizes.     

Topp–Leone normal distribution with application to increasing failure rate data

In this article, we proposed a new three-parameter probability distribution, called Topp–Leone normal, for modelling increasing failure rate data. The distribution is obtained by using Topp–Leone-X family of distributions with normal as a baseline model. The basic properties including moments, quantile function, stochastic ordering and order statistics are derived here. The estimation of unknown parameters is approached by the method of maximum likelihood, least squares, weighted least squares and maximum product spacings. An extensive simulation study is carried out to compare the long-run performance of the estimators. Applicability of the distribution is illustrated by means of three real data analyses over existing distributions.     

Efficient estimation in the Pareto distribution

The maximum likelihood estimation (MLE) of the probability density function (pdf) and cumulative distribution function (CDF) are derived for the Pareto distribution. It has been shown that MLEs are more efficient than uniform minimum variance unbiased estimators of pdf and CDF.     

LIKELIHOOD MOMENT ESTIMATION FOR THE GENERALIZED PARETO DISTRIBUTION

Traditional methods for estimating parameters in the generalized Pareto distribution have theoretical and computational defects. The moment estimator and the probability‐weighted moment estimator have low asymptotic efficiencies. They may not exist or may give nonsensical estimates. The maximum likelihood estimator, which sometimes does not exist, is asymptotically efficient, but its computation is complex and has convergence problems. The likelihood moment estimator is proposed, which is computationally easy and has high asymptotic efficiency.     

Wrapped Lindley distribution

In this article, we introduce a new circular distribution to be called as wrapped Lindley distribution and derive expressions for characteristic function, trigonometric moments, coefficients of skewness, and kurtosis. Method of maximum likelihood estimation is used for the estimation of parameters. We carry out a simulation study to show that the obtained maximum likelihood estimator is consistent. The proposed model is also applied to a real-life dataset, and its performance is compared with that of wrapped exponential distribution.     

Poisson–exponential distribution: different methods of estimation

In this study, we present different estimation procedures for the parameters of the Poisson–exponential distribution, such as the maximum likelihood, method of moments, modified moments, ordinary and weighted least-squares, percentile, maximum product of spacings, Cramer–von Mises and the Anderson–Darling maximum goodness-of-fit estimators and compare them using extensive numerical simulations. We showed that the Anderson–Darling estimator is the most efficient for estimating the parameters of the proposed distribution. Our proposed methodology was also illustrated in three real data sets related to the minimum, average and the maximum flows during October at São Carlos River in Brazil demonstrating that the PE distribution is a simple alternative to be used in hydrological applications.     

Exact expressions for the weights used in least-squares regression estimation for the log-logistic and Weibull distribution

Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting, and this technique outperforms the maximum likelihood (ML) estimation often in small samples. This leads to a highly heteroskedastic regression problem. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples, the ML performs best, but it is shown that in smaller samples, the weighted regression outperforms the ML estimation with respect to bias and mean square error.     

Identifiability and estimation of recursive max‐linear models

We address the identifiability and estimation of recursive max‐linear structural equation models represented by an edge‐weighted directed acyclic graph (DAG). Such models are generally unidentifiable and we identify the whole class of DAG s and edge weights corresponding to a given observational distribution. For estimation, standard likelihood theory cannot be applied because the corresponding families of distributions are not dominated. Given the underlying DAG, we present an estimator for the class of edge weights and show that it can be considered a generalized maximum likelihood estimator. In addition, we develop a simple method for identifying the structure of the DAG. With probability tending to one at an exponential rate with the number of observations, this method correctly identifies the class of DAGs and, similarly, exactly identifies the possible edge weights.     

Opportunities of the minimum Anderson–Darling estimator as a variant of the maximum likelihood method

We reveal that the minimum Anderson–Darling (MAD) estimator is a variant of the maximum likelihood method. Furthermore, it is shown that the MAD estimator offers excellent opportunities for parameter estimation if there is no explicit formulation for the distribution model. The computation time for the MAD estimator with approximated cumulative distribution function is much shorter than that of the classical maximum likelihood method with approximated probability density function. Additionally, we research the performance of the MAD estimator for the generalized Pareto distribution and demonstrate a further advantage of the MAD estimator with an issue of seismic hazard analysis.