Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

Bayesian estimation for the Pareto income distribution

The Bayes estimators of the Gini index, the mean income and the proportion of the population living below a prescribed income level are obtained in this paper on the basis of censored income data from a pareto income distribution. The said estimators are obtained under the assumptions of a two-parameter exponential prior distribution and the usual squared error loss function. This work is also extended to the case when the income data are grouped and the exact incomes for the individuals in the population are not available. The method for the assessment of the hyperparameters is also outlined. Finally, the results are generalized for the doubly truncated gamma prior distribution. Now deceased.     

Reducing bias of the maximum-likelihood estimation for the truncated Pareto distribution

The Pareto distribution is an important distribution in statistics, which has been widely used in finance, physics, hydrology, geology, astronomy, and so on. Even though the parameter estimation for the Pareto distribution has been well established in the literature, the estimation problem for the truncated Pareto distribution becomes complex. This article investigates the bias and mean-squared error of the maximum-likelihood estimation for the truncated Pareto distribution, and some useful results are obtained.     

On estimation of the exponentiated Pareto distribution under different sample schemes

Bayes and classical estimators have been obtained for a two-parameter exponentiated Pareto distribution for when samples are available from complete, type I and type II censoring schemes. Bayes estimators have been developed under a squared error loss function as well as under a LINEX loss function using priors of non-informative type for the parameters. It has been seen that the estimators obtained are not available in nice closed forms, although they can be easily evaluated for a given sample by using suitable numerical methods. The performances of the proposed estimators have been compared on the basis of their simulated risks obtained under squared error as well as under LINEX loss functions.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

Parameter estimation for generalized Pareto distribution by generalized probability weighted moment-equations

The generalized Pareto distribution (GPD) has been widely used to model exceedances over a threshold. This article generalizes the method of generalized probability weighted moments, and applies this method to estimate the parameters of GPD. The estimator is computationally easy. Some asymptotic results of this method are provided. Two simulations are carried out to investigate the behavior of this method and to compare them with other methods suggested in the literature. The simulation results show that the performance of the proposed method is better than some other methods. Finally, this method is applied to analyze a real-life data.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

On detecting outliers in the Pareto distribution

In this paper, we introduce two new statistics for detecting outliers in the Pareto distribution. These new statistics are the extension of the statistics for detecting outliers in exponential and gamma distributions. In fact, we compare the power of our test statistics with the other statistics and select the best test statistic for detecting outliers in the Pareto distribution. Finally, numerical examples of different insurance claims are used to see the performance of the test.     

Parameter estimation for three-parameter generalized Pareto distribution by weighted non linear least squares

Generalized Pareto distribution (GPD) is widely used to model exceedances over thresholds. In this paper, we propose a new method, called weighted non linear least squares (WNLS), to estimate the parameters of the three-parameter GPD. Some asymptotic results of the proposed method are provided. An extensive simulation is carried out to evaluate the finite sample behaviour of the proposed method and to compare the behaviour with other methods suggested in the literature. The simulation results show that WNLS outperforms other methods in general situations. Finally, the WNLS is applied to analysis the real-life data.     

Classical estimation of hazard rate and mean residual life functions of Pareto distribution

Abstract

In this paper we find the maximum likelihood estimates (MLEs) of hazard rate and mean residual life functions (MRLF) of Pareto distribution, their asymptotic non degenerate distribution, exact distribution and moments. We also discuss the uniformly minimum variance unbiased estimate (UMVUE) of hazard rate function and MRLF. Finally, two numerical examples with simulated data and real data set, are presented to illustrate the proposed estimates.     

Comments on two papers concerning estimation of the parameters of the Pareto distribution in the presence of outliers

In this paper, we examine and correct various results relating to estimation of a Pareto distribution in the presence of outliers according to a model introduced by Dixit and Jabbari Nooghabi (2011) [1] and further studied by Dixit and Jabbari Nooghabi (2011) [2]. In particular, Dixit and Jabbari Nooghabi (2011) [2] state that the maximum likelihood estimators for the parameters appearing in their model do not exist. We show that these estimators can in fact exist, and we present and illustrate a method for determining them when they do. Two numerical illustrations using actual insurance data are included.     

Efficient estimation of the PDF and the CDF of the inverse Rayleigh distribution

In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.     

On the mixtures of Weibull and Pareto (IV) distribution: An alternative to Pareto distribution

Finite mixture models have provided a reasonable tool to model various types of observed phenomena, specially those which are random in nature. In this article, a finite mixture of Weibull and Pareto (IV) distribution is considered and studied. Some structural properties of the resulting model are discussed including estimation of the model parameters via expectation maximization (EM) algorithm. A real-life data application exhibits the fact that in certain situations, this mixture model might be a better alternative than the rival popular models.     

Extended generalised Pareto models for tail estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.     

Estimation of the exponential Pareto II distribution parameters

In this article, we estimate the parameters of exponential Pareto II distribution by two new methods. The first one is based on the principle of maximum entropy (POME) and the second is by Kullback–Leibler divergence of survival function (KLS). Monte Carlo simulated data are used to evaluate these methods and compare them with the maximum likelihood method. Finally, we fit this distribution to a set of real data by estimation procedures.     

Entropy-based goodness-of-fit tests for the Pareto I distribution

In this article, having observed the generalized order statistics in a sample, we construct a test for the hypothesis that the underlying distribution is the Pareto I distribution. The Shannon entropy of generalized order statistics is used to test the null hypothesis.