A generalization for two-sided power distributions and adjusted method of moments

A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a new class of discrete distributions on {0, 1, …, k}. In addition, a new numerical method for parameter estimation using moments is given.     

Robust fitting of claim severity distributions and the method of trimmed moments

Many quantities arising in non-life insurance depend on claim severity distributions, which are usually modeled assuming a parametric form. Obtaining good estimates of the quantities, therefore, reduces to having good estimates of the model parameters. However, the notion of ‘good estimate’ depends on the problem at hand. For example, the maximum likelihood estimators (MLEs) are efficient, but they generally lack robustness. Since outliers are common in insurance loss data, it is therefore important to have a method that allows one to balance between efficiency and robustness. Guided by this philosophy, in the present paper we suggest a general estimation method that we call the method of trimmed moments (MTM). This method is appropriate for various model-fitting situations including those for which a close fit in one or both tails of the distribution is not required. The MTM estimators can achieve various degrees of robustness, and they also allow the decision maker to easily see the actions of the estimators on the data, which makes them particularly appealing. We illustrate these features with detailed theoretical analyses and simulation studies of the MTM estimators in the case of location–scale families and several loss distributions such as lognormal and Pareto. As a further illustration, we analyze a real data set concerning hurricane damages in the United States from 1925 to 1995.     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

Inferences on correlation coefficients of bivariate log-normal distributions

This article considers inference on correlation coefficients of bivariate log-normal distributions. We developed generalized confidence intervals and hypothesis tests for the correlation coefficients, and extended the results to compare two independent correlations. Simulation studies show that the suggested methods work well. Two practical examples are used to illustrate the application of the proposed methods.     

Inferences on the common mean of several heterogeneous log-normal distributions

We consider the problem of making inferences on the common mean of several heterogeneous log-normal populations. We apply the parametric bootstrap (PB) approach and the method of variance estimate recovery (MOVER) to construct confidence intervals for the log-normal common mean. We then compare the performances of the proposed confidence intervals with the existing confidence intervals via an extensive simulation study. Simulation results show that our proposed MOVER and PB confidence intervals can be recommended generally for different sample sizes and number of populations.     

Conditionally least-informative distributions: the case of the location parameter

We extend a basic result of Huber's on least favorable distributions to the setting of conditional inference, using an approach based on the notion of log-Gâteaux differentiation and perturbed models. Whereas Huber considered intervals of fixed width for location parameters and their average coverage rates, we study error models having longest confidence intervals, conditional on the location configuration of the sample. Our version of the problem does not have a global solution, but one that changes from configuration to configuration. Asymptotically, the conditionally least-informative shape minimizes the conditional Fisher information. We characterize the asymptotic solution within Huber's contamination model.     

Comparison between method of moments and entropy regularization algorithm applied to parameter estimation for mixed-Weibull distribution

Mixed-Weibull distribution has been used to model a wide range of failure data sets, and in many practical situations the number of components in a mixture model is unknown. Thus, the parameter estimation of a mixed-Weibull distribution is considered and the important issue of how to determine the number of components is discussed. Two approaches are proposed to solve this problem. One is the method of moments and the other is a regularization type of fuzzy clustering algorithm. Finally, numerical examples and two real data sets are given to illustrate the features of the proposed approaches.     

Some characterizations of distributions by truncated moments

In this paper we extend GUPTA'S (1975) resLilt and show that the constant value of a truncated moment characterizes the exponential distribution. Similar results are prov¬ed in the discrete case and it is shown that two consecutive factorial moments are enough to determine the distribution. However, under mild conditions, the constancy of one fac¬torial moment is enough to guarantee that the distribution is geometric. Considering the truncation on the right, a general method of obtaining the distribution, whenever £(h(X) | Xj) is known, is fxmbitud in tnr continuous and in the discrete case. Several of the known characterization theorems toilow trivially trorn our results     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

Inferences on the difference and ratio of the means of two inverse Gaussian distributions

Methods for interval estimation and hypothesis testing about the ratio of two independent inverse Gaussian (IG) means based on the concept of generalized variable approach are proposed. As assessed by simulation, the coverage probabilities of the proposed approach are found to be very close to the nominal level even for small samples. The proposed new approaches are conceptually simple and are easy to use. Similar procedures are developed for constructing confidence intervals and hypothesis testing about the difference between two independent IG means. Monte Carlo comparison studies show that the results based on the generalized variable approach are as good as those based on the modified likelihood ratio test. The methods are illustrated using two examples.     

Estimation for distributions with monotone likelihood ratio:case of vector valued parameter

An extension of a result about the estimation in Karlin and Rubin is given for the following case:The sample space, the parameter space and the decision space are subsets of a multi-dimensional Euclidean space, there is defined a suitable partial ordering in each of spaces, and a probability distribution has monotone likelihood ratio with respect to the partial orderings (see Ishii, 1976). In the special case when the loss function is quadratic a simple proof of a result in Karlin and Rubin is given. Stein's estimators are discussed as examples.     

The average likelihood and a fiducial approximation: one parameter members of the generalized gamma distributions

The average likelihood, defined as the integral of the like-lihood function over the parameter space, has been used as a criterion for model selection The form of the average likelihood considered uses a uniform prior. An approximation is presented based on fiducial distributions. The sampling distributions of the average likelihood and its fiducial approximation are derived for cases of sampling from one parameter members of the general-ized gamma distributions.     

The two-color urn model: recursions for the moments

For the two-color reinforcement-depletion urn model, with balancing reinforcement and depletion held constant over cycles, a recursive formula is given from which all factorial moments (for white balls, for example) can be determined. When the reinforcement of each color is positive, the stationary distribution of white balls (infinite number of cycles) turns out to be determined by three parameters. namely (i) the total number of balls in the urn, (ii) the richness of the reinforcement, or ratio of white ball reinforcement to total reinforcement, and (iii) the size of the white ball reinforcement. In addition, the distribution mimics the binomial (with less variance and skewness (√β₁:) ) and from formulas for the exact first four moments rapidly approaches normality. On the basis of the few cases studied, an approximating Gram-Charlier distribution with a binomial nucleus is only moderately successful     

Minimax estimators for the lower-bounded scale parameter of a location-scale family of distributions

This article is concerned with the minimax estimation of a scale parameter under the quadratic loss function where the family of densities is location-scale type. We obtain results for the case when the scale parameter is bounded below by a known constant. Implications for the estimation of a lower-bounded scale parameter of an exponential distribution are presented under unknown location. Furthermore, classes of improved minimax estimators are derived for the restricted parameter using the Integral Expression for Risk Difference (IERD) approach of Kubokawa (1994 Kubokawa, T. (1994). A unified approach to improving equivariant estimators. Ann. Stat. 22:290–299.[Crossref], [Web of Science ®] , [Google Scholar]). These classes are shown to include some existing estimators from literature.     

Remarks on the 'Bayesian' method of moments

Zellner has proposed a novel methodology for estimating structural parameters and predicting future observables based on two moments of a subjective distribution and the application of the maximum entropy principle-all in the absence of an explicit statistical model or likelihood function for the data. He calls his procedure the 'Bayesian method of moments' (BMOM). In a recent paper in this journal, Green and Strawderman applied the BMOM to a model for slash pine plantations. It is our view that there are inconsistencies between BMOM and Bayesian (conditional) probability, as we explain in this paper.     

Explicit formulas for the cumulants and the vector-valued odd moments of the multivariate linearly skewed elliptical distributions

In this letter explicit expressions are derived for the cumulants and the vector-valued odd moments of the multivariate linearly skewed elliptical family of distributions. The general calculations of such moments are described by multivariate integrals which complicate the calculations. We show how such multivariate computations can be projected into a univariate framework, which extremely simplifies the computations.     

Estimation of the smallest scale parameter of two-parameter exponential distributions

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.