Ioannis A. Koutrouvelis

Estimation in the Pearson type 3 distribution

Although the Pearson type 3 (P3) is one of the basic models in statistical hydrology, its use to model untransformed data has been restrained because of difficulties encountered in fitting this distribution by traditional methods. In this paper an adaptive estimation procedure of mixed moments for the P3 family is introduced which is based on several fractional moments of the exponentially transformed data and the mean of the original data. The procedure is easy to implement in small samples and is valid over the entire parameter space. Explicit formulae for the variances and covariances of parameter estimators and of the variance of the T-year event are derived. In addition, two variants of the new procedure are compared with two versions of the method of moments and a version of the method of conditional moments via Monte Carlo simulation. With samples generated from P3 populations, it is found that one of the variants of the new procedure is the best overall method in estimating 100-year flood events, and the other variant is best in estimating the median and 10-year low-flow events. The good performance of these two variants is also observed in samples generated from alternatives to P3 distributions. A modification of the procedure is also introduced and investigated when a prior assumption of positive skewness is adopted.

Estimation in the three-parameter inverse Gaussian distribution

A mixed moments method for the estimation of parameters in the three-parameter inverse Gaussian distribution (IG3) is introduced. The method is an adaptive iterative procedure, which combines the method of moments with a regression method based on the empirical moment generating function. Monte Carlo results indicate that the new procedure is more efficient than alternative estimation methods (including the maximum likelihood) over large portions of the parameter space with samples of small or moderate size. Asymptotic results are also obtained and may be used to draw approximate inferences with small samples. Two data sets are used to illustrate estimation and testing procedures and to construct exploratory graphs for the appropriateness of the IG3 model.

Testing for stability based on the empirical characteristic funstion with applications to financial data

Stable distributions have been widely employed in the last thirty five years to model financial asset returns and other economic variables. However, very few methods have been proposed to test the composite hypothesis of stability and most of these on merely an ad hoc basis. This article presents tests of stability which possess a large-sample level of significance. All tests use the empirical characteristic function. The empirical level and power of the tests are investigated in a series of Monte Carlo experiments which employ models that are popular in the financial literature. The application of the proposed methods to real data leads to reconsideration of prior beliefs regarding the distribution of certain types of financial data.

Estimation in the three-parameter gamma distribution based on the empirical moment generation function

The empirical moment generating function is used for the estimation of the shape, scale, and location parameters of a three-parameter gamma distribution. The proposed method is valid over the entire parameter space and avoids the difficulties associated with maximum likelihood estimation when the sample has a very large positive skewness. In addition, finite sample results from a simulation study indicate that the new procedure is more accurate than recently proposed modifications of both moment and maximum likelihood methods over important portions of the parameters space.

Testing the Fit of Gamma Distributions Using the Empirical Moment Generating Function

ABSTRACT This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.

Cumulant Plots for Assessing the Gamma Distribution

This article introduces graphical procedures for assessing the fit of the gamma distribution. The procedures are based on a standardized version of the cumulant generating function. Plots with bands of 95% simultaneous confidence level are developed by utilizing asymptotic and finite-sample results. The plots have linear scales and do not rely on the use of tables or values of special functions. Further, it is found through simulation, that the goodness-of-fit test implied by these plots compares favorably with respect to power to other known tests for the gamma distribution in samples drawn from lognormal and inverse Gaussian distributions.

Theory & Methods: Estimating the parameters of Poisson-exponential models

This paper proposes two methods of estimation for the parameters in a Poisson-exponential model. The proposed methods combine the method of moments with a regression method based on the empirical moment generating function. One of the methods is an adaptation of the mixed-moments procedure of Koutrouvelis & Canavos (1999). The asymptotic distribution of the estimator obtained with this method is derived. Finite-sample comparisons are made with the maximum likelihood estimator and the method of moments. The paper concludes with an exploratory-type analysis of real data based on the empirical moment generating function.

Cumulant plots and goodness-of-fit tests for the inverse Gaussian distribution

This paper uses a standardized version of the logarithm of the empirical moment generating function in order to construct plots for assessing the appropriateness of the inverse Gaussian distribution. Variability is added to the plots by utilizing asymptotic and finite-sample results. The plots have linear scales and do not rely on the use of tables or special functions. In addition, they are equivalent to a goodness-of-fit test whose critical values are obtained from fitted equations involving the sample size and the estimated shape parameter of the inverse Gaussian distribution. Three data sets are used to illustrate the plots. A similar test is also proposed whose critical values are found through parametric bootstrap. An extensive simulation study shows that the new tests maintain good stability in level and high power across a wider range of distributions and sample sizes than other tests.

Percentile Estimation in Inverse Gaussian Distributions

This paper presents procedures for percentile estimation in the three-parameter inverse Gaussian (IG3) and the two-parameter inverse Gaussian (IG2) distributions. All procedures require first the estimation of distribution parameters and second the computation of the desired quantile at the estimated parameters. Parameter estimation is accomplished by maximum likelihood (ML) or a mixed moments (MXM) method. A Newton-Rahpson (NR) procedure is used for inverting the CDF. Simulation and asymptotic results are given for the resulting estimators. Two sets of real data are used to illustrate the procedures.

TESTING THE FIT TO GENERALIZED POISSON DISTRIBUTIONS BASED ON AN EMPIRICAL TRANSFORM

Generalized Poisson distributions appear as applied-research models in many fields. For example in reliability, the total amount of wear of items, the hazard rate and the total time to failure can be modeled after a Generalized Poisson distribution. Methods of Statistical Inference for such distributions have been scarce since the corresponding distribution functions come in complicated, often not closed form, expressions. In this article, we present a method for testing the goodness-of-fit to any specified member of the family of Generalized Poisson distributions. The proposed method utilizes the general form of the moment generating function of Generalized Poisson distributions. The asymptotic distribution of the test statistics is derived when the parameters of the generalizing distribution are assumed known as well as unknown. The performance of the procedures is investigated by employing real and simulated data.