Simos G. Meintanis

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.

Permutation tests for homogeneity based on the empirical characteristic function

In this paper, omnibus tests are proposed for testing the homogeneity of two populations. The tests are based on weighted integrals involving the empirical characteristic function. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight functions tends to infinity, the test statistics approach limit values which are related to moment differences between the two populations. The test procedure is based on resampling from the permutation distribution of the test statistic. The new tests are compared with other omnibus tests for homogeneity via a Monte Carlo procedure.

A Class of Omnibus Tests for the Laplace Distribution based on the Empirical Characteristic Function

Abstract In this paper a class of goodness-of-fit tests for the Laplace distribution is proposed. The tests are based on a weighted integral involving the empirical characteristic function. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity the test statistics approach limit values. In a particular case the resulting limit statistic is related to the first nonzero component of Neymans smooth test for this distribution. The new tests are compared with other omnibus tests for the Laplace distribution.

Tests for the multivariate k-sample problem based on the empirical characteristic function

Tests for the multivariate k-sample problem are considered. The tests are based on the weighted L2 distance between empirical characteristic functions, and afford an interesting interpretation in terms of a corresponding test statistic based on the L2 distance of pairs of non-parametric density estimators. Depending on the choice of weighting, a corresponding Dirac-type weight function reduces the test to a normalised version of the L2 distance between the sample means of the k populations. Theoretical and computational issues are considered, while the finite-sample implementation based on the permutation distribution of the test statistic shows that the new test performs well in comparison with alternative procedures of the change-point type.

Tests for normal mixtures based on the empirical characteristic function

A goodness-of-fit test for two-component homoscedastic and homothetic mixtures of normal distributions is proposed. The tests are based on a weighted L2-type distance between the empirical characteristic function and its population counterpart, where in the latter, parameters are replaced by consistent estimators. Consequently, the resulting tests are consistent against general alternatives. When moment estimation is employed and as the decay of the weight function tends to infinity the test statistics approach limit values, which are related to the first nonvanishing moment equation. The new tests are compared via simulation to other omnibus tests for mixtures of normal distributions, and are applied to several real data sets.

Fourier methods for testing multivariate independence

Recently a power study of some popular tests for bivariate independence based on ranks has been conducted. An alternative class of tests appropriate for testing not only bivariate, but also multivariate independence is developed, and their small-sample performance is studied. The test statistics employ the familiar equation between the joint characteristic function and the product of component characteristic functions, and may be written in a closed form convenient for computer implementation. Simulations on a distribution-free version of the new test statistic show that the proposed method compares well to standard methods of testing independence via the empirical distribution function. The methods are applied to multivariate observations incorporating data from several major stock-market indices. Issues pertaining to the theoretical properties of the new test are also addressed.

Omnibus tests for the error distribution in the linear regression model

Test procedures are constructed for testing the goodness-of-fit of the error distribution in the regression context. The test statistic is based on an L 2-type distance between the characteristic function of the (assumed) error distribution and the empirical characteristic function of the residuals. The asymptotic null distribution as well as the behavior of the test statistic under contiguous alternatives is investigated, while the issue of the choice of suitable estimators has been particularly emphasized. Theoretical results are accompanied by a simulation study.

Inference procedures for the Birnbaum-Saunders distribution and its generalizations

Goodness-of-fit tests are constructed for the two-parameter Birnbaum-Saunders distribution in the case where the parameters are unknown and are therefore estimated from the data. With each test the procedure starts by computing efficient estimators of the parameters. Then the data are transformed to normality and normality tests are applied on the transformed data, thereby avoiding reliance on parametric asymptotic critical values or the need for bootstrap computations. Two classes of tests are considered, the first class being the classical tests based on the empirical distribution function, while the other class utilizes the empirical characteristic function. All methods are extended to cover the case of generalized three-parameter Birnbaum-Saunders distributions.

Goodness-of-fit testing by transforming to normality: comparison between classical and characteristic function-based methods

Chen and Balakrishnan [Chen, G. and Balakrishnan, N., 1995, A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, 154–161] proposed an approximate method of goodness-of-fit testing that avoids the use of extensive tables. This procedure first transforms the data to normality, and subsequently applies the classical tests for normality based on the empirical distribution function, and critical points thereof. In this paper, we investigate the potential of this method in comparison to a corresponding goodness-of-fit test which instead of the empirical distribution function, utilizes the empirical characteristic function. Both methods are in full generality as they may be applied to arbitrary laws with continuous distribution function, provided that an efficient method of estimation exists for the parameters of the hypothesized distribution.

Tests for independence in non-parametric heteroscedastic regression models

Consistent procedures are constructed for testing independence between the regressor and the error in non-parametric regression models. The tests are based on the Fourier formulation of independence, and utilize the joint and the marginal empirical characteristic functions of the regressor and of estimated residuals. The asymptotic null distribution as well as the behavior of the test statistic under alternatives is investigated. A simulation study compares bootstrap versions of the proposed tests to corresponding procedures utilizing the empirical distribution function.