Goodness-of-fit test for density estimation

It is often necessary to test whether X,…, X_n are from a certain density f(x) or not. Most test statistics such as the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling statistics are based on the empirical distribution function F(x). In this paper we suggest a test statistic based on the integrated squared error of the kernel density estimator. We derive the asymptotic distribution of the statistic under the null and alternative hypothesis. Some simulation results for power comparisons are also given.     

A new goodness of fit test for the equality of multinomial cell probabilities versus trend alternatives

In this paper, a new test statistic is presented for testing the null hypothesis of equal multinomial cell probabilities versus various trend alternatives. Exact asymptotic critical values are obtained, The power of the test is compared with several other statistics considered by Choulakian et al (1995), The test is shown to have better power for certain trend alternatives.     

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.     

MINIMUM DISTANCE and INDUCED STATISTICS FOR TESTING INEQUALITIES

Minimum distance statistics for testing inequality restrictions on the mean vector of a multivariate normal population are considered and are shown to be equivalent to certain induced (or intersection) test statistics. Tests based on these statistics are shown to be admissible.     

A Class of Goodness-of-fit Tests Based on Transformation

There is an increasing number of goodness-of-fit tests whose test statistics measure deviations between the empirical characteristic function and an estimated characteristic function of the distribution in the null hypothesis. With the aim of overcoming certain computational difficulties with the calculation of some of these test statistics, a transformation of the data is considered. To apply such a transformation, the data are assumed to be continuous with arbitrary dimension, but we also provide a modification for discrete random vectors. Practical considerations leading to analytic formulas for the test statistics are studied, as well as theoretical properties such as the asymptotic null distribution, validity of the corresponding bootstrap approximation, and consistency of the test against fixed alternatives. Five applications are provided in order to illustrate the theory. These applications also include numerical comparison with other existing techniques for testing goodness-of-fit.     

The union-intersection principle applied to the test of dimensionality

In the first section Anderson-Rao-Fujikoshi's test statistics for testing the hypothesis of dimensionality are reviewed and then Olkin-Tomsky's generalized union-intersection principle is applied to show that a new class of test statistics for testing the hypothesis of dimensionality are derived which includes the likelihood ratio test statistics, the trace test statistics and a version of ROY'S maximum root test statistics.     

Logistic Regression and Discriminant Analysis by Ordinary Least Squares

If the observations for fitting a polytomous logistic regression model satisfy certain normality assumptions, the maximum likelihood estimates of the regression coefficients are the discriminant function estimates. This article shows that these estimates, their unbiased counterparts, and associated test statistics for variable selection can be calculated using ordinary least squares regression techniques, thereby providing a convenient method for fitting logistic regression models in the normal case. Evidence is given indicating that the discriminant function estimates and test statistics merit wider use in nonnormal cases, especially in exploratory work on large data sets.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

Some Results on the Control of the False Discovery Rate under Dependence

Abstract.  Controlling the false discovery rate (FDR) is a powerful approach to multiple testing, with procedures developed with applications in many areas. Dependence among the test statistics is a common problem, and many attempts have been made to extend the procedures. In this paper, we show that a certain degree of dependence is allowed among the test statistics, when the number of tests is large, with no need for any correction. We then suggest a way to conservatively estimate the proportion of false nulls, both under dependence and independence, and discuss the advantages of using such estimators when controlling the FDR.     

On the asymptotic normality of rank tests for independence

We show that the commonly used linear rank tests for independence have certain continuity properties with respect to the score functions which are applied to the ranks. Using these properties, we derive the asymptotic normality of the test statistics under general conditions. These conditions are closely related to those under which simple linear rank statistics are known to be asymptotically normal.     

Tests for high-dimensional covariance matrices using the theory of U-statistics

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p?n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.     

Tests of Symmetry in Three-dimensional Contingency Tables Based on Phi-divergence Statistics

In this paper we introduce a family of test statistics for testing complete symmetry in three-dimensional contingency tables based on phi- divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistics as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to show that some new statistics offer an attractive alternative to the classical Pearson and likelihood ratio test statistics for this problem of complete symmetry.     

Model Checks for Generalized Linear Models

In this paper we propose and study non-parametric tests for the validity of (composite) Generalized Linear Models with a given parametric link structure, which are based on certain empirical processes marked by the residuals. When properly transformed to their innovation part the resulting test statistics are distribution-free. The method perfectly adapts to a situation, when also the input vector follows a dimension reducing model.     

Computing asymptotic p-values for EDF tests

In this paper the problem of computingp-values for the asymptotic distribution of certain goodness-of-fit test statistics based on the empirical distribution is approached via quadrature. Through examples it is shown that this approach can lead to considerable time savings over the standard practice of discretizing the underlying eigenvalue problem.     

Multivariate analysis with an autoregressive covariance model

This paper addresses the problem of testing the multivariate linear hypothesis when the errors follow an antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as a nonstationary autoregressive model of general order. Three test statistics are derived that provide analogs to three commonly used MANOVA statistics: Wilks' Lambda, the Lawley-Hotelling Trace, and Pillai's Trace. Formulas are given for each of these statistics that show how they can be obtained From any statistical computing package that calculates the usual MANOVA statistics. These antedependent statistics would be appropriate in analyzing certain multivariate data sets in which repeated measurements are taken on the same subjects over a period of time.     

A Class of Efficient Non-parametric K-sample Tests for Multivariate Counting Processes with Frailties

A K -sample testing problem is studied for multivariate counting processes with time-dependent frailty. Asymptotic distributions and efficiency of a class of non-parametric test statistics are established for certain local alternatives. The concept of efficiency is to show that for every non-parametric test in this class, there is a parametric submodel for which the optimal test has the same asymptotic power as the non-parametric one. The theory is applied to analyse a diabetic retinopathy study data set. A simulation study is also presented to illustrate the theory     

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints.

In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

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.     

GNR, MGR, and exact misspeclfication testing

The Gauss-Newton regression (GNR) is widely used to compute Lagrange multiplier statistics. A regression described by Milliken and Graybill yields an exact F test in a certain class of nonlinear models which are linear under the null. This paper shows that the Milliken-Graybill regression is a GNR. Hence one interpretation of Milliken-Graybill is that they identified a class of nonlinear models for which the GNR yields an exact test.