Tests for the multivariate two-sample problem based on empirical probability measures

Nonparametric tests are proposed for the equality of two unknown p-variate distributions. Empirical probability measures are defined from samples from the two distributions and used to construct test statistics as the supremum of the absolute differences between empirical probabilities, the supremum being taken over all possible events. The test statistics are truly multivariate in not requiring the artificial ranking of multivariate observations, and they are distribution-free in the general p-variate case. Asymptotic null distributions are obtained. Powers of the proposed tests and a competitor are examined by Monte Carlo techniques.     

Comparisons of some tests for validation of stochastic simulation models

Properties of proposed parametric and nonparametric tests for the problem of validating stochastic simulation models are discussed. Asymptotic efficiencies for various types of alternatives are developed for some of the tests. Numerical procedures along with simulation methods are used to evaluate the adequacy of normal approximations to the null distributions of the statistics and the small-sample power of the tests.     

Second order asymptotic and non-asymptotic optimality properties of combined tests

Let p independent test statistics be available to test a null hypothesis concerned with the same parameter. The p are assumed to be similar tests. Asymptotic and non-asymptotic optimality properties of combined tests are studied. The asymptotic study centers around two notions. The first is Bahadur efficiency. The second is based on a notion of second order comparisons. The non-asymptotic study is concerned with admissibility questions. Most of the popular combining methods are considered along with a method not studied in the past. Among the results are the following: Assume each of the p statistics has the same Bahadur slope. Then the combined test based on the sum of normal transforms, is asymptotically best among all tests studied, by virtue of second order considerations. Most of the popular combined tests are inadmissible for testing the noncentrality parameter of chi-square, t, and F distributions. For chi-square a combined test is offered which is admissible, asymptotically optimal (first order), asymptotically optimal (second order) among all tests studied, and for which critical values are obtainable in special cases. Extensions of the basic model are given.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

This paper proposes a class of non‐parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t₀’ than G. Using Hoeffding's two‐sample U‐statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution‐free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

Asymptotic normality of MRPP statistics from invariance principles of u-statistics

Multi-response permutation procedures (MRPP) were recently introduced to test differences between a priori classified groups of objects ( Mielke, Berry Johnson, 1976; Mielke, 1979 ). The null distributions of the MRPP statistics were initially conjectured to be asymptotically normal for some specified conditions within the setting of a sequence of finite populations due to Madow ( 1948 ).

Asymptotic normality of a class of MRPP statistics (under the null hypothesis) is shown in two cases: (i) the setting which considers the populations to be the samples resulting from sequential independent identically distributed (i.i.d.) sampling (sampling from infinite populations) and (ii) the setting of a sequence of increasingly large finite populations (sampling from finite populations). The results are direct applications of the weak convergence of a U-statistic process in the i.i.d. case to a Brownian motion (Bhattacharyya and Sen, 1977) and of the weak convergence of a U-statistic process in the finite populations case to a Brownian bridge (Sen, 1972). The conditions are milder for the i.i.d. case than for the finite populations case. However, neither case provides a restriction of a practical consequence in applications of MRPP. In either case, convergence is shown to depend on the asymptotic ratios of the group sizes to the population size.     

Testing for spherical symmetry via the empirical characteristic function

Kolmogorov–Smirnov-type and Cramér–von Mises-type goodness-of-fit tests are proposed for the null hypothesis that the distribution of a random vector X is spherically symmetric. The test statistics utilize the fact that X has a spherical symmetric distribution if, and only if, the characteristic function of X is constant over surfaces of spheres centred at the origin. Both tests come in convenient forms that are straightforwardly applicable with the computer. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. Since both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of X, a conditional Monte Carlo procedure is used to actually carry out the tests. Results on the behaviour of the test in finite-samples are included along with a real-data example.     

A note on a family of criteria for evaluating test statistics

ABSTRACT

In noting that the usual criteria for choosing an optimal test, Uniform Power and Local Power are at opposite ends of a spectrum of dominance criteria, a complete “Power Dominance” family of criteria for classifying and choosing optimal tests on the basis of their power characteristics is identified, wherein successive orders of dominance attach increasing weight to power close to the null hypothesis. Indices of the extent to which a preferred test has superior power characteristics over other members in its class, and an index of the proximity of a test to the envelope function of alternative tests are also provided. The ideas are exemplified using various optimal test statistics for Normal and Laplace population distributions.     

Test of fit for Marshall–Olkin distributions with applications

Marshall and Olkin [1967. A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44], introduced a bivariate distribution with exponential marginals, which generalizes the simple case of a bivariate random variable with independent exponential components. The distribution is popular under the name ‘Marshall–Olkin distribution’, and has been extended to the multivariate case. L2-type statistics are constructed for testing the composite null hypothesis of the Marshall–Olkin distribution with unspecified parameters. The test statistics utilize the empirical Laplace transform with consistently estimated parameters. Asymptotic properties pertaining to the null distribution of the test statistic and the consistency of the test are investigated. Theoretical results are accompanied by a simulation study, and real-data applications.     

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.     

Multivariate generalizations of jonckheere's test for ordered alternatives

In many dose-response studies, each of several independent groups of animals is treated with a different dose of a substance. Many response variables are then measured on each animal. The distributions of the response variables may be nonnormal, and Jonckheere's (1954) test for ordered alternatives in the one-way layout is sometimes used to test whether the level of a single variable increases with increasing dose. In some applications, however, it is important to consider a set of response variables simultaneously. For instance, an increase in each of certain enzymes in the blood serum may suggest liver damage. To test whether these enzyme levels increase with increasing dose, it may be preferable to consider these enzymes as a group, rather than individually.

I propose two multivariate generalizations of Jonckheere's univariate test. Each multivariate test statistic is a function of coordinate-wise Jonckheere statistics—one a sum, the other a quadratic form. The sum statistic can be used to test the alternative hypothesis that each variable is stochastically increasing with increasing dose. The quadratic form statistic is designed for the more general alternative hypothesis that each variable is stochastically ordered with increasing dose.

For each of these two alternatives, I also propose a multivariate generalization of a normal theory test described by Puri (1965). I examine the asymptotic distributions of the four test statistics under the null hypothesis and under translation alternatives and compare each distribution-free test to the corresponding normal theory test in terms of asymptotic relative efficiency.

The multivariate Jonckheere tests are illustrated using does-response data from a subchronic toxicology study carried out by the National Toxicology Program. Four groups of ten male rats each were treated with increasing doses of vinylidene flouride, and the serum enzymes SDH, SGOT, and SGPT were measured. A comparison of univariate Jonckheere tests on each variable, bivariate tests on SDH and SGOT, and multivariate tests on all three variables gives insight into the behavior of the various procedures.     

The asymptotic distribution of the correlation coefficient in testing fit to the exponential distribution

The asymptotic distribution of certain tests of fit to the exponential distribution is obtained. The tests are based on regression of the order statistics on their expectations under a standard exponential distribution. Asymptotic normality at the rate (log n)^1/2 is obtained for a family of statistics including the correlation coefficient.     

Tests on the Equality of Two Multiple Correlation Coefficients

Asymptotic tests are suggested for testing the equality of two multiple correlation coefficients calculated from a single sample from a multivariate normal distribution. An F test is possible only when the two dependent variables coincide and one set of independent variables is a subset of the second set. Tests are compared by simulation for situations in which the F test is inapplicable. Special attention is paid to cases in which asymptotic normality of the test statistics does not hold.     

On the asymptotic behaviour of location-scale invariant Bickel–Rosenblatt tests

Location-scale invariant Bickel–Rosenblatt goodness-of-fit tests (IBR tests) are considered in this paper to test the hypothesis that f, the common density function of the observed independent d-dimensional random vectors, belongs to a null location-scale family of density functions. The asymptotic behaviour of the test procedures for fixed and non-fixed bandwidths is studied by using an unifying approach. We establish the limiting null distribution of the test statistics, the consistency of the associated tests and we derive its asymptotic power against sequences of local alternatives. These results show the asymptotic superiority, for fixed and local alternatives, of IBR tests with fixed bandwidth over IBR tests with non-fixed bandwidth.     

Asymptotic spacings theory with applications to the two-sample problem

The asymptotic distribution theory of test statistics which are functions of spacings is studied here. Distribution theory under appropriate close alternatives is also derived and used to find the locally most powerful spacing tests. For the two-sample problem, which is to test if two independent samples are from the same population, test statistics which are based on “spacing-frequencies” (i.e., the numbers of observations of one sample which fall in between the spacings made by the other sample) are utilized. The general asymptotic distribution theory of such statistics is studied both under the null hypothesis and under a sequence of close alternatives.     

Decisions on Seasonal Unit Roots

Decisions on the presence of seasonal unit roots in economic time series are commonly taken on the basis of statistical hypothesis tests. Some of these tests have absence of unit roots as the null hypothesis, while others use unit roots as their null. Following a suggestion by Hylleberg (1995) to combine such tests in order to reach a clearer conclusion, we evaluate the merits of such test combinations on the basis of a Bayesian decision setup. We find that the potential gains over a pure application of the most common test due to Hylleberg et al. (1990) can be small.     

On the asymptotic distribution of an estimate of the change point in a failure rate

The failure rate r(t) is assumed to have the shape of the"first"part of the"bathtub"model, i.e.r(t) is non-increasing for t<r and is constant for t> r. Asymptotic distribution of one of the estimates proposed earlier has been investigated in this paper. This leads to a test for the hypothesis HQ r<r ₀ vs H :r>r (where TQ > 0). Asymptotic expression for the power of this test under Pitman alternatives is derived. Some simulations are reported.     

Goodness-of-fit tests for location–scale families based on random distance

For location–scale families, we consider a random distance between the sample order statistics and the quasi sample order statistics derived from the null distribution as a measure of discrepancy. The conditional qth quantile and expectation of the random discrepancy on the given sample are chosen as test statistics. Simulation results of powers against various alternatives are illustrated under the normal and exponential hypotheses for moderate sample size. The proposed tests, especially the qth quantile tests with a small or large q, are shown to be more powerful than other prominent goodness-of-fit tests in most cases.