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.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

On similarity and independence properties of composite edf goodness-of-fit tests

Many test statistics for classical simple goodness-of-fit hypothesis testing problems are distancemeasures between the distribution function of the null hypothesis distributipn and the empirical distribution function sometimes called EDF tests. If a composite parametric null hypothesis is considered in place of the simple null hypothesis, then a test statistic can be obtained from each EDF test by replacing the known distribution function of the simple problem by the Rao-Blackwell estimating distribution function. In this note we use known results to show that these Rao-Blackwell-EDF test statistics have distributions that do not depend upon parameter values, and hence that these tests are independent of a complete sufficient statistic for the parameters.     

Tests for Comparing Mark-Specific Hazards and Cumulative Incidence Functions

It is of interest in some applications to determine whether there is a relationship between a hazard rate function (or a cumulative incidence function) and a mark variable which is only observed at uncensored failure times. We develop nonparametric tests for this problem when the mark variable is continuous. Tests are developed for the null hypothesis that the mark-specific hazard rate is independent of the mark versus ordered and two-sided alternatives expressed in terms of mark-specific hazard functions and mark-specific cumulative incidence functions. The test statistics are based on functionals of a bivariate test process equal to a weighted average of differences between a Nelson-Aalen-type estimator of the mark-specific cumulative hazard function and a nonparametric estimator of this function under the null hypothesis. The weight function in the test process can be chosen so that the test statistics are asymptotically distribution-free. Asymptotically correct critical values are obtained through a simple simulation procedure. The testing procedures are shown to perform well in numerical studies, and are illustrated with an AIDS clinical trial example. Specifically, the tests are used to assess if the instantaneous or absolute risk of treatment failure depends on the amount of accumulation of drug resistance mutations in a subject's HIV virus. This assessment helps guide development of anti-HIV therapies that surmount the problem of drug resistance.     

A general theory of hypothesis testing based on rankings

In nonparametric statistics, a hypothesis testing problem based on the ranks of the data gives rise to two separate permutation sets corresponding to the null and to the alternative hypothesis, respectively. A modification of Critchlow's unified approach to hypothesis testing is proposed. By defining the distance between permutation sets to be the average distance between pairs of permutations, one from each set, various test statistics are derived for the multi-sample location problem and the two-way layout. The asymptotic distributions of the test statistics are computed under both the null and alternative hypotheses. Some comparisons are made on the basis of the asymptotic relative efficiency.     

Testing asymptotic independence in bivariate extremes

Bivariate extreme value condition (see (1.1) below) includes the marginal extreme value conditions and the existence of the (extreme) dependence function. Two cases are of interest: asymptotic independence and asymptotic dependence. In this paper, we investigate testing the existence of the dependence function under the null hypothesis of asymptotic independence and present two suitable test statistics. Small simulations are studied and the application for a real data is shown. The other case with the null hypothesis of asymptotic dependence is already investigated.     

Tests for time series of counts based on the probability-generating function

We propose testing procedures for the hypothesis that a given set of discrete observations may be formulated as a particular time series of counts with a specific conditional law. The new test statistics incorporate the empirical probability-generating function computed from the observations. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is included as well as real-data examples.     

Goodness-of-fit tests for one-shot device accelerated life testing data

In this article, we develop a formal goodness-of-fit testing procedure for one-shot device testing data, in which each observation in the sample is either left censored or right censored. Such data are also called current status data. We provide an algorithm for calculating the nonparametric maximum likelihood estimate (NPMLE) of the unknown lifetime distribution based on such data. Then, we consider four different test statistics that can be used for testing the goodness-of-fit of accelerated failure time (AFT) model by the use of samples of residuals: a chi-square-type statistic based on the difference between the empirical and expected numbers of failures at each inspection time; two other statistics based on the difference between the NPMLE of the lifetime distribution obtained from one-shot device testing data and the distribution specified under the null hypothesis; as a final statistic, we use White's idea of comparing two estimators of the Fisher Information (FI) to propose a test statistic. We then compare these tests in terms of power, and draw some conclusions. Finally, we present an example to illustrate the proposed tests.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Robust testing with generalized partial linear models for longitudinal data

By approximating the nonparametric component using a regression spline in generalized partial linear models (GPLM), robust generalized estimating equations (GEE), involving bounded score function and leverage-based weighting function, can be used to estimate the regression parameters in GPLM robustly for longitudinal data or clustered data. In this paper, score test statistics are proposed for testing the regression parameters with robustness, and their asymptotic distributions under the null hypothesis and a class of local alternative hypotheses are studied. The proposed score tests reply on the estimation of a smaller model without the testing parameters involved, and perform well in the simulation studies and real data analysis conducted in this paper.     

Some asymptotic results for hypothesis testing in multivariate failure time data

This paper presents a class of generalized Wald, generalized score and generalized likelihood ratio statistics for hypothesis testing and model selection for multivariate failure time data. These statistics are based on a marginal hazard model with a common baseline hazard function. The large sample distributions of these statistics are examined. It is shown that the proposed test statistics follow asymptotically a weighted sum of independent χ₁² distributions.     

Non parametric hypothesis tests for comparing reliability functions

In reliability and related disciplines, comparing reliability functions of two (or more) aging processes is a crucial step in the process of determining reliability and understanding an aging process. The aim of this paper is to propose a non parametric statistical methodology to compare two populations based on their mean residual life function and expected inactivity time function. We introduce some novel hypothesis testing procedures that involve both Cramér–von Mises- and Kolmogorov–Smirnov-type test statistics and their decision rules are constructed based on the asymptotic distributions of these test statistics and bootstrapping method. We study the practical behavior of the proposed testing procedures extensively through simulations. The results reveal that the proposed hypothesis testing procedures perform efficiently in identifying small and large differences. Two real-life examples are discussed to demonstrate the practical utility of the tests.     

How Important Is the Effect of Rounding in Goodness-of-Fit

Abstract

In the area of goodness-of-fit there is a clear distinction between the problem of testing the fit of a continuous distribution and that of testing a discrete distribution. In all continuous problems the data is recorded with a limited number of decimals, so in theory one could say that the problem is always of a discrete nature, but it is a common practice to ignore discretization and proceed as if the data is continuous. It is therefore an interesting question whether in a given problem of test of fit, the “limited resolution” in the observed recorded values may be or may be not of concern, if the analysis done ignores this implied discretization. In this article, we address the problem of testing the fit of a continuous distribution with data recorded with a limited resolution. A measure for the degree of discretization is proposed which involves the size of the rounding interval, the dispersion in the underlying distribution and the sample size. This measure is shown to be a key characteristic which allows comparison, in different problems, of the amount of discretization involved. Some asymptotic results are given for the distribution of the EDF (empirical distribution function) statistics that explicitly depend on the above mentioned measure of degree of discretization. The results obtained are illustrated with some simulations for testing normality when the parameters are known and also when they are unknown. The asymptotic distributions are shown to be an accurate approximation for the true finite n distribution obtained by Monte Carlo. A real example from image analysis is also discussed. The conclusion drawn is that in the cases where the value of the measure for the degree of discretization is not “large”, the practice of ignoring discreteness is of no concern. However, when this value is “large”, the effect of ignoring discreteness leads to an exceded number of rejections of the distribution tested, as compared to what would be the number of rejections if no rounding is taking into account. The error made in the number of rejections might be huge.     

Robust inference from multiple test statistics via permutations: a better alternative to the single test statistic approach for randomized trials

Formal inference in randomized clinical trials is based on controlling the type I error rate associated with a single pre‐specified statistic. The deficiency of using just one method of analysis is that it depends on assumptions that may not be met. For robust inference, we propose pre‐specifying multiple test statistics and relying on the minimum p‐value for testing the null hypothesis of no treatment effect. The null hypothesis associated with the various test statistics is that the treatment groups are indistinguishable. The critical value for hypothesis testing comes from permutation distributions. Rejection of the null hypothesis when the smallest p‐value is less than the critical value controls the type I error rate at its designated value. Even if one of the candidate test statistics has low power, the adverse effect on the power of the minimum p‐value statistic is not much. Its use is illustrated with examples. We conclude that it is better to rely on the minimum p‐value rather than a single statistic particularly when that single statistic is the logrank test, because of the cost and complexity of many survival trials. Copyright © 2013 John Wiley & Sons, Ltd.     

Tests for Coefficients in High‐dimensional Additive Hazard Models

We consider hypothesis testing problems for low‐dimensional coefficients in a high dimensional additive hazard model. A variance reduced partial profiling estimator (VRPPE) is proposed and its asymptotic normality is established, which enables us to test the significance of each single coefficient when the data dimension is much larger than the sample size. Based on the p‐values obtained from the proposed test statistics, we then apply a multiple testing procedure to identify significant coefficients and show that the false discovery rate can be controlled at the desired level. The proposed method is also extended to testing a low‐dimensional sub‐vector of coefficients. The finite sample performance of the proposed testing procedure is evaluated by simulation studies. We also apply it to two real data sets, with one focusing on testing low‐dimensional coefficients and the other focusing on identifying significant coefficients through the proposed multiple testing procedure.     

Maximizing a Family of Optimal Statistics over a Nuisance Parameter with Applications to Genetic Data Analysis

In this article, a simple algorithm is used to maximize a family of optimal statistics for hypothesis testing with a nuisance parameter not defined under the null hypothesis. This arises from genetic linkage and association studies and other hypothesis testing problems. The maximum of optimal statistics over the nuisance parameter space can be used as a robust test in this situation. Here, we use the maximum and minimum statistics to examine the sensitivity of testing results with respect to the unknown nuisance parameter. Examples from genetic linkage analysis using affected sub pairs and a candidate-gene association study in case-parents trio design are studied.     

A Family of Goodness‐of‐Fit Tests for Copulas Based on Characteristic Functions

A general class of rank statistics based on the characteristic function is introduced for testing goodness‐of‐fit hypotheses about the copula of a continuous random vector. These statistics are defined as L ₂ weighted functional distances between a nonparametric estimator and a semi‐parametric estimator of the characteristic function associated with a copula. It is shown that these statistics behave asymptotically as degenerate V ‐statistics of order four and that the limit distributions have representations in terms of weighted sums of independent chi‐square variables. The consistency of the tests against general alternatives is established and an asymptotically valid parametric bootstrap is suggested for the computation of the critical values of the tests. The behaviour of the new tests in small and moderate sample sizes is investigated with the help of simulations and compared with a competing test based on the empirical copula. Finally, the methodology is illustrated on a five‐dimensional data set.     

Inference for longitudinal data from complex sampling surveys: An approach based on quadratic inference functions

We propose a survey weighted quadratic inference function method for the analysis of data collected from longitudinal surveys, as an alternative to the survey weighted generalized estimating equation method. The procedure yields estimators of model parameters, which are shown to be consistent and have a limiting normal distribution. Furthermore, based on the inference function, a pseudolikelihood ratio type statistic for testing a composite hypothesis on model parameters and a statistic for testing the goodness of fit of the assumed model are proposed. We establish their asymptotic distributions as weighted sums of independent chi‐squared random variables and obtain Rao‐Scott corrections to those statistics leading to a chi‐squared distribution, approximately. We examine the performance of the proposed methods in a simulation study.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

OPTIMAL TESTS OF SIGNIFICANCE

To perform a test of significance of a null hypothesis, a test statistic is chosen which is expected to be small if the hypothesis is false. Then the significance level of the test for an observed sample is the probability that the test statistic, under the assumptions of the hypothesis, is as small, or smaller than, its observed value. A "good" test statistic is taken to be one which is stochastically small when the null hypothesis is false. Optimal test statistics are defined using this criterion and the relationship of these methods to the Neyman-Pearson theory of hypothesis testing is considered.