Goodness‐of‐fit tests for distributions estimated from complex survey data

We propose nonparametric procedures for comparing the empirical distribution function of data from a complex survey with a hypothesized parametric reference distribution. The hypothesized distribution may be fully specified, or it may be a family with the parameters to be estimated from the data. Of the procedures studied, a modification of the Cramér–von Mises test proposed by Lockhart, Spinelli & Stephens [Lockhart, Spinelli and Stephens, The Canadian Journal of Statistics 2007; 35, 125–133] is supported theoretically and performs well in two simulation studies. The methods are applied to examine the distribution of body mass index in the U.S. National Health and Nutrition Examination Survey. The Canadian Journal of Statistics 47: 409–425; 2019 © 2019 Statistical Society of Canada     

An improved test of equality of mean directions for the Langevin‐von Mises‐Fisher distribution

A multi‐sample test for equality of mean directions is developed for populations having Langevin‐von Mises‐Fisher distributions with a common unknown concentration. The proposed test statistic is a monotone transformation of the likelihood ratio. The high‐concentration asymptotic null distribution of the test statistic is derived. In contrast to previously suggested high‐concentration tests, the high‐concentration asymptotic approximation to the null distribution of the proposed test statistic is also valid for large sample sizes with any fixed nonzero concentration parameter. Simulations of size and power show that the proposed test outperforms competing tests. An example with three‐dimensional data from an anthropological study illustrates the practical application of the testing procedure.     

On a new goodness‐of‐fit process for families of copulas

A goodness‐of‐fit procedure is proposed for parametric families of copulas. The new test statistics are functionals of an empirical process based on the theoretical and sample versions of Spearman's dependence function. Conditions under which this empirical process converges weakly are seen to hold for many families including the Gaussian, Frank, and generalized Farlie–Gumbel–Morgenstern systems of distributions, as well as the models with singular components described by Durante [Durante ( 2007 ) Comptes Rendus Mathématique. Académie des Sciences. Paris, 344, 195–198]. Thanks to a parametric bootstrap method that allows to compute valid P‐values, it is shown empirically that tests based on Cramér–von Mises distances keep their size under the null hypothesis. Simulations attesting the power of the newly proposed tests, comparisons with competing procedures and complete analyses of real hydrological and financial data sets are presented. The Canadian Journal of Statistics 37: 80‐101; 2009 © 2009 Statistical Society of Canada     

Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

Goodness‐of‐fit tests for the hyperbolic distribution

The authors give tests of fit for the hyperbolic distribution, based on the Cramér‐von Mises statistic W². They consider the general case with four parameters unknown, and some specific cases where one or two parameters are fixed. They give two examples using stock price data.     

Testing multivariate uniformity: The distance‐to‐boundary method

Given a random sample taken on a compact domain S ? ^d, the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of S. The proposed test has a number of interesting properties. In particular, it is feasible and particularly suitable for high dimensional data; it is distribution free for a wide range of choices of 5; it can be adapted to the case that the support of S is unknown; and it also allows for one‐sided versions. Moreover, the results suggest that, in some cases, this procedure does not suffer from the well‐known curse of dimensionality. The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive Monte Carlo simulation study allows them to compare their methods with some alternative procedures. They conclude that the proposed test provides quite a satisfactory balance between power, computational simplicity, and adaptability to different dimensions and supports.     

A likelihood ratio test for goodness‐of‐fit of recessive and dominant models for case–control studies

Testing goodness‐of‐fit of commonly used genetic models is of critical importance in many applications including association studies and testing for departure from Hardy–Weinberg equilibrium. Case–control design has become widely used in population genetics and genetic epidemiology, thus it is of interest to develop powerful goodness‐of‐fit tests for genetic models using case–control data. This paper develops a likelihood ratio test (LRT) for testing recessive and dominant models for case–control studies. The LRT statistic has a closed‐form formula with a simple $\chi^{2}(1)$ null asymptotic distribution, thus its implementation is easy even for genome‐wide association studies. Moreover, it has the same power and optimality as when the disease prevalence is known in the population. The Canadian Journal of Statistics 41: 341–352; 2013 © 2013 Statistical Society of Canada     

On Cramér–von Mises type test based on local time of switching diffusion process

We consider a Cramér–von Mises type test for hypothesis that the observed diffusion process has sign-type trend coefficient based on empirical density function. It is shown that the limit distribution of the proposed test statistic is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process. Approximation of the threshold is given through the representation for the limit statistic.     

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.     

Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

Abstract. A right‐censored version of a U ‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U ‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U ‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.     

One‐Way anova for Functional Data via Globalizing the Pointwise F‐test

In this paper, we propose and study a new global test, namely, GPF test, for the one‐way anova problem for functional data, obtained via globalizing the usual pointwise F‐test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root‐n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one‐way anova for functional data. Via some simulation studies, it is found that in terms of size‐controlling and power, the GPF test is comparable with two existing tests adopted for the one‐way anova problem for functional data. A real data example illustrates the GPF test.     

Sample size determination in fixed‐dose combination drug studies

In this paper, the application of the intersection–union test method in fixed‐dose combination drug studies is discussed. An approximate sample size formula for the problem of testing the efficacy of a combination drug using intersection–union tests is proposed. The sample sizes obtained from the formula are found to be reasonably accurate in terms of attaining the target power 1?β for a specified β. Copyright © 2003 John Wiley & Sons, Ltd.     

Cramér‐von Mises regression

Consider a linear regression model with unknown regression parameters β₀ and independent errors of unknown distribution. Block the observations into q groups whose independent variables have a common value and measure the homogeneity of the blocks of residuals by a Cramér‐von Mises q‐sample statistic T_q(β). This statistic is designed so that its expected value as a function of the chosen regression parameter β has a minimum value of zero precisely at the true value β₀. The minimizer β of T_q(β) over all β is shown to be a consistent estimate of β₀. It is also shown that the bootstrap distribution of T_q(β₀) can be used to do a lack of fit test of the regression model and to construct a confidence region for β₀     

CVX‐based algorithms for constructing various optimal regression designs

CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada     

Bayesian Analysis of the Proportional Hazards Model with Time‐Varying Coefficients

We study a Bayesian analysis of the proportional hazards model with time‐varying coefficients. We consider two priors for time‐varying coefficients – one based on B‐spline basis functions and the other based on Gamma processes – and we use a beta process prior for the baseline hazard functions. We show that the two priors provide optimal posterior convergence rates (up to the term) and that the Bayes factor is consistent for testing the assumption of the proportional hazards when the two priors are used for an alternative hypothesis. In addition, adaptive priors are considered for theoretical investigation, in which the smoothness of the true function is assumed to be unknown, and prior distributions are assigned based on B‐splines.     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.     

Non‐Parametric Change‐Point Tests for Long‐Range Dependent Data

Abstract. We propose a non‐parametric change‐point test for long‐range dependent data, which is based on the Wilcoxon two‐sample test. We derive the asymptotic distribution of the test statistic under the null hypothesis that no change occurred. In a simulation study, we compare the power of our test with the power of a test which is based on differences of means. The results of the simulation study show that in the case of Gaussian data, our test has only slightly smaller power minus.3pt than the ‘difference‐of‐means’ test. For heavy‐tailed data, our test outperforms the ‘difference‐of‐means’ test.     

Variance Shifts,Structural Breaks,and Stationarity Tests

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modifications of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér–von Mises distributions. One such modification is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modified statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level. The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, inflation, and exchange rate data series.     

Model Checks in Inverse Regression Models with Convolution‐Type Operators

Abstract. We consider the problem of testing parametric assumptions in an inverse regression model with a convolution‐type operator. An L ₂ ‐type goodness‐of‐fit test is proposed which compares the distance between a parametric and a non‐parametric estimate of the regression function. Asymptotic normality of the corresponding test statistic is shown under the null hypothesis and under a general non‐parametric alternative with different rates of convergence in both cases. The feasibility of the proposed test is demonstrated by means of a small simulation study. In particular, the power of the test against certain types of alternative is investigated. Finally, an empirical example is provided, in which the proposed methods are applied to the determination of the shape of the luminosity profile of the elliptical galaxy NGC 5017.     

The marked empirical process to test nonlinear time series against a large class of alternatives when the random vectors are nonstationary and absolutely regular

In this paper, we propose a method for testing absolutely regular and possibly nonstationary nonlinear time-series, with application to general AR-ARCH models. Our test statistic is based on a marked empirical process of residuals which is shown to converge to a Gaussian process with respect to the Skohorod topology. This testing procedure was first introduced by Stute [Nonparametric model checks for regression, Ann. Statist. 25 (1997), pp. 613–641] and then widely developed by Ngatchou-Wandji [Weak convergence of some marked empirical processes: Application to testing heteroscedasticity, J. Nonparametr. Stat. 14 (2002), pp. 325–339; Checking nonlinear heteroscedastic time series models, J. Statist. Plann. Inference 133 (2005), pp. 33–68; Local power of a Cramer-von Mises type test for parametric autoregressive models of order one, Compt. Math. Appl. 56(4) (2008), pp. 918–929] under more general conditions. Applications to general AR-ARCH models are given.