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     

Testing fit for the grouped exponential distribution

Grouped data can often arise due to the lack of resolution of the measurement instruments; they also arise when data are deliberately rounded to a certain accuracy and are presented, say, in the form of a histogram. The author uses statistics of the Cramér‐von Mises type to test for the exponential distribution when data are grouped.     

On the use of priors in goodness‐of‐fit tests

Priors are introduced into goodness‐of‐fit tests, both for unknown parameters in the tested distribution and on the alternative density. Neyman–Pearson theory leads to the test with the highest expected power. To make the test practical, we seek priors that make it likely a priori that the power will be larger than the level of the test but not too close to one. As a result, priors are sample size dependent. We explore this procedure in particular for priors that are defined via a Gaussian process approximation for the logarithm of the alternative density. In the case of testing for the uniform distribution, we show that the optimal test is of the U‐statistic type and establish limiting distributions for the optimal test statistic, both under the null hypothesis and averaged over the alternative hypotheses. The optimal test statistic is shown to be of the Cramér–von Mises type for specific choices of the Gaussian process involved. The methodology when parameters in the tested distribution are unknown is discussed and illustrated in the case of testing for the von Mises distribution. The Canadian Journal of Statistics 47: 560–579; 2019 © 2019 Statistical Society of Canada     

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.     

Cramér-von Mises tests of fit for the Poisson distribution

Goodness-of-fit tests based on the Cramér-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A², give good overall tests of fit. The statistic A² will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.     

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     

Estimators Based on Data‐Driven Generalized Weighted Cramér‐von Mises Distances under Censoring – with Applications to Mixture Models

Abstract. Estimators based on data‐driven generalized weighted Cramér‐von Mises distances are defined for data that are subject to a possible right censorship. The function used to measure the distance between the data, summarized by the Kaplan–Meier estimator, and the target model is allowed to depend on the sample size and, for example, on the number of censored items. It is shown that the estimators are consistent and asymptotically multivariate normal for every p dimensional parametric family fulfiling some mild regularity conditions. The results are applied to finite mixtures. Simulation results for finite mixtures indicate that the estimators are useful for moderate sample sizes. Furthermore, the simulation results reveal the usefulness of sample size dependent and censoring sensitive distance functions for moderate sample sizes. Moreover, the estimators for the mixing proportion seem to be fairly robust against a ‘symmetric’ contamination model even when censoring is present.     

Goodness‐of‐fit tests for linear regression models with missing response data

The authors show how to test the goodness‐of‐fit of a linear regression model when there are missing data in the response variable. Their statistics are based on the L₂ distance between nonparametric estimators of the regression function and a ‐consistent estimator of the same function under the parametric model. They obtain the limit distribution of the statistics and check the validity of their bootstrap version. Finally, a simulation study allows them to examine the behaviour of their tests, whether the samples are complete or not.     

A Model Specification Test For GARCH(1,1) Processes

We provide a consistent specification test for generalized autoregressive conditional heteroscedastic (GARCH (1,1)) models based on a test statistic of Cramér‐von Mises type. Because the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model‐based (semiparametric) bootstrap method to approximate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finite sample behaviour of the test by some simulations.     

Goodness‐of‐fit tests for parametric models based on biased samples

The authors study the problem of checking the adequacy of a parametric model for a distribution using several possibly censored weight biased samples. They discuss identifiability problems related to the underlying distribution and the distributions of the biased samples. They propose a test statistic based on the supremum of the weighted aggregated martingale residual processes from a number of such samples. Both numerical and graphical procedures are discussed, which the authors apply to do model checking for oil exploration drilling data.     

On a cramér-von mises type statistic for testing bivariate independence

A Cramér-von Mises type statistic for testing bivariate independence, proposed by Hoeffding (1948) and by Blum, Kiefer, and Rosenblatt (1961), is examined in greater detail. The statistic is decomposed into components in the manner of Durbin and Knott (1972), and the components are shown to be related to linear rank statistics. Asymptotic power properties of the Hoeffding statistic and its components in testing for independence with bivariate normal random observations are described; a Monte Carlo study comparing these statistics with other nonparametric statistics for bivariate independence is also reported.     

Goodness‐of‐fit methods for matched case‐control studies

The authors propose graphical and numerical methods for checking the adequacy of the logistic regression model for matched case‐control data. Their approach is based on the cumulative sum of residuals over the covariate or linear predictor. Under the assumed model, the cumulative residual process converges weakly to a centered Gaussian limit whose distribution can be approximated via computer simulation. The observed cumulative residual pattern can then be compared both visually and analytically to a certain number of simulated realizations of the approximate limiting process under the null hypothesis. The proposed techniques allow one to check the functional form of each covariate, the logistic link function as well as the overall model adequacy. The authors assess the performance of the proposed methods through simulation studies and illustrate them using data from a cardiovascular study.     

An integral formula for the distribution of self-normalized Gaussian random samples

Given i.i.d. Gaussian random variables and after standardizing the sample by subtracting the sample mean and dividing it by the sample deviation, we obtain an integral formula for the distribution of these self-normalized variables. Using geometrical arguments, we obtain the distribution of each and the joint distribution of two of them. These formulas can be used to calculate the expected value of the particular type of Cramér von Mises statistic to test normality.     

Exact Goodness‐of‐Fit Testing for the Ising Model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness‐of‐fit test for the finite‐lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness‐of‐fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness‐of‐fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.     

A Note on the Large Sample Properties of Estimators Based on Generalized Linear Models for Correlated Pseudo‐observations

Pseudo‐values have proven very useful in censored data analysis in complex settings such as multi‐state models. It was originally suggested by Andersen et al., Biometrika, 90, 2003, 335 who also suggested to estimate standard errors using classical generalized estimating equation results. These results were studied more formally in Graw et al., Lifetime Data Anal., 15, 2009, 241 that derived some key results based on a second‐order von Mises expansion. However, results concerning large sample properties of estimates based on regression models for pseudo‐values still seem unclear. In this paper, we study these large sample properties in the simple setting of survival probabilities and show that the estimating function can be written as a U‐statistic of second order giving rise to an additional term that does not vanish asymptotically. We further show that previously advocated standard error estimates will typically be too large, although in many practical applications the difference will be of minor importance. We show how to estimate correctly the variability of the estimator. This is further studied in some simulation studies.     

Goodness‐of‐Fit Tests for Bivariate and Multivariate Skew‐Normal Distributions

Abstract. Goodness‐of‐fit tests are proposed for the skew‐normal law in arbitrary dimension. In the bivariate case the proposed tests utilize the fact that the moment‐generating function of the skew‐normal variable is quite simple and satisfies a partial differential equation of the first order. This differential equation is estimated from the sample and the test statistic is constructed as an L ₂ ‐type distance measure incorporating this estimate. Extension of the procedure to dimension greater than two is suggested whereas an effective bootstrap procedure is used to study the behaviour of the new method with real and simulated data.     

Tests of serial independence based on Kendall's process

The authors propose new rank statistics for testing the white noise hypothesis in a time series. These statistics are Cramér‐von Mises and Kolmogorov‐Smirnov functionals of an empirical distribution function whose mean is related to a serial version of Kendall's tau through a linear transform. The authors determine the asymptotic behaviour of the underlying serial process and the large‐sample distribution of the proposed statistics under the null hypothesis of white noise. They also present simulation results showing the power of their tests.