Testing for bivariate normality using the empirical distribution function

The problem of testing for bivariate normality using the empirical distribution function is considered. A Cramér-von Mises type statistic is defined and asymptotic percentage points for this statistic given. This involves solving a two-dimensional homogeneous integral equation. Unfortunately the Cramér-von Mises statistic is not invariant under orthogonal transformations of the data so that an invariant statistic is developed. Approximations for the distribution of this statistic are found by Monte Carlo. Applications of the statistics are given. It is shown that the statistics are particularly sensitive to certain kinds of pattern in the data and they could be useful in data analysis apart from providing a formal test of bivariate normality     

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.     

A note on the nonparametric test based on theL 1-version of the Cramér-von Mises statistic

We consider the test based on theL ₁-version of the Cramér-von Mises statistic for the nonparametric two-sample problem. Some quantiles of the exact distribution under H₀ of the test statistic are computed for small sample sizes. We compare the test in terms of power against general alternatives to other two-sample tests, namely the Wilcoxon rank sum test, the Smirnov test and the Cramér-von Mises test in the case of unbalanced small sample sizes. The computation of the power is rather complicated when the sample sizes are unequal. Using Monte Carlo power estimates it turns out that the Smirnov test is more sensitive to non stochastically ordered alternatives than the new test. And under location-contamination alternatives the power estimates of the new test and of the competing tests are equal.     

A monte carlo analysis of two spectral tests of the martingale hypothesis

Summary The size, power, and robustness properties of the Kolmogorov-Smirnov and Cramér-von Mises spectral tests of the martingale (difference) hypothesis are investigated by Monte Carlo methods. The results highlight a marked superiority of the Cramér-von Mises with respect to the Kolmogorov-Smirnov test. The paper also shows that the Cramér-von Mises test is simple to compute, more general and more powerful than other converntionally used tests.     

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.     

Cramér-von Mises statistics for discrete distributions

Cramér-von Mises statistics are developed for use in testing for discrete distributions, and tables are given for tests for the discrete uniform distribution.     

Jarque–Bera Test and its Competitors for Testing Normality – A Power Comparison

For testing normality we investigate the power of several tests, first of all, the well-known test of Jarque & Bera (1980) and furthermore the tests of Kuiper (1960) and Shapiro & Wilk (1965) as well as tests of Kolmogorov–Smirnov and Cramér-von Mises type. The tests on normality are based, first, on independent random variables (model I) and, second, on the residuals in the classical linear regression (model II). We investigate the exact critical values of the Jarque–Bera test and the Kolmogorov–Smirnov and Cramér-von Mises tests, in the latter case for the original and standardized observations where the unknown parameters μ and σ have to be estimated. The power comparison is carried out via Monte Carlo simulation assuming the model of contaminated normal distributions with varying parameters μ and σ and different proportions of contamination. It turns out that for the Jarque–Bera test the approximation of critical values by the chi-square distribution does not work very well. The test is superior in power to its competitors for symmetric distributions with medium up to long tails and for slightly skewed distributions with long tails. The power of the Jarque–Bera test is poor for distributions with short tails, especially if the shape is bimodal – sometimes the test is even biased. In this case a modification of the Cramér-von Mises test or the Shapiro–Wilk test may be recommended.     

Robustness and power of modified Lepage,Kolmogorov-Smirnov and Crame´r-von Mises two-sample tests

For the two-sample problem with location and/or scale alternatives, as well as different shapes, several statistical tests are presented, such as of Kolmogorov-Smirnov and Cramér-von Mises type for the general alternative, and such as of Lepage type for location and scale alternatives. We compare these tests with the t-test and other location tests, such as the Welch test, and also the Levene test for scale. It turns out that there is, of course, no clear winner among the tests but, for symmetric distributions with the same shape, tests of Lepage type are the best ones whereas, for different shapes, Cramér-von Mises type tests are preferred. For extremely right-skewed distributions, a modification of the Kolmogorov-Smirnov test should be applied.     

Testing independence based on Bernstein empirical copula and copula density

In this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.     

A NONPARAMETRIC TEST OF SYMMETRY VERSUS ASYMMETRY FOR RANKED-SET SAMPLES

This paper introduces a nonparametric test of symmetry for ranked-set samples to test the asymmetry of the underlying distribution. The test statistic is constructed from the Cramér-von Mises distance function which measures the distance between two probability models. The null distribution of the test statistic is established by constructing symmetric bootstrap samples from a given ranked-set sample. It is shown that the type I error probabilities are stable across all practical symmetric distributions and the test has high power for asymmetric distributions.     

The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples

A test based on the studentized empirical characteristic function calculated in a single point is derived. An empirical power comparison is made between this test and tests like the Epps–Pulley, Shapiro–Wilks, Anderson–Darling and other tests for normality. It is shown to outperform the more complicated Epps-Pulley test based on the empirical characteristic function and a Cramér-von Mises type expression in a simulation study. The test performs especially good in large samples and the derived test statistic has an asymptotic normal distribution which is easy to apply.     

Relative efficiencies of goodness-of-fit procedures with truncated data

Efficiency properties of the Kolmogorov-Smirnov, Kuiper, Cramér-von Mises, and Anderson-Darling statistics with truncated data are investigated. For these statistics, the approximate slopes are determined, and the equivalence of ratios of limiting approximate slopes to limiting Pitman efficiencies is established. Efficiencies for certain parametric families of distributions incorporating both location and scale alternatives are evaluated as the amount of truncation increases.     

Testing Symmetry of the Error Distribution in Nonlinear Heteroscedastic Models

Using the empirical characteristic function, we derive a Cramér-von Mises test for symmetry of the error distribution in a class of nonlinear parametric heteroscedastic models. We study the convergence of the residual-based empirical distribution function. We establish a functional limit theorem for an empirical process of residuals, and investigate the asymptotic null distribution function of our test statistic. A simulation experiment is conducted to evaluate small-sample performances of our test.     

Goodness-of-fit tests for parametric models in censored regression

The authors propose a goodness-of-fit test for parametric regression models when the response variable is right-censored. Their test compares an estimation of the error distribution based on parametric residuals to another estimation relying on nonparametric residuals. They call on a bootstrap mechanism in order to approximate the critical values of tests based on Kolmogorov-Smirnov and Cramér-von Mises type statistics. They also present the results of Monte Carlo simulations and use data from a study about quasars to illustrate their work.     

On the asymptotics of randomness statistics

Kiefer (1959) studied the asymptotics of q-sample Cramér-von Mises nonparametric statistics when q is fixed and the sample sizes tend to infinity. Here we prove the asymptotic normality of such statistics when the sample sizes stay fixed or small while the number of samples, q, becomes large.     

Tests of fit for exponentiality based on a characterization via the mean residual life function

We study two new omnibus goodness of fit tests for exponentiality, each based on a characterization of the exponential distribution via the mean residual life function. The limiting null distributions of the tests statistics are the same as the limiting null distributions of the Kolmogorov-Smirnov and Cramér-von Mises statistics proposed when testing the simple hypothesis that the distribution of the sample variables is uniform on the interval [0, 1]. Work supported by the Deutsche Forschungsgemeinschaft     

Some nonparametric methods for changepoint problems

A general model for changepoint problems is discussed from a nonparametric viewpoint. The test statistics introduced are based on Cramér-von Mises functionals of certain processes and are shown to converge in distribution to corresponding Gaussian functionals (under the assumption of no change in distribution, H₀). We also demonstrate how the distribution of the limiting Gaussian functionals may be tabulated. Finally, properties of the tests under the alternative hypothesis of exactly one changepoint occurring are studied, and some examples are given.     

Asymptotic properties of a goodness-of-fit test based on maximum correlations

We study the efficiency properties of the goodness-of-fit test based on the Q _n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q _n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.     

EDF tests of goodness of fit for transform-both-sides models

Very often in regression analysis, a particular functional form connecting known covariates and unknown parameters is either suggested by previous work or demanded by theoretical considerations so that the deterministic part of the responses has a known form. However, the underlying error structure is often less well understood. In this case, the transform-both-sides (TBS) models are appropriate. In this paper we generalize the usual TBS models and develop tests to assess goodness of fit when fitting TBS or GTBS models. Parameter estimation is discussed, and tests based on the Cramér-von Mises statistic and the Anderson-Darling statistic are presented with a table suitable for finite-sample applications.     

Robust weighted Cramér-von Mises Estimators of location,with minimax variance in ϵ-contamination neighbourhoods

We construct weighted Cramér-von Mises location estimators which are asymptotically normally distributed throughout an ?e-contamination neighbourhood of a given, strongly unimodal distribution function, and which minimize the maximum asymptotic variance in such neighbourhoods. Applications to the estimation of a normal or logistic mean are given.