A nonparametric test for bivariate circular symmetry based on the empirical cdf

A nonparametric test for circular symmetry about 0 in a continuous bivariate distribution is proposed. The test is of the von Mises type, based on the empirical cdf of the sample, expressed in polar co-ordinates. However, the test is independent of the choice of the polar axis. The asymptotic form of the test statistic is obtained by considering the weak convergence of the empirical process to a limiting Gaussian process. The asymptotic distribution of the test statistic is found explicitly, both under the null hypothesis and under simple alternatives. The test is shown to be consistent against all alternatives.     

A weighted cramer-von mises statistic derived from a decomposition of the anderson-darling statistic

Two goodness of fit statistics with asymmetric weight function are derived from a decomposition of the Anderson-Darling statistic, For each one, the asymptotic null distribution is found for a simple null hypothesis and some upper percentties are calculated. The asymptotic power of the tests are obtained for some contiguous alternatives around a normal null hypothesis. The tests allow the user to choose to which tail to give more weight and it is intended to be used for that purpose. Therefore it should be not considered as a competitor of the classical goodness of fit tests.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.     

Testing nonparametric statistical functionals with applications to rank tests

The present paper discusses how nonparametric tests can be deduced from statistical functionals. Efficient and asymptotically most powerful maximin tests are derived. Their power function is calculated under implicit alternatives given by the functional for one – and two – sample testing problems. It is shown that the asymptotic power function does not depend on the special implicit direction of the alternatives but only on quantities of the functional. The present approach offers a nonparametric principle how to construct common rank tests as the Wilcoxon test, the log rank test, and the median test from special two-sample functionals. In addition it is shown that studentized permutation tests yield asymptotically valid tests for certain extended null hypotheses given by functionals which are strictly larger than the common i.i.d. null hypothesis. As example tests concerning the von Mises functional and the Wilcoxon two-sample test are treated.     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Testing Covariate Effects in Aalen's Linear Hazard Model

The performance of tests in Aalen's linear regression model is studied using asymptotic power calculations and stochastic simulation. Aalen's original least squares test is compared to two modifications: a weighted least squares test with correct weights and a test where the variance is re-estimated under the null hypothesis. The test with re-estimated variance provides the highest power of the tests for the setting of this paper, and the gain is substantial for covariates following a skewed distribution like the exponential. It is further shown that Aalen's choice for weight function with re-estimated variance is optimal in the one-parameter case against proportional alternatives.     

Seasonality Tests

This article modifies and extends the test against nonstationary stochastic seasonality proposed by Canova and Hansen. A simplified form of the test statistic in which the nonparametric correction for serial correlation is based on estimates of the spectrum at the seasonal frequencies is considered and shown to have the same asymptotic distribution as the original formulation. Under the null hypothesis, the distribution of the seasonality test statistics is not affected by the inclusion of trends, even when modified to allow for structural breaks, or by the inclusion of regressors with nonseasonal unit roots. A parametric version of the test is proposed, and its performance is compared with that of the nonparametric test using Monte Carlo experiments. A test that allows for breaks in the seasonal pattern is then derived. It is shown that its asymptotic distribution is independent of the break point, and its use is illustrated with a series on U.K. marriages. A general test against any form of permanent seasonality, deterministic or stochastic, is suggested and compared with a Wald test for the significance of fixed seasonal dummies. It is noted that tests constructed in a similar way can be used to detect trading-day effects. An appealing feature of the proposed test statistics is that under the null hypothesis, they all have asymptotic distributions belonging to the Cramér–von Mises family.     

Specification testing in nonparametric AR‐ARCH models

In this paper, an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. Tests for the model assumption of independence of innovations from past time series values are suggested. Tests based on weighted L²‐distances of empirical characteristic functions are considered as well as a Cramér–von Mises‐type test. The asymptotic distributions under the null hypothesis of independence are derived, and the consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested, and its performance is shown in a simulation study.     

Data-Transformation and Test of Fit for the Generalized Pareto Hypothesis

In this article, we consider Crámer–von Mises type goodness-of-fit statistics for the Generalized Pareto law. The tests involve a certain transformation of the original observations, which, at least in the case of completely specified null distribution, may be viewed as transforming to uniformity and comparing the resulting moments of arbitrary positive order to those of a uniform distribution. The method is shown to be consistent, and the asymptotic null distribution of the test statistic is derived. Simulation results indicate that the proposed test compares well with standard methods based on the empirical distribution function.     

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.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

Testing for pairwise serial independence via the empirical distribution function

Built on Skaug and Tjøstheim's approach, this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags, where the number of lags increases with the sample size and different lags are assigned different weights. Typically, the more recent information receives a larger weight. The test has some appealing attributes. It is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases. Although the test statistic is a weighted sum of degenerate Cramér–von Mises statistics, it has a null asymptotic N (0, 1) distribution. The test statistic and its limit distribution are invariant to any order preserving transformation. The test applies to time series whose distributions can be discrete or continuous, with possibly infinite moments. Finally, the test statistic only involves ranking the observations and is computationally simple. It has the advantage of avoiding smoothed nonparametric estimation. A simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests.     

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.     

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.     

Local Power Analyses of Goodness-of-fit Tests for Copulas

Abstract.  The asymptotic behaviour of several goodness-of-fit statistics for copula families is obtained under contiguous alternatives. Many comparisons between a Cramér–von Mises functional of the empirical copula process and new moment-based goodness-of-fit statistics are made by considering their associated asymptotic local power curves. It is shown that the choice of the estimator for the unknown parameter can have a significant influence on the power of the Cramér–von Mises test and that some of the moment-based statistics can provide simple and efficient goodness-of-fit methods.     

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.     

A family of test statistics for trend change in mean residual life with unknown turning point using censored data

In this paper, we develop a family of test statistics for testing exponentiality against increasing then decreasing mean residual life alternatives to accommodate the randomly censored data when neither the change point nor the proportion is known. We establish the asymptotic null distribution of test statistics. Monte Carlo simulations are conducted to investigate the speed of convergence of test statistics to the asymptotic null distribution and study the performance of test statistics by the power of tests.     

Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis

This paper introduces a general framework for testing hypotheses about the structure of the mean function of complex functional processes. Important particular cases of the proposed framework are as follows: (1) testing the null hypothesis that the mean of a functional process is parametric against a general alternative modelled by penalized splines; and (2) testing the null hypothesis that the means of two possibly correlated functional processes are equal or differ by only a simple parametric function. A global pseudo‐likelihood ratio test is proposed, and its asymptotic distribution is derived. The size and power properties of the test are confirmed in realistic simulation scenarios. Finite‐sample power results indicate that the proposed test is much more powerful than competing alternatives. Methods are applied to testing the equality between the means of normalized δ‐power of sleep electroencephalograms of subjects with sleep‐disordered breathing and matched controls.     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

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.