Cramer-von Mises Type Test Research Articles

Testing conditional quantile independence with functional covariate.

We propose a new non-parametric conditional independence test for a scalar response and a functional covariate over a continuum of quantile levels. We build a Cramer-von Mises type test statistic based on an empirical process indexed by random projections of the functional covariate, effectively avoiding the "curse of dimensionality" under the projected hypothesis, which is almost surely equivalent to the null hypothesis. The asymptotic null distribution of the proposed test statistic is obtained under some mild assumptions. The asymptotic global and local power properties of our test statistic are then investigated. We specifically demonstrate that the statistic is able to detect a broad class of local alternatives converging to the null at the parametric rate. Additionally, we recommend a simple multiplier bootstrap approach for estimating the critical values. The finite-sample performance of our statistic is examined through several Monte Carlo simulation experiments. Finally, an analysis of an EEG data set is used to show the utility and versatility of our proposed test statistic.

Testing similarity between first-order intensities of spatial point processes. A comparative study

Testing whether two spatial point processes have the same spatial distribution is an important task that can be addressed from different perspectives. A Kolmogorov-Smirnov test with asymptotic calibration and a Cramer von Mises type test with bootstrap calibration have recently been developed to compare the first-order intensity of two observed patterns. Motivated by common practice in epidemiological studies, we introduce a regression test based on the relative risk function with two alternative bootstrap calibrations. This paper compares the performance of these nonparametric tests through both an intensive simulation study, and the application to wildfire and crime data. The three tests provide good calibrations of the null hypothesis for simulated Poisson and non-Poisson spatial point processes, but the Cramer von Mises and regression tests outperform the cost-efficient Kolmogorov-Smirnov test in terms of power. In the real data analysis we have seen that the Kolmogorov-Smirnov test does not detect differences between spatial point patterns when dealing with sparse data. In view of these results, it would be preferable using the Cramer von Mises or regression tests despite their higher computational demand.

Model diagnostics of parametric Tobit model based on cumulative residuals

In this paper, we investigate the adequate check of the parametric Tobit model. A Cramer–Von Mises type test statistic is constructed, and its asymptotic properties under the null and alternative hypotheses are rigorously studied. The method is effective for the adequacy check of parametric regression models with a scalar or multivariate covariate. Furthermore, it avoids the nonparametric smoothing of the regression function and the choice of the smoothing parameter. Simulation studies are conducted to compare the performance of the proposed test procedure and the existing methods in the literature. A real data set of income is analyzed by applying the proposed method.

Consistent nonparametric change point detection combining CUSUM and marked empirical processes

A weakly dependent time series regression model with multivariate covariates and univariate observations is considered, for which we develop a procedure to detect whether the nonparametric conditional mean function is stable in time against change point alternatives. Our proposal is based on a modified CUSUM type test procedure, which uses a sequential marked empirical process of residuals. We show weak convergence of the considered process to a centered Gaussian process under the null hypothesis of no change in the mean function and a stationarity assumption. This requires some sophisticated arguments for sequential empirical processes of weakly dependent variables. As a consequence we obtain convergence of Kolmogorov-Smirnov and Cramér-von Mises type test statistics. The proposed procedure acquires a very simple limiting distribution and nice consistency properties, features from which related tests are lacking. We moreover suggest a bootstrap version of the procedure and discuss its applicability in the case of unstable variances.

Generalized information matrix tests for copulas

We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White and so relate to the copula test proposed by Huang and Prokhorov. The idea is that eigenspectrum-based statements of the IM equality reduce the degrees of freedom of the test’s asymptotic distribution and lead to better size-power properties, even in high dimensions. The gains are especially pronounced for vine copulas, where additional benefits come from simplifications of score functions and the Hessian. We derive the asymptotic distribution of the generalized tests, accounting for the nonparametric estimation of the marginals and apply a parametric bootstrap procedure, valid when asymptotic critical values are inaccurate. In Monte Carlo simulations, we study the behavior of the new tests, compare them with several Cramer–von Mises type tests and confirm the desired properties of the new tests in high dimensions.

Tests for the equality of conditional variance functions in nonparametric regression

In this paper we are interested in checking whether the conditional variances are equal in k ≥ 2 location-scale models. Our procedure is fully nonparametric and is based on the comparison of the error distributions under the null hypothesis of equality of variances and without making use of this null hypothesis. We propose four test statistic based on empirical distribution functions (Kolmogorov-Smirnovand and Cramer-von Mises type test statistics) and two test statistics based on empirical characteristic functions. The limiting distributions of these six test statistics are established under the null hypothesis and under a local alternative. We show how to approximate the critical values under null using either an estimated version of the asymptotic null distribution or a bootstrap procedure. Simulation studies are conducted to assess the finite sample performance of the proposed tests. We also apply our tests to data on monthly expenditure.

Two-sample tests based on empirical Hankel transforms

We study a new Cramer-von Mises type test for the general nonparametric two-sample problem on the nonnegative half-line. The test statistic is based on the empirical Hankel transforms of the sample variables, critical values are obtained by bootstrapping. The test is shown to be consistent against each fixed alternative. A scale invariant version of the test is also considered. A power comparison with the classical Cramer-von Mises test and another new Cramer-von Mises type test is done by simulation.

On goodness-of-fit testing for ergodic diffusion process with shift parameter

A problem of goodness-of-fit test for ergodic diffusion processes is presented. In the null hypothesis the drift of the diffusion is supposed to be in a parametric form with unknown shift parameter. Two Cramer–von Mises type test statistics are studied. The first test uses the local time estimator of the invariant density, the second one uses the empirical distribution function. The unknown parameter is estimated via the maximum likelihood estimator. It is shown that the limit distribution of the two test statistics does not depend on the unknown parameter, thus both the tests are asymptotically parameter free. Some considerations on the consistency of the proposed tests and some simulation studies are also given.

Testing for the bivariate Poisson distribution

This paper studies goodness-of-fit tests for the bivariate Poisson distribution. Specifically, we propose and study several Cramer–von Mises type tests based on the empirical probability generating function. They are consistent against fixed alternatives for adequate choices of the weight function involved in their definition. They are also able to detect local alternatives converging to the null at a certain rate. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approximation and compares their powers for finite sample sizes. Extensions for testing goodness-of-fit for the multivariate Poisson distribution are also discussed.

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.

Conditional Predictive Density Evaluation in the Presence of Instabilities

We propose new methods for evaluating predictive densities. The methods include Kolmogorov-Smirnov and Cramer-von Mises-type tests for the correct specification of predictive densities robust to dynamic mis-specification. The novelty is that the tests can detect mis-specification in the predictive densities even if it appears only over a fraction of the sample, due to the presence of instabilities. Our results indicate that our tests are well sized and have good power in detecting mis-specification in predictive densities, even when it is time-varying. An application to density forecasts of the Survey of Professional Forecasters demonstrates the usefulness of the proposed methodologies.

A test for independence of two stationary infinite order autoregressive processes

This paper considers the independence test for two stationary infinite order autoregressive processes. For a test, we follow the empirical process method and construct the Cramer-von Mises type test statistics based on the least squares residuals. It is shown that the proposed test statistics behave asymptotically the same as those based on true errors. Simulation results are provided for illustration.

A Test of the Martingale Hypothesis

This paper proposes a statistical test of the martingale hypothesis. It can be used to test whether a given time series is a martingale process against certain non-martingale alternatives. The class of alternative processes against which our test has power is very general and it encompasses many nonlinear non-martingale processes which may not be detected using traditional spectrum-based or variance-ratio tests. We look at the hypothesis of martingale, in contrast with other existing methods which test for the hypothesis of martingale difference. Two different types of test are considered: one is a generalized Kolmogorov-Smirnov test and the other is a Cramer-von Mises type test. For the processes that are first-order Markovian in mean, in particular, our approach yields the test statistics that neither depend upon any smoothing parameter nor require any resampling procedure to simulate the null distributions. Their null limiting distributions are nicely characterized as functionals of a continuous stochastic process so that the critical values are easily tabulated. We prove consistency of our tests and further investigate their finite sample properties via simulation. Our tests are found to be rather powerful in moderate size samples against a wide variety of non-martingales including exponential autoregressive, threshold autoregressive, markov switching, chaotic, and some of nonstationary processes.

Goodness of Fit Tests for Multivariate Counting Process Models with Applications

In this paper, we develop some distribution‐free tests for checking the adequacy of the parametric forms of the intensity processes of a multivariate counting process model. The proposed tests, based in Khmaladze's transformations, are derived from the transforms of weighted aggregated martingale residual processes. The transformed processes converge weakly to independent Gaussian martingales under the assumed model. The distribution‐free tests, such as Kolmogorov–Smirnov and Cramer–von Mises type tests, are appropriately defined to account for deviations in each of the transformed aggregated martingale residual processes. Consistency of the tests are discussed. The tests are applicable to multiplicative intensity models such as a competing risks model as well as to non‐multiplicative intensity models such as a constant relative or excess mortality model. A small simulation study is conducted and an illustration to a real data example is provided.

A Comparison of Weighted Tests for a Slope Change in Simple Regression

Testing for a change in the slope of the simple linear regression model has many applications in bio-sciences, quality control and survival analysis. This paper compares Anderson-Darling and Erdos-Darling type test statistics which are based on the least squares change point process of Sen (1980) with the corresponding Kolmogorov-Smirnov and Cramer-von Mises type test statistics. We estimated the limiting critical values of these test statistics and conducted Monte Carlo simulation studies to compare their powers.

A test of conditional heteroscedasticity in time series

A new test of conditional heteroscedasticity for time series is proposed. The new testing method is based on a goodness of fit type test statistics and a Cramer-von Mises type test statistic. The asymptotic properties of the new test statistic is establised. The results demonstrate that such a test is consistent.

A test of fit for a semiparametric additive risk model

SUMMARY Kolmogorov-Smirnov and Cramer-von Mises type test statistics based on the standardised cumulative hazard process are proposed. It is very difficult to evaluate their asymptotic distributions, but they can be approximated by the use of the bootstrap. The advantages of the goodness-of-fit test are that arbitrary partitions of the time axis and covariate spaces are not needed for evaluating test statistics and that it has excellent consistency properties. The test is applied to data from the Mayo Clinic trial in primary biliary cirrhosis of the liver. A simulation study indicates that the proposed test is suitable for practical use.

Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions

Most Cramér -von Mises-type statistics of interest converge in distribution to for some Gaussian process {ξ(t): t ϵ I}. We present a numerically efficient method for the evaluation of ℙ(ω2≤x). First is approximated by where am and ti are the coefficients and nodes of a quadrature formula. Under suitable regularity conditions imposed upon the covariance function K of ξ, we show that the approximation error ϵm = ω2- ω2,m satisfies Var(ϵm)=O(1/m 2) This allows us to approximate ℙ(ω2≤x) byℙ(ω2,m≤x) which in turn is evaluated by inverting the characteristic function , where and K m = (K(ti,tj ),1 ≤ i,j ≤m) An application of this method is provided for tests of marginal independence for multivariate extreme values. Given a sample of size n from U = (U1,U2 ) with distribution function we test the hypothesis that θ(t) = 1 by a Cramér-von Mises-type test statistic, converging weakly as where ξ(t) is a Gaussian process with covariance function We tabulate the distribution and critical values of ω2 with a precision of 10-5. Further applications are discussed, including Anderson-Darling and Cramér-von Mises-type tests of fit when parameters are estimated.

Testing Stationarity in the Mean of Autoregressive Processes with a Nonparametric Regression Trend

In this paper, we suggest tests of stationarity in the mean of autoregressive time series versus arbitrary trend alternatives. As an intermediate, though essential, step local asymptotic normality of autoregressive models with a nonparametric regression trend is established. Moreover, a functional central limit theorem for the underlying likelihood ratio processes is derived. These results then offer a general construction principle by which every goodness of fit test (case 0), which is based on comparing the empirical distribution function and the hypothetical distribution function, corresponds to a test of stationarity in the mean of AR processes. The asymptotic power of these tests is derived. A small simulation study illustrates the performance of Kolmogorov-Smirnov and Cramer-von Mises type tests of stationarity in the mean at hand of a particular AR(2) process.

Goodness-of-fit tests for the weibull distribution with unknown parameters and heavy censoring

Goodness-of-fit tests are considered for testing the two-parameters Weibull distribution based on type II censored sampling with both parameters assumed unknown. Some extremely heavy censoring levels are considered which are useful when analyzing in-service field data with a large population and a small number of failures. Critical values are obtained by Monte Carlo simulation for Kolmogorov-Smirnov, Kuiper and Cramer-von Mises type test statistics. The approximate Snedecor’s F -distribution is verified for the Mann-Scheuer-Fertig test statistic. The two-sided Mann-Scheuer-Fertig test is also studied and found to be important. A power study is carried out for these four test statistics for both moderate and heavy censoring for a number of different alternative models.