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.     

Improved local polynomial estimation in time series regression

We propose a modification of local polynomial estimation which improves the efficiency of the conventional method when the observation errors are correlated. The procedure is based on a pre-transformation of the data as a generalization of the pre-whitening procedure introduced by Xiao et al. [(2003), ‘More Efficient Local Polynomial Estimation in Nonparametric Regression with Autocorrelated Errors’, Journal of the American Statistical Association, 98, 980–992]. While these authors assumed a linear process representation for the error process, we avoid any structural assumption. We further allow the regressors and the errors to be dependent. More importantly, we show that the inclusion of both leading and lagged variables in the approximation of the error terms outperforms the best approximation based on lagged variables only. Establishing its asymptotic distribution, we show that the proposed estimator is more efficient than the standard local polynomial estimator. As a by-product we prove a suitable version of a central limit theorem which allows us to improve the asymptotic normality result for local polynomial estimators by Masry and Fan [(1997), ‘Local Polynomial Estimation of Regression Functions for Mixing Processes’, Scandinavian Journal of Statistics, 24, 165–179]. A simulation study confirms the efficiency of our estimator on finite samples. An application to climate data also shows that our new method leads to an estimator with decreased variability.     

A nonparametric test for diagnosis of the proportionality assumption

We propose a nonparametric test for diagnosis of the proportionality assumption between hazard functions based on a functional equation. Because of involvement of censoring distribution, we consider the test procedure in an asymptotic manner and obtain the asymptotic normality for the proposed test statistic. Then we discuss the rationale of use of the functional equation for the initial effect model. Finally we compare our test with others using an example.     

Sample size determination for testing equality in Poisson frequency data under an AB/BA crossover trial

Assuming that the frequency of occurrence follows the Poisson distribution, we develop sample size calculation procedures for testing equality based on an exact test procedure and an asymptotic test procedure under an AB/BA crossover design. We employ Monte Carlo simulation to demonstrate the use of these sample size formulae and evaluate the accuracy of sample size calculation formula derived from the asymptotic test procedure with respect to power in a variety of situations. We note that when both the relative treatment effect of interest and the underlying intraclass correlation between frequencies within patients are large, the sample size calculation based on the asymptotic test procedure can lose accuracy. In this case, the sample size calculation procedure based on the exact test is recommended. On the other hand, if the relative treatment effect of interest is small, the minimum required number of patients per group will be large, and the asymptotic test procedure will be valid for use. In this case, we may consider use of the sample size calculation formula derived from the asymptotic test procedure to reduce the number of patients needed for the exact test procedure. We include an example regarding a double‐blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these sample size formulae. Copyright © 2013 John Wiley & Sons, Ltd.     

On Estimating the Integrated Co-Volatility Using Noisy High-Frequency Data with Jumps

In this article, we consider the estimation of covariation of two asset prices which contain jumps and microstructure noise, based on high-frequency data. We propose a realized covariance estimator, which combines pre-averaging method to remove the microstructure noise and the threshold method to reduce the jumps effect. The asymptotic properties, such as consistency and asymptotic normality, are investigated. The estimator allows very general structure of jumps, for example, infinity activity or even infinity variation. Simulation is also included to illustrate the performance of the proposed procedure.     

Nonparametric tests for conditional independence using conditional distributions

The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aims to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L₂ metric. We use Nadaraya–Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. We ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test which is based on a linear mean-regression, we find that VIX index predicts excess returns both at short and long horizons.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ² statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.     

Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation

Abstract.  Wang & Wells [ J. Amer. Statist. Assoc. 95 (2000) 62] describe a non-parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [ J. Amer. Statist. Assoc. 88 (1993) 1034] and later studied by Barbe et al . [ J. Multivariate Anal. 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its L ²-norm. This paper shows how to compute asymptotic p -values for various goodness-of-fit test statistics based on a non-truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness-of-fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih [ Biometrika 85 (1998) 189] for the gamma frailty family.     

Testing for Threshold Diffusion

The threshold diffusion model assumes a piecewise linear drift term and a piecewise smooth diffusion term, which constitutes a rich model for analyzing nonlinear continuous-time processes. We consider the problem of testing for threshold nonlinearity in the drift term. We do this by developing a quasi-likelihood test derived under the working assumption of a constant diffusion term, which circumvents the problem of generally unknown functional form for the diffusion term. The test is first developed for testing for one threshold at which the drift term breaks into two linear functions. We show that under some mild regularity conditions, the asymptotic null distribution of the proposed test statistic is given by the distribution of certain functional of some centered Gaussian process. We develop a computationally efficient method for calibrating the p-value of the test statistic by bootstrapping its asymptotic null distribution. The local power function is also derived, which establishes the consistency of the proposed test. The test is then extended to testing for multiple thresholds. We demonstrate the efficacy of the proposed test by simulations. Using the proposed test, we examine the evidence of nonlinearity in the term structure of a long time series of U.S. interest rates.     

A nonparametric test for proving noninferiority in clinical trials with ordered categorical data

We consider the problem of proving noninferiority when the comparison is based on ordered categorical data. We apply a rank test based on the Wilcoxon–Mann–Whitney effect where the asymptotic variance is estimated consistently under the alternative and a small‐sample approximation is given. We give the associated 100(1?α)% confidence interval and propose a formula for sample size determination. Finally, we illustrate the procedure and possible choices of the noninferiority margin using data from a clinical trial. Copyright © 2003 John Wiley & Sons, Ltd.     

A family of IDMRL tests with unknown turning point

In this paper we propose a family of tests for exponentiality against the IDMRL alternative. Here we assume that the turning point or the proportion before the turning point is unknown. We derive the asymptotic null distributions of the test statistics and obtain their asymptotic critical values based on Durbin's approximation method. A simulation study is conducted to evaluate the proposed tests.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Non parametric hypothesis tests for comparing reliability functions

In reliability and related disciplines, comparing reliability functions of two (or more) aging processes is a crucial step in the process of determining reliability and understanding an aging process. The aim of this paper is to propose a non parametric statistical methodology to compare two populations based on their mean residual life function and expected inactivity time function. We introduce some novel hypothesis testing procedures that involve both Cramér–von Mises- and Kolmogorov–Smirnov-type test statistics and their decision rules are constructed based on the asymptotic distributions of these test statistics and bootstrapping method. We study the practical behavior of the proposed testing procedures extensively through simulations. The results reveal that the proposed hypothesis testing procedures perform efficiently in identifying small and large differences. Two real-life examples are discussed to demonstrate the practical utility of the tests.     

On the asymptotic normality of functions of uniform spacings

A new proof is given for the asymptotic normality of sum functions of spacings, providing an alternative to the method of Le Cam (1958). The result is obtained under an optimal moment condition. The proof is based on a simple decomposition into a leading term, which is asymptotically normal, and a remainder term, which is shown to be negligible.     

Exact sample-size determination in testing non-inferiority under a simple crossover trial

For testing the non-inferiority (or equivalence) of an experimental treatment to a standard treatment, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of an exact test procedure proposed elsewhere for a simple crossover design, we develop an exact sample-size calculation procedure with respect to the OR of patient response rates for a desired power of detecting non-inferiority at a given nominal type I error. We note that the sample size calculated for a desired power based on an asymptotic test procedure can be much smaller than that based on the exact test procedure under a given situation. We further discuss the advantage and disadvantage of sample-size calculation using the exact test and the asymptotic test procedures. We employ an example by studying two inhalation devices for asthmatics to illustrate the use of sample-size calculation procedure developed here.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Empirical likelihood tests for two‐sample problems via nonparametric density estimation

The authors study the problem of testing whether two populations have the same law by comparing kernel estimators of the two density functions. The proposed test statistic is based on a local empirical likelihood approach. They obtain the asymptotic distribution of the test statistic and propose a bootstrap approximation to calibrate the test. A simulation study is carried out in which the proposed method is compared with two competitors, and a procedure to select the bandwidth parameter is studied. The proposed test can be extended to more than two samples and to multivariate distributions.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.