On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

A Monte Carlo comparison of studentized bootstrap and permutation tests for heteroscedastic two-sample problems

The present Monte Carlo study compares bootstrap and permutation tests for semiparametric heteroscedastic two-sample testing problems of Behrens-Fisher type. The underlying functionals to be tested are (a) the difference of the means and (b) the Wilcoxon functionalP(Y < X) which is invariant under strictly increasing transformations. The consideration leads to semiparametric modifications of Welch type tests for the Behrens-Fisher model and an extended two-sample Wilcoxon test which also works under some null hypothesis with non-exchangeable distributions. The present Monte Carlo study confirms the high quality of studentized permutation tests at finite sample size. They are typically better than tests with asymptotic critical values and for many situations and they are also better than two-sample bootstrap tests when their type I error probabilities are compared.     

Hybrid bootstrap aided unit root testing

In this paper, we propose a hybrid bootstrap procedure for augmented Dickey-Fuller (ADF) tests for the presence of a unit root. This hybrid proposal combines a time domain parametric autoregressive fit to the data and a nonparametric correction applied in the frequency domain to capture features that are possibly not represented by the parametric model. It is known that considerable size and power problems can occur in small samples for unit root testing in the presence of an MA parameter using critical values of the asymptotic Dickey-Fuller distribution. The benefit of the sieve bootstrap in this situation has been investigated by Chang and Park (J Time Ser Anal 24:379–400, 2003). They showed asymptotic validity as well as substantial improvements for small sample sizes, but the actual sizes of their bootstrap tests were still quite far away from the nominal size. The finite sample performances of our procedure are extensively investigated through Monte Carlo simulations and compared to the sieve bootstrap approach. Regarding the size of the tests, our results show that the hybrid bootstrap remarkably outperforms the sieve bootstrap.     

Inference for ordered parameters in multinomial distributions

This paper discusses inference for ordered parameters of multinomial distributions. We first show that the asymptotic distributions of their maximum likelihood estimators (MLEs) are not always normal and the bootstrap distribution estimators of the MLEs can be inconsistent. Then a class of weighted sum estimators (WSEs) of the ordered parameters is proposed. Properties of the WSEs are studied, including their asymptotic normality. Based on those results, large sample inferences for smooth functions of the ordered parameters can be made. Especially, the confidence intervals of the maximum cell probabilities are constructed. Simulation results indicate that this interval estimation performs much better than the bootstrap approaches in the literature. Finally, the above results for ordered parameters of multinomial distributions are extended to more general distribution models. This work was supported by National Natural Science Foundation of China (Grant No. 10371126)     

Statistical estimation in varying coefficient models with surrogate data and validation sampling

Varying coefficient error-in-covariables models are considered with surrogate data and validation sampling. Without specifying any error structure equation, two estimators for the coefficient function vector are suggested by using the local linear kernel smoothing technique. The proposed estimators are proved to be asymptotically normal. A bootstrap procedure is suggested to estimate the asymptotic variances. The data-driven bandwidth selection method is discussed. A simulation study is conducted to evaluate the proposed estimating methods.     

Heteroscedasticity and/or Autocorrelation Checks in Longitudinal Nonlinear Models with Elliptical and AR（1） Errors

The aim of this paper is to study the tests for variance heterogeneity and/or autocorrelation in nonlinear regression models with elliptical and AR(1) errors. The elliptical class includes several symmetric multivariate distributions such as normal, Student-t, power exponential, among others. Several diagnostic tests using score statistics and their adjustment are constructed. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score statistics, are studied. The properties of test statistics are investigated through Monte Carlo simulations. A data set previously analyzed under normal errors is reanalyzed under elliptical models to illustrate our test methods.     

On the performance of block-bootstrap continuously updated GMM for a class of non-linear conditional moment models

In the context of the continuously updated generalized-methods-of-moments (GMM), this study evaluates the finite sample properties of Wald- and criterion-based bootstrap inference for a class of models defined by non-linear conditional moment functions. This work provides simulation evidence that validates the moving block-bootstrap (MBB) as an alternative to asymptotic approximation for robust finite sample GMM inference. The study considers data generating processes with highly non-linear conditional moment functions, weak instruments, and near failure of the identification condition. In the absence of a consensus on best practice when identification is weak, Monte Carlo results of this study are encouraging to the empirical researchers. For criterion-based tests, the MBB performs fairly well in reducing the error in the rejection frequency that occurs when first-order asymptotic critical values are used. In particular, it is possible to improve finite sample inference by inverting bootstrap Wald-type statistics which are commonly used in practice The bootstrap percentile- $t$ confidence intervals performed better than the asymptotic confidence intervals but only marginally in weakly identified specifications with high non-linear moment functions.     

Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points

The main purpose of the present paper is to establish the asymptotic properties of pseudo maximum likelihood estimators of the parameters of a multiple change-point model in the multivariate copula models when marginal distributions are unspecified but the copula function is parametrized. A pseudo likelihood ratio-type statistic is proposed for testing a sequence of observations for no change in the copula parameter against possible changes. Finally, a weighted bootstrap procedure that aims at evaluating the limiting distributions is examined.     

线性回归模型Bootstrap LM-Error检验的水平扭曲

基于OLS估计残差,本文将Bootstrap方法用于空间误差相关性LM-Error检验,综合考虑Bootstrap模拟抽样次数、空间衔接结构以及样本量,研究并比较空间误差相关Bootstrap LM-Error检验与渐近检验的水平扭曲。大量Monte Carlo实验结果显示,当模型误差不满足独立正态分布的假设条件时,空间误差相关LM-Error渐近检验的水平扭曲较大,采用Bootstrap方法可以较好地降低该水平扭曲;不管模型误差是否满足独立正态分布的假设条件,Bootstrap方法均能够有效地降低LMError渐近检验的水平扭曲。     

The bootstrap: Some large sample theory and connections with robustness

The bootstrap, discussed by Efron (1979, 1981), is a powerful tool for the nonparametric estimation of sampling distributions and asymptotic standard errors. We demonstrate consistency of the bootstrap distribution estimates for a general class of robust differentiable statistical functionals. Our conditions for consistency of the bootstrap are variants of previously considered criteria for robustness of the associated statistics. A general example shows that, for almost any location statistic, consistency of the bootstrap variance estimator requires a tail condition on the distribution from which samples are taken. A modification of Efron's estimator of standard error is shown to circumvent this problem.     

Heteroscedasticity checks for regression models

For checking on heteroscedasticity in regression models, a unified approach is proposed to constructing test statistics in parametric and nonparametric regression models. For nonparametric regression, the test is not affected sensitively by the choice of smoothing parameters which are involved in estimation of the nonparametric regression function. The limiting null distribution of the test statistic remains the same in a wide range of the smoothing parameters. When the covariate is one-dimensional, the tests are, under some conditions, asymptotically distribution-free. In the high-dimensional cases, the validity of bootstrap approximations is investigated. It is shown that a variant of the wild bootstrap is consistent while the classical bootstrap is not in the general case, but is applicable if some extra assumption on conditional variance of the squared error is imposed. A simulation study is performed to provide evidence of how the tests work and compare with tests that have appeared in the literature. The approach may readily be extended to handle partial linear, and linear autoregressive models.     

Tests for independence in non-parametric heteroscedastic regression models

Consistent procedures are constructed for testing independence between the regressor and the error in non-parametric regression models. The tests are based on the Fourier formulation of independence, and utilize the joint and the marginal empirical characteristic functions of the regressor and of estimated residuals. The asymptotic null distribution as well as the behavior of the test statistic under alternatives is investigated. A simulation study compares bootstrap versions of the proposed tests to corresponding procedures utilizing the empirical distribution function.     

Bootstrapping generalizedU-processes andV-processes and their applications in projection pursuit

In this paper, we study bootstrap approximation for generalizedU-processes (GUP) indexed by a class of functions. Under mild conditions we obtain that the asymptotic distributions of bootstrapping generalizedU-processes (BGUP) are the same as those of GUP almost surely. As a result, the asymptotic properties of bootstrap approximation for PP generalizedU-processes (BPPGUP) are obtained. In addition we have derived bootstrap approximation for generalizedV-processes (GVP). Thus, we can use BGUP or bootstrapping GVP (BGVP) to simulate GUP and GVP.This project is supported by the National Natural Science Foundation of China and the Science Foundation of Educational Committee of Guizhou.     

BOOTSTRAPPING GENERALIZED U－PROCESSES AND V－PROCESSES AND THEIR APPLICATIONS IN PROJECTION PURSUIT

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Bootstrapping Pseudolikelihood Models for Clustered Binary Data

Asymptotic properties of the parametric bootstrap procedure for maximum pseudolikelihood estimators and hypothesis tests are studied in the general framework of associated populations. The technique is applied to the analysis of toxicological experiments which, based on pseudolikelihood inference for clustered binary data, fits into this framework. It is shown that the bootstrap approximation can be used as an interesting alternative to the classical asymptotic distribution of estimators and test statistics. Finite sample simulations for clustered binary data models confirm the asymptotic theory and indicate some substantial improvements.     

用投影寻踪自助法进行多元分布函数的拟合优度检验

文中记 F 为 q 维分布函数,P 是概率测度,P_F 是由分布函数 F 所规定的概率测度.在进行统计推断时,常常需要知道统计量 R(X_1,…,X_n;F)的分布 J_n(x,F)=P_F(R(X_1,(?),X_n;F)≤x),其中 X_1,…,X_n i i d～F,i i d 表独立同分布,或者用 R(X_1,…,X_n;F)的极限分布 J(x,F).但是 J_n(x,F)和 J(x,F)经常与 F 有关,即使 F 知道,J_n(x,F)和 J(x,F)的确切表达式大多是不知道的.倘若 F 未知,就更难知道 J_n(x,F)     

Testing the equality of error distributions from k independent GARCH models

In this paper we study the problem of testing the null hypothesis that errors from $k$ independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the $k$ residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance in terms of empirical size and power of the three-sample test based on the Wilcoxon and Van der Waerden score generating functions in finite samples. The results demonstrate that the two proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated. As an illustrative example, the tests are applied to daily individual stock returns of the New York Stock Exchange data.     

Diagnosing Bootstrap Success

We show that convergence of intuitive bootstrap distributions to the correct limit distribution is equivalent to a local asymptotic equivariance property of estimators and to an asymptotic independence property in the bootstrap world. The first equivalence implies that bootstrap convergence fails at superefficiency points in the parameter space. However, superefficiency is only a sufficient condition for bootstrap failure. The second equivalence suggests graphical diagnostics for detecting whether or not the intuitive bootstrap is trustworthy in a given data analysis. Both criteria for bootstrap convergence are related to Hájek's (1970, Zeit. Wahrscheinlichkeitsth., 14, 323-330) formulation of the convolution theorem and to Basu's (1955, Sankhy, 15, 377-380) theorem on the independence of an ancillary statistic and a complete sufficient statistic.     

Validation tests for the innovation distribution in INAR time series models

Goodness-of-fit tests are proposed for the innovation distribution in INAR models. The test statistics incorporate the joint probability generating function of the observations. Special emphasis is given to the INAR(1) model and particular instances of the procedures which involve innovations from the general family of Poisson stopped-sum distributions. A Monte Carlo power study of a bootstrap version of the test statistic is included as well as a real data example. Generalizations of the proposed methods are also discussed.