线性混合效应模型预测值关于白噪声偏离的敏感性分析

本文应用Banerjee和Magnus于1999年提出的局部敏感性分析方法讨论了线性混合效应模型中预测值关于误差项白噪声偏离的敏感性问题,提出了敏感性度量统计量,并在AR(1)和MA(1)误差项条件下数值模拟了这些统计量的表现.结果表明,预测值在AR(1)和MA(1)误差项下一般都不具有明显的敏感性;特别是在MA(1)下预测值对误差项的白噪声偏离通常具有很好的稳健性.     

Testing for the change of the mean-reverting parameter of an autoregressive model with stationary Gaussian noise

The likelihood ratio test for a change in the mean-reverting parameter of a first order autoregressive model with stationary Gaussian noise is considered. The test statistic converges in distribution to the Gumbel extreme value distribution under the null hypothesis of no change-point for a large class of covariance structures including long-memory processes as the fractional Gaussian noise.

     

Robustness of Stein-type estimators under a non-scalar error covariance structure

The Stein-rule (SR) and positive-part Stein-rule (PSR) estimators are two popular shrinkage techniques used in linear regression, yet very little is known about the robustness of these estimators to the disturbances’ deviation from the white noise assumption. Recent studies have shown that the OLS estimator is quite robust, but whether this is so for the SR and PSR estimators is less clear as these estimators also depend on the F statistic which is highly susceptible to covariance misspecification. This study attempts to evaluate the effects of misspecifying the disturbances as white noise on the SR and PSR estimators by a sensitivity analysis. Sensitivity statistics of the SR and PSR estimators are derived and their properties are analyzed. We find that the sensitivity statistics of these estimators exhibit very similar properties and both estimators are extremely robust to MA(1) disturbances and reasonably robust to AR(1) disturbances except for the cases of severe autocorrelation. The results are useful in light of the rising interest of the SR and PSR techniques in the applied literature.     

Testing for no effect via splines

A spline-based test statistic for a constant mean function is proposed based on the penalized residual sum-of-squares difference between the null model and a B-spline model in which the regression function is approximated with P-splines approach. When the number of knots is fixed, the limiting null distribution of the test statistic is shown to be the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. A smoothing parameter is selected by setting a specified value equal to the expected value of the test statistic under the null hypothesis. Simulation experiments are conducted to study the proposed spline-based test statistic’s finite-sample properties.     

Modified cramer-von mises goodness-of-fit tests for spectral distribution functions

Modifications to the Cramer-von Mises goodness-of-fit test statistic for spectral distributions are discussed. The modifications consist of inserting weight functions into the usual sto¬chastic integral for the test statistic. Conditions on the weight function are given under which the integral of the weighted square of the difference between the empirical and theoretical spectral distribution functions converges in distribution to the corresponding integral of a process related to Brownian Motion. The distributions of the test statistic under certain alternatives to the null hypothesis are also discussed. A discussion is given of the large sample distributions for weight function of the form ψ(t) = at ^k ,k < –2.     

Approximation by randomly weighting method in censored regression model

Censored regression (“Tobit”) models have been in common use, and their linear hypothesis testings have been widely studied. However, the critical values of these tests are usually related to quantities of an unknown error distribution and estimators of nuisance parameters. In this paper, we propose a randomly weighting test statistic and take its conditional distribution as an approximation to null distribution of the test statistic. It is shown that, under both the null and local alternative hypotheses, conditionally asymptotic distribution of the randomly weighting test statistic is the same as the null distribution of the test statistic. Therefore, the critical values of the test statistic can be obtained by randomly weighting method without estimating the nuisance parameters. At the same time, we also achieve the weak consistency and asymptotic normality of the randomly weighting least absolute deviation estimate in censored regression model. Simulation studies illustrate that the performance of our proposed resampling test method is better than that of central chi-square distribution under the null hypothesis. This work was supported by National Natural Science Foundation of China (Grant No. 10471136), PhD Program Foundation of the Ministry of Education of China, and Special Foundations of the Chinese Academy of Sciences and University of Science and Technology of China     

Test for a specified signal when the noise covariance matrix is unknown

In the univariate case it is well known that the one sided t test is uniformly most powerful for the null hypothesis against all one sided alternatives. Such a property does not easily extend to the multivariate case. In this paper, a test derived for the hypothesis that the mean of a vector random variable is zero against specified alternatives, when the covariance matrix is unknown. This test depends on the given alternatives and is more powerful than Hotelling's T². The results are derived both for real and complex vector observations and under normal and spherical distributions. The properties of the proposed tests are investigated in detail when a single alternative is specified.     

A novel trace test for the mean parameters in a multivariate growth curve model

A trace test for the mean parameters of the growth curve model is proposed. It is constructed using the restricted maximum likelihood followed by an estimated likelihood ratio approach. The statistic reduces to the Lawley-Hotelling trace test for the Multivariate Analysis of Variance (MANOVA) models. Our test statistic is, therefore, a natural extension of the classical trace test to GMANOVA models. We show that the distribution of the test under the null hypothesis does not depend on the unknown covariance matrix Σ. We also show that the distributions under the null and alternative hypotheses can be represented as sums of weighted central and non-central chi-square random variables, respectively. Under the null hypothesis, the Satterthwaite approximation is used to get an approximate critical point. A novel Satterthwaite type approximation is proposed to obtain an approximate power. A simulation study is performed to evaluate the performance of our proposed test and numerical examples are provided as illustrations.     

Two step-down tests for equality of covariance matrices

The classical problem of testing the equality of the covariance matrices from k ? 2 p-dimensional normal populations is reexamined. The likelihood ratio (LR) statistic, also called Bartlett’s statistic, can be decomposed in two ways, corresponding to two distinct component-wise decompositions of the null hypothesis in terms of the covariance matrices or precision matrices, respectively. The factors of the LR statistic that appear in these two decompositions can be interpreted as conditional and unconditional LR statistics for the component-wise null hypotheses, and their mutual independence under the null hypothesis allows the determination of the overall significance level.     

A test of the hypothesis of partial common principal components

A test of the equality of the first h eigenvectors of covariance matrices of several populations is constructed without the assumption that the sampled distributions are Gaussian. It is proved that the test statistic is asymptotically chi-square distributed. In this general setting, an explicit formula for column space of the asymptotic covariance matrix of the sample eigenvectors is derived and the rank of this matrix is computed. An essential assumption in deriving the asymptotic distribution of the presented test statistic is the existence of the finite fourth moments and the simplicity of the h largest eigenvalues of population covariance matrices, which makes possible to use the formulas for derivatives of eigenvectors of symmetric matrices.     

Asymptotic distribution of the LR statistic for equality of the smallest eigenvalues in high-dimensional principal component analysis

This paper deals with the distribution of the LR statistic for testing the hypothesis that the smallest eigenvalues of a covariance matrix are equal. We derive an asymptotic null distribution of the LR statistic when the dimension p and the sample size N approach infinity, while the ratio p/N converging on a finite nonzero limit c(0,1). Numerical simulations revealed that our approximation is more accurate than the classical chi-square-type approximation as p increases in value.     

Hypothesis tests for high-dimensional covariance structures

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

     

Testing inference in heteroskedastic fixed effects models

This paper considers the issue of performing testing inference in fixed effects panel data models under heteroskedasticity of unknown form. We use numerical integration to compute the exact null distributions of different quasi-t test statistics and compare them to their limiting counterpart. The test statistics use different heteroskedasticity-consistent standard errors. Our results reveal that the asymptotic approximation is usually poor in small samples when the test statistic is based on the covariance matrix estimator proposed by Arellano (1987). The quality of the approximation is greatly increased when the standard error is obtained using other heteroskedasticity-consistent estimators, most notably the CHC4 estimator. Our results also reveal that the performance of Arellano’s test improves considerably when standard errors are computed using restricted residuals.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.     

Conditional properties of Bayesian interval estimates

Consider the construction of an interval estimate for a scalar parameter of interest in the presence of orthogonal nuisance parameters. A conditional prior density on the parameter of interest that is proportional to the square root of its information element, generates one-sided Bayes intervals that are approximately confidence intervals as well, having coverage error of orderO(1/n), wheren is the sample size. We show that the frequency property of these intervals also holds conditionally on a locally ancillary statistic near the true parameter value.     

Testing equality between several populations covariance operators

In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert–Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set.     

Numerical Evaluation of the Evans Function by Magnus Integration

We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane. In this situation, a substantial portion of the computational effort—the numerical evaluation of the iterated integrals which appear in the Magnus series—can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated a priori. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers. AMS subject classification (2000) 65F20     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.