On posterior consistency in nonparametric regression problems

We provide sufficient conditions to establish posterior consistency in nonparametric regression problems with Gaussian errors when suitable prior distributions are used for the unknown regression function and the noise variance. When the prior under consideration satisfies certain properties, the crucial condition for posterior consistency is to construct tests that separate from the outside of the suitable neighborhoods of the parameter. Under appropriate conditions on the regression function, we show there exist tests, of which the type I error and the type II error probabilities are exponentially small for distinguishing the true parameter from the complements of the suitable neighborhoods of the parameter. These sufficient conditions enable us to establish almost sure consistency based on the appropriate metrics with multi-dimensional covariate values fixed in advance or sampled from a probability distribution. We consider several examples of nonparametric regression problems.     

A bootstrap approach to model checking for linear models under length-biased data

In this paper, we propose two bootstrap-based model checking tests for a parametric linear model when data are affected by length-bias. These tests are based on the measure of the discrepancy between nonparametric and parametric estimators for the regression function when the data are drawn under a length-biased mechanism. We consider two different discrepancy measures: the supremum and the integral of the quadratic difference between the parametric and nonparametric estimators.     

Nonparametric methods for unbalanced multivariate data and many factor levels

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small n_i case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.     

Nonparametric tests for homogeneity of scale against ordered alternatives

Summary In this paper the nonparametric several sample scale problem is considered and some tests are proposed for the hypothesis of homogeneity versus ordered alternatives. These tests are based on statistics that are weighted linear combinations of Sugiura (1965,Osaka J. Math.,2, 385–426) type statistics proposed for testing homogeneity of scale against the omnibus alternative. For each class of test statistics suggested, the member with maximum Pitman efficiency is identified. The optimal statistics are compared with their parametric and nonparametric competitors.     

DIMENSION-REDUCTION TYPE TESTFOR LINEARITY OF ASTOCHASTIC REGRESSION MODEL

1.IntroductionLinearregressionmodelsarewidelyusedinstatisticalanalysisofexperimentalandobservationaldata,thatis,oneoftenemploysastandardlinearmodely=or K: E,a.s.,(1.1)todostatisticalanalysis,whereydenotesascalaroutcomevariableand2denotesaP-dimensionalcolumnvectorofregressorvariables.Thismodelmeansthattheprojectionofthepdimensionalexplanatory2ontotheone-dimensionalsubspaceadZcapturesalltheinformationweneedtoknowabouttheoutcomevariabley.Thisisadimension-reductionmodel.Hencewemayreachthegoalofd…     

Tests d'hypothèses sur les coefficients de Fourier d'une fonction de régression non linéaire

We consider the nonparametric regression model with regression function ƒ. We construct tests for hypotheses on Fourier coefficients of ƒ in a given band of frequencies. Such tests can be used in particular to compare two noisy signals in a frequency band. The test statistics, we use, are based on the empirical Fourier coefficients of the regression function ƒ.     

Effciency comparison between mean and log-rank tests for recurrent event time data

Recurrent event time data are common in biomedical follow-up studies, in which a study subject may experience repeated occurrences of an event of interest. In this paper, we evaluate two popular nonparametric tests for recurrent event time data in terms of their relative effciency. One is the log-rank test for classical survival data and the other a more recently developed nonparametric test based on comparing mean recurrent rates. We show analytically that, somewhat surprisingly, the log-rank test that onl...     

Two tests for heteroscedasticity in nonparametric regression

In this paper, two new tests for heteroscedasticity in nonparametric regression are presented and compared. The first of these tests consists in first estimating nonparametrically the unknown conditional variance function and then using a classical least-squares test for a general linear model to test whether this function is a constant. The second test is based on using an overall distance between a nonparametric estimator of the conditional variance function and a parametric estimator of the variance of the model under the assumption of homoscedasticity. A bootstrap algorithm is used to approximate the distribution of this test statistic. Extended versions of both procedures in two directions, first, in the context of dependent data, and second, in the case of testing if the variance function is a polynomial of a certain degree, are also described. A broad simulation study is carried out to illustrate the finite sample performance of both tests when the observations are independent and when they are dependent.     

Testing Serial Independence via Density-Based Measures of Divergence

This article reviews some nonparametric serial independence tests based on measures of divergence between densities. Among others, the well-known Kullback–Leibler, Hellinger, Tsallis, and Rosenblatt divergences are analyzed. Moreover, their copula-based version is taken into account. Via a wide simulation study, the performances of the considered serial independence tests are compared under different settings. Both single-lag and multiple-lag testing procedures are investigated to find out the best “omnibus” solution.     

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.     

Rank analogues of the likelihood ratio test for an ordered alternative in a two-way layout

Summary Distribution-free tests for no treatment effect against the simple order alternative in a two-way layout with equal number of observations per cell are considered. The nonparametric test statistics are constructed by the rank analogues of the likelihood ratio test statistic assuming normality (i) based on within-block rankings and (ii) based on combined rankings of all the observations after alignment within each block. The exact distributions are given and large sample properties are investigated. The asymptotic power of the test (i) as the number of observations per eell tends to infinity can be satisfied enough, and in the case that the number of blocks tends to infinity, the asymptotic power of the test (ii) is almost higher than that of the test (i). Also these rank tests are compared with linear rank tests and it is shown that these proposed tests are robust by a table.     

On nonparametric tests for symmetry inRm

This paper considers the problem for testing symmetry of a distribution inR ^m based on the empirical distribution function. Limit theorems which play important roles for investigating asymptotic behavior of such tests are obtained. The limit processes of the theorems are multiparameter Wiener process. Based on the limit theorems, nonparametric tests are proposed whose asymptotic distributions are functionals of a multiparameter standard Wiener process. The tests are compared asymptotically with each other in the sense of Bahadur.     

Stochastic integrals of empirical-type processes with applications to censored regression

Motivated by the analysis of linear rank estimators and the Buckley-James nonparametric EM estimator in censored regression models, we study herein the asymptotic properties of stochastic integrals of certain two-parameter empirical processes. Applications of these results on empirical processes and their stochastic integrals to the asymptotic analysis of censored regression estimators are also given.     

Goodness-of-fit tests for a semiparametric model under random double truncation

Doubly truncated data are commonly encountered in areas like medicine, astronomy, economics, among others. A semiparametric estimator of a doubly truncated random variable may be computed based on a parametric specification of the distribution function of the truncation times. This semiparametric estimator outperforms the nonparametric maximum likelihood estimator when the parametric information is correct, but might behave badly when the assumed parametric model is far off. In this paper we introduce several goodness-of-fit tests for the parametric model. The proposed tests are investigated through simulations. For illustration purposes, the tests are also applied to data on the induction time to acquired immune deficiency syndrome for blood transfusion patients.     

非参数模型均值函数结构变点的Bootstrap检测

本文检测非参数回归模型均值函数结构变点,针对均值函数跃度的长期均值为零时,基于残量的CUSUM统计量对均值函数结构变点检验无效的问题,本文提出了一种基于均值函数的核估计的检验统计量,得到统计量在原假设和备择假设下的极限分布,并构造Bootstrap方法对非参数回归模型均值函数结构变点进行检验,证明了检验和估计的一致性;模拟结果表明本文方法明显优于已有方法。     

Nonparametric lack-of-fit tests for parametric mean-regression models with censored data

We developed two kernel smoothing based tests of a parametric mean-regression model against a nonparametric alternative when the response variable is right-censored. The new test statistics are inspired by the synthetic data and the weighted least squares approaches for estimating the parameters of a (non)linear regression model under censoring. The asymptotic critical values of our tests are given by the quantiles of the standard normal law. The tests are consistent against fixed alternatives, local Pitman alternatives and uniformly over alternatives in Hölder classes of functions of known regularity.     

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.     

A class of multi-sample nonparametric tests for panel count data

This paper considers the problem of multi-sample nonparametric comparison of mean functions of point processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct a class of nonparametric test statistics based on the integrated weighted differences between the estimated mean functions of the point processes. The asymptotic distributions of the proposed statistics are rigorously derived when the monotonicity assumptions for weight processes are removed, and their finite-sample properties are examined through Monte Carlo simulations. The simulation results show that the proposed methods are good for practical use and are slightly powerful than the existing tests. A set of panel count data from a cancer study is analyzed and presented as an illustrative example.     

Runs tests for assessing volatility forecastability in financial time series

In this work we refine a nonparametric methodology firstly applied in Christoffersen and Diebold [Review of Economics and Statistics 82 (2000) 12] for assessing volatility forecastability in financial time series based on discretization and on the use of runs tests. Empirical results are provided for SP500 and MIB30 indexes that lead naturally to a discretized one-period Markov chain. The results are confirmed with other persistence measures and their robustness is studied via numerical simulation.     

A nonparametric version of Wilks’ lambda—Asymptotic results and small sample approximations

We propose a nonparametric version of Wilks’ lambda (the multivariate likelihood ratio test) and investigate its asymptotic properties under the two different scenarios of either large sample size or large number of samples. For unbalanced samples, a weighted and an unweighted variant are introduced. The unweighted variant of the proposed test appears to be novel also in the normal-theory context.The theoretical results are supplemented by a simulation study with parameter settings that are motivated by clinical and agricultural data, considering in particular the performance for small sample sizes, small number of samples, and varying dimensions. Inference methods based on the asymptotic sampling distribution and a small sample approximation are compared to permutation tests and to other parametric and nonparametric procedures. Application of the proposed method is illustrated by examples.