Simultaneous confidence band and hypothesis test in generalised varying-coefficient models

Generalised varying-coefficient models (GVC) are very important models. There are a considerable number of literature addressing these models. However, most of the existing literature are devoted to the estimation procedure. In this paper, we systematically investigate the statistical inference for GVC, which includes confidence band as well as hypothesis test. We establish the asymptotic distribution of the maximum discrepancy between the estimated functional coefficient and the true functional coefficient. We compare different approaches for the construction of confidence band and hypothesis test. Finally, the proposed statistical inference methods are used to analyse the data from China about contraceptive use there, which leads to some interesting findings.     

Bootstrap,wild bootstrap,and asymptotic normality

Summary We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.This work has been supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 Stochastische Mathematische Modelle     

TESTS OF COVARIANCE MATRIX BY USING PROJECTION PURSUIT AND BOOTSTRAP METHOD

Testing equality of covariance matrix has long been an interesting issue in statistics inference, To overcome the sparseness of data points in high-dimensional space and deal with the general cases, the author suggests several projection pursuit type statistics. Some results on the limiting distidbutions of the statistics are obtained. Some properties of bootstrap approximation are investigated. Furthermore, for computational reasons an approximation for the statistics based on number-theoretic roethod is applied. Several simulation experiments are performed.     

On nondifferentiable functions and the bootstrap

Summary We investigate a class of statistical problems, where usual bootstrap methods fail, and discuss two alternative solutions. In particular, a stochastic procedure for constructing confidence sets is proposed. Special applications are the eigenvalues of a covariance matrix and minimum distance functionals.Work supported by the Miller Institute for Basic Research in Science, Berkeley     

Enhanced one-sided confidence regions for a multivariate location parameter

If a one-sided test for a multivariate location parameter is inverted, the resulting confidence region may have an unpleasant shape. In particular, if the null and alternative hypothesis are both composite and complementary, the confidence region usually does not resemble the alternative parameter region in shape, but rather a reflected version of the null parameter region.We illustrate this effect and show one possibility of obtaining confidence regions for the location parameter that are smaller and have a more suitable shape for the type of problems investigated. This method is based on the closed testing principle applied to a family of nested hypotheses.     

Diagnostic checking for multivariate regression models

Diagnostic checking for multivariate parametric models is investigated in this article. A nonparametric Monte Carlo Test (NMCT) procedure is proposed. This Monte Carlo approximation is easy to implement and can automatically make any test procedure scale-invariant even when the test statistic is not scale-invariant. With it we do not need plug-in estimation of the asymptotic covariance matrix that is used to normalize test statistic and then the power performance can be enhanced. The consistency of NMCT approximation is proved. For comparison, we also extend the score type test to one-dimensional cases. NMCT can also be applied to diverse problems such as a classical problem for which we test whether or not certain covariables in linear model has significant impact for response. Although the Wilks lambda, a likelihood ratio test, is a proven powerful test, NMCT outperforms it especially in non-normal cases. Simulations are carried out and an application to a real data set is illustrated.     

On a new multivariate two-sample test

In this paper we propose a new test for the multivariate two-sample problem. The test statistic is the difference of the sum of all the Euclidean interpoint distances between the random variables from the two different samples and one-half of the two corresponding sums of distances of the variables within the same sample. The asymptotic null distribution of the test statistic is derived using the projection method and shown to be the limit of the bootstrap distribution. A simulation study includes the comparison of univariate and multivariate normal distributions for location and dispersion alternatives. For normal location alternatives the new test is shown to have power similar to that of the t- and T²-Test.     

Oracally efficient spline smoothing of nonlinear additive autoregression models with simultaneous confidence band

Under weak conditions of smoothness and mixing, we propose spline-backfitted spline (SBS) estimators of the component functions for a nonlinear additive autoregression model that is both computationally expedient for analyzing high dimensional large time series data, and theoretically reliable as the estimator is oracally efficient and comes with asymptotically simultaneous confidence band. Simulation evidence strongly corroborates with the asymptotic theory.     

Robust semiparametric M-estimation and the weighted bootstrap

M-estimation is a widely used technique for statistical inference. In this paper, we study properties of ordinary and weighted M-estimators for semiparametric models, especially when there exist parameters that cannot be estimated at the convergence rate. Results on consistency, rates of convergence for all parameters, and consistency and asymptotic normality for the Euclidean parameters are provided. These results, together with a generic paradigm for studying semiparametric M-estimators, provide a valuable extension to previous related research on semiparametric maximum-likelihood estimators (MLEs). Although penalized M-estimation does not in general fit in the framework we discuss here, it is shown for a great variety of models that many of the forgoing results still hold, including the consistency and asymptotic normality of the Euclidean parameters. For semiparametric M-estimators that are not likelihood based, general inference procedures for the Euclidean parameters have not previously been developed. We demonstrate that our paradigm leads naturally to verification of the validity of the weighted bootstrap in this setting. For illustration, several examples are investigated in detail. The new M-estimation framework and accompanying weighted bootstrap technique shed light on a universal way of investigating semiparametric models.     

Asymptotic confidence intervals for Poisson regression

Let (X,Y) be a R^d×N₀-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L₁ distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

Bartlett correctable two-sample adjusted empirical likelihood

We propose a two-sample adjusted empirical likelihood (AEL) to construct confidence regions for the difference of two d-dimensional population means. This method eliminates the non-definition of the usual two-sample empirical likelihood (EL) and is shown to be Bartlett correctable. We further show that when the adjustment level is half the Bartlett correction factor for the usual two-sample EL, the two-sample AEL has the same high-order precision as the EL with Bartlett correction. To enhance the performance of the two-sample AEL with adjustment level being half the Bartlett correction factor, we propose a less biased estimate of the Bartlett correction factor. The efficiency of the proposed method is illustrated by simulations and a real data example.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.     

Estimation for a marginal generalized single-index longitudinal model

In this paper, we suggest an estimating equations based approach to study a general single-index model with a given out-layer link for longitudinal data and treat the classical one as its special case. Within a wide range of bandwidths which is for estimating the inner-layer nonparametric link, the root-n consistency of the estimator of the index can be proved. The estimation efficiency can be achieved even when there is an infinite-dimensional nuisance parameter to be estimated. The performance of the new method is assessed through the comparison with other existing methods and through an application to an epileptic seizure study.     

Consistent tuning parameter selection in high dimensional sparse linear regression

An exhaustive search as required for traditional variable selection methods is impractical in high dimensional statistical modeling. Thus, to conduct variable selection, various forms of penalized estimators with good statistical and computational properties, have been proposed during the past two decades. The attractive properties of these shrinkage and selection estimators, however, depend critically on the size of regularization which controls model complexity. In this paper, we consider the problem of consistent tuning parameter selection in high dimensional sparse linear regression where the dimension of the predictor vector is larger than the size of the sample. First, we propose a family of high dimensional Bayesian Information Criteria (HBIC), and then investigate the selection consistency, extending the results of the extended Bayesian Information Criterion (EBIC), in Chen and Chen (2008) to ultra-high dimensional situations. Second, we develop a two-step procedure, the SIS+AENET, to conduct variable selection in p>n situations. The consistency of tuning parameter selection is established under fairly mild technical conditions. Simulation studies are presented to confirm theoretical findings, and an empirical example is given to illustrate the use in the internet advertising data.     

Nonconcave penalized inverse regression in single-index models with high dimensional predictors

In this paper we aim to estimate the direction in general single-index models and to select important variables simultaneously when a diverging number of predictors are involved in regressions. Towards this end, we propose the nonconcave penalized inverse regression method. Specifically, the resulting estimation with the SCAD penalty enjoys an oracle property in semi-parametric models even when the dimension, p_n, of predictors goes to infinity. Under regularity conditions we also achieve the asymptotic normality when the dimension of predictor vector goes to infinity at the rate of p_n=o(n^1/3) where n is sample size, which enables us to construct confidence interval/region for the estimated index. The asymptotic results are augmented by simulations, and illustrated by analysis of an air pollution dataset.     

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.     

Scan clustering: A false discovery approach

We present a method that scans a random field for localized clusters while controlling the fraction of false discoveries. We use a kernel density estimator as the test statistic and adjust for the bias in this estimator by a method we introduce in this paper. We also show how to combine information across multiple bandwidths while maintaining false discovery control.     

Multivariate nonparametric tests in a randomized complete block design

In this paper multivariate extensions of the Friedman and Page tests for the comparison of several treatments are introduced. Related unadjusted and adjusted treatment effect estimates for the multivariate response variable are also found and their properties discussed. The test statistics and estimates are analogous to the traditional univariate methods. In test constructions, the univariate ranks are replaced by multivariate spatial ranks (J. Nonparam. Statist. 5 (1995) 201). Asymptotic theory is developed to provide approximations for the limiting distributions of the test statistics and estimates. Limiting efficiencies of the tests and treatment effect estimates are found in the multivariate normal and t distribution cases. The tests are rotation invariant only, but affine invariant versions can be easily constructed. The theory is illustrated by an example.