核实数据下删失线性EV模型的经验似然推断

本文考虑协变量带有误差的删失线性回归模型,借助于核实数据,对回归系数构造了两种经验对数似然比统计量,证明了所提出的估计的经验对数似然比统计量渐近收敛到一个自由度为1的独立χ2变量的加权和;而经调整后所得的调整的经验对数似然比统计量具有渐近标准χ2p分布,所得结果可以用来构造未知参数的置信域,通过模拟研究在置信域的精度及其平均区间长度大小方面进行了比较。     

核实数据下响应变量缺失的线性EV模型经验似然推断

考虑响应变量随机缺失而协变量带有误差的线性模型,借助于核实数据和借补方法,构造了回归系数的两种经验似然比,证明了所提出的估计的经验对数似然比渐近于一个自由度为1的独立χ2变量的加权和;而经调整后所得的调整经验对数似然比渐近于自由度为p的χ2分布,该结果可以用来构造未知参数的置信域.此外,我们也构造了响应均值的调整经验对数似然比统计量,并证明了所提出的统计量渐近于x2分布,可用此结果构造响应均值的置信域.通过模拟研究比较了置信域的精度及其平均区间长度.     

Empirical Bayes estimation in a multiple linear regression model

Summary Estimation of the vector β of the regression coefficients in a multiple linear regressionY=Xβ+ε is considered when β has a completely unknown and unspecified distribution and the error-vector ε has a multivariate standard normal distribution. The optimal estimator for β, which minimizes the overall mean squared error, cannot be constructed for use in practice. UsingX, Y and the information contained in the observation-vectors obtained fromn independent past experiences of the problem, (empirical Bayes) estimators for β are exhibited. These estimators are compared with the optimal estimator and are shown to be asymptotically optimal. Estimators asymptotically optimal with rates nearO(n ⁻¹) are constructed. Supported in part by a Natural Sciences and Engineering Research Council of Canada grant.     

Empirical likelihood for linear models under negatively associated errors

In this paper, we discuss the construction of the confidence intervals for the regression vector β in a linear model under negatively associated errors. It is shown that the blockwise empirical likelihood (EL) ratio statistic for β is asymptotically χ²-type distributed. The result is used to obtain an EL based confidence region for β.     

Empirical likelihood for single-index models

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).     

On the Asymptotics of Fekete-Type Points for Univariate Radial Basis Interpolation

Suppose that K ^d is compact and that we are given a function fC(K) together with distinct points x_iK, 1in. Radial basis interpolation consists of choosing a fixed (basis) function g : ⁺→ and looking for a linear combination of the translates g(|x−x_j|) which interpolates f at the given points. Specifically, we look for coefficients c_j such that has the property that F(x_i)=f(x_i), 1in. The Fekete-type points of this process are those for which the associated interpolation matrix [g(|x_i−x_j|)]_1i,jn has determinant as large as possible (in absolute value). In this work, we show that, in the univariate case, for a broad class of functions g, among all point sequences which are (strongly) asymptotically distributed according to a weight function, the equally spaced points give the asymptotically largest determinant. This gives strong evidence that the Fekete points themselves are indeed asymptotically equally spaced.     

Randomly censored partially linear single-index models

This paper proposes a method for estimation of a class of partially linear single-index models with randomly censored samples. The method provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure-time models for survival analysis. The estimation procedure involves three stages: first, transform the censored data into synthetic data or pseudo-responses unbiasedly; second, obtain quasi-likelihood estimates of the regression coefficients in both linear and single-index components by an iteratively algorithm; finally, estimate the unknown nonparametric regression function using techniques for univariate censored nonparametric regression. The estimators for the regression coefficients are shown to be jointly root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as all the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodology.     

k-NN METHOD IN PARTIAL LINEAR MODEL UNDER RANDOM CENSORSHIP

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Asymptotic results in segmented multiple regression

This paper studies the asymptotic behavior of the least squares estimators in segmented multiple regression. For a model with more than one partitioning variable, each of which has one or more change-points, we study the asymptotic properties of the estimated change-points and regression coefficients. Using techniques in empirical process theory, we prove the consistency of the least squares estimators and also establish the asymptotic normality of the estimated regression coefficients. For the estimated change-points, we obtain their consistency at the rates of or 1/n, with or without continuity constraints, respectively. The change-points estimated under the continuity constraints are also shown to asymptotically have a multivariate normal distribution. For the case where the regression mean functions are not assumed to be continuous at the change-points, the asymptotic distribution of the estimated change-points involves a step function process, whose distribution does not follow a well-known distribution.     

Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q x 1 responses, the bounded and fixed (or adaptive) p × q regressors Zi and the general link function, under the most general assumption on the minimum eigenvalue of ZiZ'i,the moment condition on responses as weak as possible and the other mild regular conditions, we prove that the maximum quasi-likelihood estimates for the regression parameter vector are asymptotically normal and strongly consistent.     

Automatic Smoothing Spline Projection Pursuit

Abstract

A highly flexible nonparametric regression model for predicting a response y given covariates {x_k}^d _k=1 is the projection pursuit regression (PPR) model ? = h(x) = β₀ + Σ_jβ_jf_j(α^T _jx) where the f_j , are general smooth functions with mean 0 and norm 1, and Σ^d _k=1α² _kj=1. The standard PPR algorithm of Friedman and Stuetzle (1981) estimates the smooth functions f_j using the supersmoother nonparametric scatterplot smoother. Friedman's algorithm constructs a model with M _max linear combinations, then prunes back to a simpler model of size M ≤ M _max, where M and M _max are specified by the user. This article discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients α_j, the amount of smoothing in each direction, and the number of terms M and M _max are determined to optimize a single generalized cross-validation measure.     

Empirical likelihood-based inference in a partially linear model for longitudinal data

A partially linear model with longitudinal data is considered, empirical likelihood to infer- ence for the regression coefficients and the baseline function is investigated, the empirical log-likelihood ratios is proven to be asymptotically chi-squared, and the corresponding confidence regions for the pa- rameters of interest are then constructed. Also by the empirical likelihood ratio functions, we can obtain the maximum empirical likelihood estimates of the regression coefficients and the baseline function, and prove the asymptotic normality. The numerical results are conducted to compare the performance of the empirical likelihood and the normal approximation-based method, and a real example is analysed.     

Bootstrapping Autoregressive and Moving Average Parameter Estimates of Infinite Order Vector Autoregressive Processes

We consider anr-dimensional multivariate time series {y_t, tZ} which is generated by an infinite order vector autoregressive process. We show that a bootstrap procedure which works by generating time series replicates via an estimated finitek-order vector autoregressive process (k→∞ at an appropriate rate with the sample size) gives asymptotically valid approximations to the joint distribution of the growing set of estimated autoregressive coefficients and to the corresponding set of estimated moving average coefficients (impuls responses).     

Infinite T?plitz–Lipschitz matrices and operators

We introduce a class of infinite matrices (A_ss￠, s, s￠ ? \mathbbZ^d){(A_{ss\prime}, s, s\prime \in \mathbb{Z}^d)} , which are asymptotically (as |s| + |s′| → ∞) close to Hankel–T?plitz matrices. We prove that this class forms an algebra, and that flow-maps of nonautonomous linear equations with coefficients from the class also belong to it.     

Discrete time parametric models with long memory and infinite variance

We study a large class of infinite variance time series that display long memory. They can be represented as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a stable distribution with index 1 < α < 2 (stable fractional ARIMA is a particular example). Assume that the coefficients of the linear process depend on an unknown parameter vector β which is to be estimated from a series of length n. We show that a Whittle-type estimator β_n for β is consistent (β_n converges to the true value β₀ in probability as n → ∞), and, under some additional conditions, we characterize the limiting distribution of the rescaled differences (n/logn)^1/ga(β_n − β₀).     

Bounds for the Concentration of Asymptotically Normal Statistics

Let X₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal bound is derived for the concentration function of an arbitrary real-valued statistic T = T (X ₁,...,X _n ) for which ET² < . Applications are given for Wilcoxon"s rank-sum statistic, U-statistics, Student"s statistic, the two-sample Student statistic and linear regression.     

Transformations with Improved Chi-Squared Approximations

Suppose that a nonnegative statistic T is asymptotically distributed as a chi-squared distribution with f degrees of freedom, χ²_f, as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of χ²_f up to the order O(n⁻¹). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as χ²_f up to O(n⁻¹), or with errors of terms of O(n⁻²). Bartlett-type corrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed by using their asymptotic expansions up to O(n⁻¹). The purpose of the present paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of χ²_f up to O(n⁻¹). It may be noted that the proposed transformations can be applied to a wide class of statistics whether their asymptotic expansions are available or not. A numerical study of some test statistics that are not a log-likelihood ratio statistic is discribed. It is shown that the proposed transformations of these statistics give a larger improvement to the chi-squared approximation than do the Bartlett corrections. Further, it is seen that the proposed approximations are comparable with the approximation based on an Edgeworth expansion.     

Estimation of Mean and Covariance Operator of Autoregressive Processes in Banach Spaces

The autoregressive model in a Banach space (ARB) contains many continuous time processes used in practice, for example, processes that satisfy linear stochastic differential equations of order k, a very particular case being the Ornstein–Uhlenbeck process. In this paper we study empirical estimators for ARB processes. In particular we show that, under some regularity conditions, the empirical mean is asymptotically optimal with respect to a.s. convergence and convergence of order 2. Limit in distribution and the law of the iterated logarithm are also presented. Concerning the empirical covariance operator we note that, if (X _n, n ∈ &Zopf;) is ARB then (X _n ⊗ X _n, n ∈ &Zopf;) is AR in a suitable space of linear operators. This fact allows us to interpret the empirical covariance operator as a sample mean of an AR and to derive similar results for it. This revised version was published online in August 2006 with corrections to the Cover Date.     

Empirical likelihood analysis of longitudinal data involving within-subject correlation

In this paper we use profile empirical likelihood to construct confidence regions for regression coefficients in partially linear model with longitudinal data. The main contribution is that the within-subject correlation is considered to improve estimation efficiency. We suppose a semi-parametric structure for the covariances of observation errors in each subject and employ both the first order and the second order moment conditions of the observation errors to construct the estimating equations. Although there are nonparametric estimators, the empirical log-likelihood ratio statistic still tends to a standard ?? _p ² variable in distribution after the nuisance parameters are profiled away. A data simulation is also conducted.     

On measures of uniformly distributed sequences and Benford's law

The metric theory of uniform distribution of sequences is complemented by considering product measures with not necessarily identical factors. A necessary and sufficient condition is given under which a general product measure assigns the value one to the set of uniformly distributed sequences. For a stationary random product measure, almost all sequences are uniformly distributed with probability one. The discrepancy is estimated byN ^–1/2log³ N for sufficiently largeN. Thus the metric predominance of uniformly distributed sequences is stated, and a further explanation for Benford's law is provided. The results can also be interpreted as estimates of the empirical distribution function for non-identical distributed samples.