Nonparametric two‐step regression estimation when regressors and error are dependent

This paper considers estimation of the function g in the model Y_t = g(X_t ) + ?_t when E(?_t|Xt) ≠ 0 with nonzero probability. We assume the existence of an instrumental variable Z_t that is independent of ?_t, and of an innovation ηt = X_t — E(X_t|Z_t). We use a nonparametric regression of X_t on Z_t to obtain residuals η_t, which in turn are used to obtain a consistent estimator of g. The estimator was first analyzed by Newey, Powell & Vella (1999) under the assumption that the observations are independent and identically distributed. Here we derive a sample mean‐squared‐error convergence result for independent identically distributed observations as well as a uniform‐convergence result under time‐series dependence.     

Bayesian analysis of skew-t multivariate null intercept measurement error model

The multivariate skew-t distribution (J Multivar Anal 79:93–113, 2001; J R Stat Soc, Ser B 65:367–389, 2003; Statistics 37:359–363, 2003) includes the Student t, skew-Cauchy and Cauchy distributions as special cases and the normal and skew–normal ones as limiting cases. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis of repeated measures, pretest/post-test data, under multivariate null intercept measurement error model (J Biopharm Stat 13(4):763–771, 2003) where the random errors and the unobserved value of the covariate (latent variable) follows a Student t and skew-t distribution, respectively. The results and methods are numerically illustrated with an example in the field of dentistry.     

Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

The estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Mean square error and efficiency of the least squares estimator over interval constraints

We derive the mean square error of an interval constrained least squares estimator (INCLS) for a regression model. We then show that the INCLS estimator dominates, in mean square error, the unconstrained least squares estimator provided the regression residuais are normally distri'iiuted and Ynat Yrie imposed coii-

straint is satisfied or nearly satisfied.     

Calculating the mvue of the fraction nonconforming for the bivariate normal

The minimum variance unbiased estimator of the proportion lying outside an m-dimensional rectangle for multivariate normal populations was derived by Baillie (1987a, b). The estimator is a natural extension of a univariate estimator widely used in acceptance sampling. Computation of the multivariate estimator is nontrivial; one must integrate a multivariate density over the intersection of an m-dimensional ellipsoid and an m-dimensional rectangle. We propose an algorithm for the bivariate case which involves a one-dimensional numerical integration and calls to routines for either an incomplete beta function or a Student's t cumulative distribution function     

A Note on parameter and standard error estimation in adaptive robust regression

This paper introduces practical methods of parameter and standard error estimation for adaptive robust regression where errors are assumed to be from a normal/independent family of distributions. In particular, generalized EM algorithms (GEM) are considered for the two cases of t and slash families of distributions. For the t family, a one step method is proposed to estimate the degree of freedom parameter. Use of empirical information is suggested for standard error estimation. It is shown that this choice leads to standard errors that can be obtained as a by-product of the GEM algorithm. The proposed methods, as discussed, can be implemented in most available nonlinear regression programs. Details of implementation in SAS NLIN are given using two specific examples.     

Risk comparison of the Stein-rule estimator in a linear regression model with omitted relevant regressors and multivariate<Emphasis Type="Italic">t</Emphasis> errors under the Pitman nearness criterion

In this paper we consider a linear regression model with omitted relevant regressors and multivariatet error terms. The explicit formula for the Pitman nearness criterion of the Stein-rule (SR) estimator relative to the ordinary least squares (OLS) estimator is derived. It is shown numerically that the dominance of the SR estimator over the OLS estimator under the Pitman nearness criterion can be extended to the case of the multivariatet error distribution when the specification error is not severe. It is also shown that the dominance of the SR estimator over the OLS estimator cannot be extended to the case of the multivariatet error distribution when the specification error is severe. This research is partially supported by the Grants-in-Aid for 21st Century COE program.     

Estimation and hypothesis testing in multivariate linear regression models under non normality

This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modified maximum likelihood estimation method that provides the estimator, called modified maximum likelihood estimator (MMLE), in closed form. These estimators are shown to be unbiased, efficient, and robust as compared to the widely used least square estimators (LSEs). Also, the tests based upon MMLEs are found to be more powerful than the similar tests based upon LSEs.     

A class of weighted multivariate distributions related to doubly truncated multivariate t-distribution

This article introduces a class of weighted multivariate t-distributions, which includes the multivariate generalized Student t and multivariate skew t as its special members. This class is defined as the marginal distribution of a doubly truncated multivariate generalized Student t-distribution and studied from several aspects such as weighting of probability density functions, inequality constrained multivariate Student t-distributions, scale mixtures of multivariate normal and probabilistic representations. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.     

Improved set estimators for the coefficients of a linear model when the error distribution is spherically symmetric with unknown variances

The usual confidence set for p (p ≥ 3) coefficients of a linear model is known to be dominated by the James-Stein confidence sets under the assumption of spherical symmetric errors with known variance (Hwang and Chen 1986). For the same confidence-set problem but for the unknown-variance case, naturally one replaces the unknown variance by an estimator. For the normal case, many previous studies have shown numerically that the resultant James-Stein confidence sets dominate the resultant usual confidence sets, i.e., the F confidence sets. In this paper we provide a further asymptotic justification, and we discover the same advantage of the James-Stein confidence sets for normal error as well as spherically symmetric error.     

Modified first-difference estimator in a panel data model with unobservable factors both in the errors and the regressors when the time dimension is small

Panel data models with factor structures in both the errors and the regressors have received considerable attention recently. In these models, the errors and the regressors are correlated and the standard estimators are inconsistent. This paper shows that, for such models, a modified first-difference estimator (in which the time and the cross-sectional dimensions are interchanged) is consistent as the cross-sectional dimension grows but the time dimension is small. Although the estimator has a non standard asymptotic distribution, t and F tests have standard asymptotic distribution under the null hypothesis.     

Prediction distribution for a linear regression model with multivariate student-t error distribution

The distribution(s) of future response(s) given a set of data from an informative experiment is known as prediction distribution. The paper derives the prediction distribution(s) from a linear regression model with a multivari-ate Student-t error distribution using the structural relations of the model. We observe that the prediction distribution(s) are multivariate t-variate(s) with degrees of freedom which do not depend on the degrees of freedom of the error distribution.     

JAMES-STEIN RULE ESTIMATORS IN LINEAR REGRESSION MODELS WITH MULTIVARIATE-t DISTRIBUTED ERROR

This paper considers estimation of β in the regression model y =Xβ+μ, where the error components in μ have the jointly multivariate Student-t distribution. A family of James-Stein type estimators (characterised by nonstochastic scalars) is presented. Sufficient conditions involving only X are given, under which these estimators are better (with respect to the risk under a general quadratic loss function) than the usual minimum variance unbiased estimator (MVUE) of β. Approximate expressions for the bias, the risk, the mean square error matrix and the variance-covariance matrix for the estimators in this family are obtained. A necessary and sufficient condition for the dominance of this family over MVUE is also given.     

On the sampling performance of an inequality pre-test estimator of the regression error variance under LINEX loss

We consider the estimation of the error variance of a linear regression model where prior information is available in the form of an (uncertain) inequality constraint on the coefficients. Previous studies on this and other related problems use the squared error loss in comparing estimator’s performance. Here, by adopting the asymmetric LINEX loss function, we derive and numerically evaluate the exact risks of the inequality constrained estimator and the inequality pre-test estimator which results after a preliminary test for an inequality constraint on the coefficients. The risks based on squared error loss are special cases of our results, and we draw appropriate comparisons.     

MSE performance and minimax regret significance points for a HPT estimator under a multivariate t error distribution

In this article, assuming that the error terms follow a multivariate t distribution,we derive the exact formulae forthe moments of the heterogeneous preliminary test (HPT) estimator proposed by Xu (2012b Xu, H. (2012b). MSE performance and minimax regret significance points for a HPT estimator when each individual regression coefficient is estimated. Commun. Stat. Theory Methods 42:2152–2164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We also execute the numerical evaluation to investigate the mean squared error (MSE) performance of the HPT estimator and compare it with those of the feasible ridge regression (FRR) estimator and the usual ordinary least squared (OLS) estimator. Further, we derive the optimal critical values of the preliminary F test for the HPT estimator, using the minimax regret function proposed by Sawa and Hiromatsu (1973 Sawa, T., Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis. Econometrica 41:1093–1101.[Crossref], [Web of Science ®] , [Google Scholar]). Our results show that (1) the optimal significance level (α*) increases as the degrees of freedom of multivariate t distribution (ν₀) increases; (2) when ν₀ ? 10, the value of α* is close to that in the normal error case.     

A study on the least squares estimator of multivariate isotonic regression function

Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X₁,…,X_d)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.     

A new heteroskedasticity-consistent covariance matrix estimator for the linear regression model

The assumption that all random errors in the linear regression model share the same variance (homoskedasticity) is often violated in practice. The ordinary least squares estimator of the vector of regression parameters remains unbiased, consistent and asymptotically normal under unequal error variances. Many practitioners then choose to base their inferences on such an estimator. The usual practice is to couple it with an asymptotically valid estimation of its covariance matrix, and then carry out hypothesis tests that are valid under heteroskedasticity of unknown form. We use numerical integration methods to compute the exact null distributions of some quasi-t test statistics, and propose a new covariance matrix estimator. The numerical results favor testing inference based on the estimator we propose.     

Inference on C pk for autocorrelated data in the presence of random measurement errors

The present paper examines the properties of the C _pk estimator when observations are autocorrelated and affected by measurement errors. The underlying reason for this choice of subject matter is that in industrial applications, process data are often autocorrelated, especially when sampling frequency is not particularly low, and even with the most advanced measuring instruments, gauge imprecision needs to be taken into consideration. In the case of a first-order stationary autoregressive process, we compare the statistical properties of the estimator in the error case with those of the estimator in the error-free case. Results indicate that the presence of gauge measurement errors leads the estimator to behave differently depending on the entity of error variability.