Series Estimation in Partially Linear In‐Slide Regression Models

Abstract. The partially linear in‐slide model (PLIM) is a useful tool to make econometric analyses and to normalize microarray data. In this article, by using series approximations and a least squares procedure, we propose a semiparametric least squares estimator (SLSE) for the parametric component and a series estimator for the non‐parametric component. Under weaker conditions than those imposed in the literature, we show that the SLSE is asymptotically normal and that the series estimator attains the optimal convergence rate of non‐parametric regression. We also investigate the estimating problem of the error variance. In addition, we propose a wild block bootstrap‐based test for the form of the non‐parametric component. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on a set of economical data is also illustrated.     

Linear least squares estimates and nonlinear means

The consistency and asymptotic normality of a linear least squares estimate of the form (X'X)^-X'Y when the mean is not Xβ is investigated in this paper. The least squares estimate is a consistent estimate of the best linear approximation of the true mean function for the design chosen. The asymptotic normality of the least squares estimate depends on the design and the asymptotic mean may not be the best linear approximation of the true mean function. Choices of designs which allow large sample inferences to be made about the best linear approximation of the true mean function are discussed.     

The exact density and distribution functions of the inequality constrained and pre-test estimators

We consider a bivariate normal linear regression model with an inequality restriction imposed on one of the regression coefficients. The exact analytical expressions for the density and distribution functions of the inequality constrained and pre-test estimators are derived and numerically evaluated. The implications of using the inequality constrained and pre-test estimators in confidence interval construction are also discussed and explored.     

FITTING SPATIAL CORRELATION MODELS: APPROXIMATING A LIKELIHOOD APPROXIMATION

The paper considers a class of spatial correlation models (stationary Gaussian processes) which includes (spatial) conditional autoregressive, simultaneous autoregressive, moving average and direct covariance models. Given observations on a finite rectangular lattice, a likelihood approximation for estimating the parameters in the spectral density of the model is discussed. The approximation consists of applying the trapezoidal rule, with a her grid of frequencies than the usual Fourier frequencies, to compute the integral in an appraximation due to Whittle (1954) and later modified by Guyon (1984). With this approximation, a Fisher scoring type algorithm has a simple form and in some casea reduces to iteratively reweighted least squares. Methods for computing the unbiased two-dimensional periodogram required by the method are presented and the accuracy of the approximation is discussed. The asymptotic distribution of the parameter estimates computed from the likelihood approximation is also given.     

Additional approximations to the distribution of a structural coefficient estimator for the case of two included endogenous variables

Four new approximations t o the exact distribution of the two-stage l e a s t squares estimator of astructuralcoefficient for

the case of two included endogeneous variables are introduced and compared with the others in the literatur e . Two of the new approximations are based on the Pearson distribution and are found to be adequate throughout the parameter space. A normal approximation using exact moments and an approximation based on the saddlepoint method (Holly and Phillips,1979) are found to be

poor for a wide range of parameter values.     

Note on a linear model with errors in the variables and with trend

The least squares estimate of the slope parameter of a simple linear model with errors in the variables is typically biased. However the bias vanishes asymptotically for increasing sample size if the regressor variable follows a linear trend. For this case asymptotic expansion formulas for bias and variance of the least squares estimator are derived from exact expressions presented by Richardson and Wu (1970) and certain bounds to these expressions given by Friedmann (1990).     

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.     

ASYMPTOTIC LIKELIHOOD APPROXIMATIONS USING A PARTIAL LAPLACE APPROXIMATION

Elimination of a nuisance variable is often non‐trivial and may involve the evaluation of an intractable integral. One approach to evaluate these integrals is to use the Laplace approximation. This paper concentrates on a new approximation, called the partial Laplace approximation, that is useful when the integrand can be partitioned into two multiplicative disjoint functions. The technique is applied to the linear mixed model and shows that the approximate likelihood obtained can be partitioned to provide a conditional likelihood for the location parameters and a marginal likelihood for the scale parameters equivalent to restricted maximum likelihood (REML). Similarly, the partial Laplace approximation is applied to the t‐distribution to obtain an approximate REML for the scale parameter. A simulation study reveals that, in comparison to maximum likelihood, the scale parameter estimates of the t‐distribution obtained from the approximate REML show reduced bias.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

Optimal designs for minimising covariances among parameter estimators in a linear model

We construct approximate optimal designs for minimising absolute covariances between least‐squares estimators of the parameters (or linear functions of the parameters) of a linear model, thereby rendering relevant parameter estimators approximately uncorrelated with each other. In particular, we consider first the case of the covariance between two linear combinations. We also consider the case of two such covariances. For this we first set up a compound optimisation problem which we transform to one of maximising two functions of the design weights simultaneously. The approaches are formulated for a general regression model and are explored through some examples including one practical problem arising in chemistry.     

Building and Fitting Non‐Gaussian Latent Variable Models via the Moment‐Generating Function

Abstract. For certain classes of hierarchical models, it is easy to derive an expression for the joint moment‐generating function (MGF) of data, whereas the joint probability density has an intractable form which typically involves an integral. The most important example is the class of linear models with non‐Gaussian latent variables. Parameters in the model can be estimated by approximate maximum likelihood, using a saddlepoint‐type approximation to invert the MGF. We focus on modelling heavy‐tailed latent variables, and suggest a family of mixture distributions that behaves well under the saddlepoint approximation (SPA). It is shown that the well‐known normalization issue renders the ordinary SPA useless in the present context. As a solution we extend the non‐Gaussian leading term SPA to a multivariate setting, and introduce a general rule for choosing the leading term density. The approach is applied to mixed‐effects regression, time‐series models and stochastic networks and it is shown that the modified SPA is very accurate.     

DISTRIBUTION OF THE LEAST SQUARES ESTIMATOR IN A FIRST-ORDER AUTOREGRESSIVE MODEL

This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. A uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. In the literature, there exist a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.     

Bent‐cable regression with autoregressive noise

Motivated by time series of atmospheric concentrations of certain pollutants the authors develop bent‐cable regression for autocorrelated errors. Bent‐cable regression extends the popular piecewise linear (broken‐stick) model, allowing for a smooth change region of any non‐negative width. Here the authors consider autoregressive noise added to a bent‐cable mean structure, with unknown regression and time series parameters. They develop asymptotic theory for conditional least‐squares estimation in a triangular array framework, wherein each segment of the bent cable contains an increasing number of observations while the autoregressive order remains constant as the sample size grows. They explore the theory in a simulation study, develop implementation details, apply the methodology to the motivating pollutant dataset, and provide a scientific interpretation of the bent‐cable change point not discussed previously. The Canadian Journal of Statistics 38: 386–407; 2010 © 2010 Statistical Society of Canada     

Simple and accurate inference for the mean of the gamma model

The two-parameter gamma model is widely used in reliability, environmental, medical and other areas of statistics. It has a two-dimensional sufficient statistic, and a two-dimensional parameter which can be taken to describe shape and mean. This makes it closely comparable to the normal model, but it differs substantially in that the exact distribution for the minimal sufficient statistic is not available. Some recently developed asymptotic theory is used to derive an approximation for observed levels of significance and confidence intervals for the mean parameter of the model. The approximation is as easy to apply as first-order methods, and substantially more accurate.     

Empirical Likelihood for Censored Linear Regression

In this paper we investigate the empirical likelihood method in a linear regression model when the observations are subject to random censoring. An empirical likelihood ratio for the slope parameter vector is defined and it is shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the empirical likelihood to the linear regression model first studied by Owen (1991) if there is no censoring present. Some simulation studies are presented to compare the empirical likelihood method with the normal approximation based method proposed in Lai et al. (1995). It was found that the empirical likelihood method performs much better than the normal approximation method.     

Assessing Influence on the Liu Estimates in Linear Regression Models

The Liu estimator has been developed as an alternative to the ordinary least squares estimator in the presence of collinearity among the elements of regressors in linear regression models. We present the DFFITS and different versions of the Cook distance analogous to the ones given for the ordinary linear regression models of each individual observation on the Liu estimates. We suggest a version of the Cook distance based on one-step approximation. The mean shift outlier model for the Liu regression has also been investigated. Moreover, using the Sherman-Morrison-Woodbury theorem, we find approximate versions of the DFFITS and the Cook distance. The proposed diagnostics are evaluated on two data sets and yield promising results.     

Statistical inference in partially time-varying coefficient models

A partially time-varying coefficient time series model is introduced to characterize the nonlinearity and trending phenomenon. To estimate the regression parameter and the nonlinear coefficient function, the profile least squares approach is applied with the help of local linear approximation. The asymptotic distributions of the proposed estimators are established under mild conditions. Meanwhile, the generalized likelihood ratio test is studied and the test statistics are demonstrated to follow asymptotic χ²-distribution under the null hypothesis. Furthermore, some extensions of the proposed model are discussed and several numerical examples are provided to illustrate the finite sample behavior of the proposed methods.     

Adaptive ridge estimator in a linear regression model with spherically symmetric error under constraint

Abstract

We consider adaptive ridge regression estimators in the general linear model with homogeneous spherically symmetric errors. A restriction on the parameter of regression is considered. We assume that all components are non negative (i.e. on the positive orthant). For this setting, we produce under general quadratic loss such estimators whose risk function dominates that of the least squares provided the number of regressors in the least fore.     

Component selection norms for principal components regression

Multicollinearity or near exact linear dependence among the vectors of regressor variables in a multiple linear regression analysis can have important effects on the quality of least squares parameter estimates. One frequently suggested approach for these problems is principal components regression. This paper investigates alternative variable selection procedures and their implications for such an analysis.     

Relative Risk Regression for Binary Outcomes: Methods and Recommendations

Relative risks are often considered preferable to odds ratios for quantifying the association between a predictor and a binary outcome. Relative risk regression is an alternative to logistic regression where the parameters are relative risks rather than odds ratios. It uses a log link binomial generalised linear model, or log‐binomial model, which requires parameter constraints to prevent probabilities from exceeding 1. This leads to numerical problems with standard approaches for finding the maximum likelihood estimate (MLE), such as Fisher scoring, and has motivated various non‐MLE approaches. In this paper we discuss the roles of the MLE and its main competitors for relative risk regression. It is argued that reliable alternatives to Fisher scoring mean that numerical issues are no longer a motivation for non‐MLE methods. Nonetheless, non‐MLE methods may be worthwhile for other reasons and we evaluate this possibility for alternatives within a class of quasi‐likelihood methods. The MLE obtained using a reliable computational method is recommended, but this approach requires bootstrapping when estimates are on the parameter space boundary. If convenience is paramount, then quasi‐likelihood estimation can be a good alternative, although parameter constraints may be violated. Sensitivity to model misspecification and outliers is also discussed along with recommendations and priorities for future research.