Generalized least squares with misspecified serial correlation structures

Summary. The regression literature contains hundreds of studies on serially correlated disturbances. Most of these studies assume that the structure of the error covariance matrix Ω is known or can be estimated consistently from data. Surprisingly, few studies investigate the properties of estimated generalized least squares (GLS) procedures when the structure of Ω is incorrectly identified and the parameters are inefficiently estimated. We compare the finite sample efficiencies of ordinary least squares (OLS), GLS and incorrect GLS (IGLS) estimators. We also prove new theorems establishing theoretical efficiency bounds for IGLS relative to GLS and OLS. Results from an exhaustive simulation study are used to evaluate the finite sample performance and to demonstrate the robustness of IGLS estimates vis-à-vis OLS and GLS estimates constructed for models with known and estimated (but correctly identified) Ω. Some of our conclusions for finite samples differ from established asymptotic results.     

On approximate least squares estimators of parameters of one-dimensional chirp signal

Chirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.     

MSE-matrix superiority of the mixed over the least squares estimator in the presence of outliers

In his recent paper, Ali (1991) has shown that the mixed regression estimator, when data contain mean-shift or variance inflation outliers, is uniformly superior to the ordinary least squares estimator in terms of scalar-valued mean square error. However, when using the matrix-valued mean square error criterion, this dominance fails to hold in general. The subsequent investigation gives a complete characterization of the situation where the mixed estimator is superior to the LS-estimator when the comparison is made with respect to this stronger MSE-property. Vice versa, the LS-estimator never dominates the mixed estimator relative to this criterion.     

On the equality of usual and amemiya's partially generalized least squares estimator

Equivalent conditions are derived for the equality of GLSE (generalized least squares estimator) and partially GLSE (PGLSE), the latter introduced by Amemiya (1983). By adopting a more general approach the ordinary least squares estimator (OLSE) can shown to be a special PGLSE. Furthcrmore, linearly restricted estimators proposed by Balestra (1983) are investigated in this context. To facilitate the comparison of estimators extensive use of oblique and orthogonal projectors is made.     

Small-sample behavior of weighted least squares in experimental design applications

In experimental design applications unbiased estimators s_i ² of the variances σ_i ² are possible. These estimators may be used in Weighted Least Squares (WLS) when estimating the parameters β. The resulting small-sample behavior is investigated in a Monte Carlo experiment. This experiment shows that an asymptotically valid covariance formula can be used if s_i ² is based on, say, at least 5 observations. The WLS estimator based on estimators s_i ² gives more accurate estimators of β, provided the σ_i ² differ by a factor, say, 10.     

Improving survey-weighted least squares regression

The weighted least squares (WLS) estimator is often employed in linear regression using complex survey data to deal with the bias in ordinary least squares (OLS) arising from informative sampling. In this paper a 'quasi-Aitken WLS' (QWLS) estimator is proposed. QWLS modifies WLS in the same way that Cragg's quasi-Aitken estimator modifies OLS. It weights by the usual inverse sample inclusion probability weights multiplied by a parameterized function of covariates, where the parameters are chosen to minimize a variance criterion. The resulting estimator is consistent for the superpopulation regression coefficient under fairly mild conditions and has a smaller asymptotic variance than WLS.     

Bayesian least squares estimates of univariate regression functions

The regression function R(?) to be estimated is assumed to have an expansion in terms of specified functions, orthogonalized vich respect to values of the explanatory variable. Relative precisions of OBSERVATION are assumed known. The estimate is the posterior linear mean of R(?) given the data. The investigator plots graphs of appropriate functions as an aid in eliciting his prior means and precisions for the coefficients in the expansion. The method is illustrated by an example using simulated data, an example in which effects of various dosages of Vitamin D are estimated, and an example in which a utility function is estimated.     

An empirical investigation of the box-cox model and a nonlinear least squares alternative

In this paper we present a generalized functional form estimator, recently developed by jeffrey Wooldridge; and then we compare it empirically to the popular Box-Cox (BC) estimator using three data sets. We begin by briefly reviewing the drawbacks of the BC estimator. We Then introduce the nonlinear lest squares (NLS) alternative of Wooldridge which retains the desirable qualities of the BC estimator without the associated theoretical problems. We continue by applying both the BC and the NLS models to data from three classic hedonic regression studies and then compare the estimation resuts-point estimates, inferences and fitted values. The estimations include a wage rate equation, and two computer hedonic regression equations, one using data from a classic study by Gregory Chow and the other using an IBM data set that formed the basis of the new official BLS computer price index.     

An empirical investigation of the box-cox model and a nonlinear least squares alternative

In this paper we present a generalized functional form estimator, recently developed by jeffrey Wooldridge; and then we compare it empirically to the popular Box-Cox (BC) estimator using three data sets. We begin by briefly reviewing the drawbacks of the BC estimator. We Then introduce the nonlinear lest squares (NLS) alternative of Wooldridge which retains the desirable qualities of the BC estimator without the associated theoretical problems. We continue by applying both the BC and the NLS models to data from three classic hedonic regression studies and then compare the estimation resuts-point estimates, inferences and fitted values. The estimations include a wage rate equation, and two computer hedonic regression equations, one using data from a classic study by Gregory Chow and the other using an IBM data set that formed the basis of the new official BLS computer price index.     

Bayes least squares linear estimation of densities

Linear, least squares statistical methods in which the "parameters" are interpreted as random variables were introduced by Whittle, and further developed by Hartigan and others. They are applied here to the problem of estimating the coefficients in an orthogonal expansion of a multivariate density, given a simple random sample.     

Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.     

Parameter estimation for three-parameter generalized Pareto distribution by weighted non linear least squares

Generalized Pareto distribution (GPD) is widely used to model exceedances over thresholds. In this paper, we propose a new method, called weighted non linear least squares (WNLS), to estimate the parameters of the three-parameter GPD. Some asymptotic results of the proposed method are provided. An extensive simulation is carried out to evaluate the finite sample behaviour of the proposed method and to compare the behaviour with other methods suggested in the literature. The simulation results show that WNLS outperforms other methods in general situations. Finally, the WNLS is applied to analysis the real-life data.     

Variance least squares estimators for multivariate linear mixed model

Non-iterative, distribution-free, and unbiased estimators of variance components by least squares method are derived for multivariate linear mixed model. A general inter-cluster variance matrix, a same-member only general inter-response variance matrix, and an uncorrelated intra-cluster error structure for each response are assumed. Projection method is suggested when unbiased estimators of variance components are not nonnegative definite matrices. A simulation study is conducted to investigate the properties of the proposed estimators in terms of bias and mean square error with comparison to the Gaussian (restricted) maximum likelihood estimators. The proposed estimators are illustrated by an application of gene expression familial study.     

Multivariate boundary kernels and a continuous least squares principle

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.     

An efficient procedure for computing minque of variance components and generalized least squares estimates of fixed effects

An efficient method for computing minimum norm quadratic unbiased estimates (MINQUE) of variance components and generalized least squares estimates of the fixed effects in the mixed model is developed. The computing algorithm uses a modification of the W transformation.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

On the unbiasedness of iterated gls estimators

We Formulate sufficienct conditions for the existonce of the expectation of iterated generalized expectation of the iterated generalized least squares estimator, which consequently guarantee its unbiasedness, The analysis is applied to the maximum likelihood estimator in the general linear model with normal disturbances, where a set of assumptions ensures convergence of the iteration as well as unbiasedness.     

The peculiar shrinkage properties of partial least squares regression

Partial least squares regression has been widely adopted within some areas as a useful alternative to ordinary least squares regression in the manner of other shrinkage methods such as principal components regression and ridge regression. In this paper we examine the nature of this shrinkage and demonstrate that partial least squares regression exhibits some undesirable properties.     

On distribution-weighted partial least squares with diverging number of highly correlated predictors

Summary.  Because highly correlated data arise from many scientific fields, we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated. For this, we first develop a distribution-weighted least squares estimator that can recover directions in the central subspace, then use the distribution-weighted least squares estimator as a seed vector and project it onto a Krylov space by partial least squares to avoid computing the inverse of the covariance of predictors. Thus, distrbution-weighted partial least squares can handle the cases with high dimensional and highly correlated predictors. Furthermore, we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares. For theoretical investigation, we obtain strong consistency and asymptotic normality when the dimension p of predictors is of convergence rate O { n ^1/2/ log ( n )} and o ( n ^1/3) respectively where n is the sample size. When there are no other constraints on the covariance of predictors, the rates n ^1/2 and n ^1/3 are optimal. We also propose a Bayesian information criterion type of criterion to estimate the dimension of the Krylov space in the partial least squares procedure. Illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non-ellipticity and works well even in 'small n –large p ' problems.     

On shrinkage least squares estimation in a parallelism problem

In a multi-sample simple regression model, generally, homogeneity of the regression slopes leads to improved estimation of the intercepts. Analogous to the preliminary test estimators, (smooth) shrinkage least squares estimators of Intercepts based on the James-Stein rule on regression slopes are considered. Relative pictures on the (asymptotic) risk of the classical, preliminary test and the shrinkage least squares estimators are also presented. None of the preliminary test and shrinkage least squares estimators may dominate over the other, though each of them fares well relative to the other estimators.