Efficiency of least squares estimators in the presence of spatial autocorrelation

The effect of spatial autocorrelation on inferences made using ordinary least squares estimation is considered. It is found, in some cases, that ordinary least squares estimators provide a reasonable alternative to the estimated generalized least squares estimators recommended in the spatial statistics literature. One of the most serious problems in using ordinary least squares is that the usual variance estimators are severely biased when the errors are correlated. An alternative variance estimator that adjusts for any observed correlation is proposed. The need to take autocorrelation into account in variance estimation negates much of the advantage that ordinary least squares estimation has in terms of computational simplicity     

An efficient and fast algorithm for estimating the frequencies of superimposed exponential signals in zero-mean multiplicative and additive noise

In this paper, we consider the superimposed exponential signals in zero-mean multiplicative and additive noise when all the noise are independently and identically distributed. We use a three-step iterative procedure to estimate the frequencies of the considered model. It is observed that the estimators are consistent and work quite well in terms of biases and mean square errors. Moreover, the convergence rate of the estimators attains O _p (N ^?3/2), which is the best convergence rate in the case of only additive noise and constant amplitude.     

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.     

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.     

Sequential estimation of the sum of sinusoidal model parameters

Estimating the parameters of the sum of a sinusoidal model in presence of additive noise is a classical problem. It is well known to be a difficult problem when the two adjacent frequencies are not well separated or when the number of components is very large. In this paper we propose a simple sequential procedure to estimate the unknown frequencies and amplitudes of the sinusoidal signals. It is observed that if there are p components in the signal then at the k  th (k?p)$(k?p)$ stage our procedure produces strongly consistent estimators of the k   dominant sinusoids. For k>p$k>p$, the amplitude estimators converge to zero almost surely. Asymptotic distribution of the proposed estimators is also established and it is observed that it coincides with the asymptotic distribution of the least squares estimators. Numerical simulations are performed to observe the performance of the proposed estimators for different sample sizes and for different models. One ECG data and one synthesized data are analyzed for illustrative purpose.     

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.     

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.     

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.     

Quantile regression via iterative least squares computations

We present an estimating framework for quantile regression where the usual L ₁-norm objective function is replaced by its smooth parametric approximation. An exact path-following algorithm is derived, leading to the well-known ‘basic’ solutions interpolating exactly a number of observations equal to the number of parameters being estimated. We discuss briefly possible practical implications of the proposed approach, such as early stopping for large data sets, confidence intervals, and additional topics for future research.     

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.     

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.     

Study of partial least squares and ridge regression methods

This article considers both Partial Least Squares (PLS) and Ridge Regression (RR) methods to combat multicollinearity problem. A simulation study has been conducted to compare their performances with respect to Ordinary Least Squares (OLS). With varying degrees of multicollinearity, it is found that both, PLS and RR, estimators produce significant reductions in the Mean Square Error (MSE) and Prediction Mean Square Error (PMSE) over OLS. However, from the simulation study it is evident that the RR performs better when the error variance is large and the PLS estimator achieves its best results when the model includes more variables. However, the advantage of the ridge regression method over PLS is that it can provide the 95% confidence interval for the regression coefficients while PLS cannot.     

On finite-sample properties of adaptive least squares regression estimates

This paper is mainly concerned with deriving finite-sample properties of least squares estimators for the regression function in a nonparametric regression situation under some simplifying assumptions such as normally distributed errors with a common known variance. The selection of basis functions to be used for the construction of an estimator may be regarded as a smoothing problem, and will usually be done in a data-dependent way, A straightforward application of a result by P. J. Kernpthorne yields that, under a squared error loss, all selection procedures are admissible. Furthermore, the minimax approach provides an interpolating estimator, which is often impractical, Therefore, within a certain class of selection procedures an optimal one is determined using the minimax regret principle. It can be seen to behave similarly to the procedure minimizing either an unbiased risk estimator or, equivalently, the C_p-criterion.     

A note on minimum average risk estimators for coefficients in linear models

In this note, we have derived a set of necessary and sufficient conditions for the biased estimators analyzed by Swamy and Mehta (1976) to be better than the generalized least squares estimator of the coefficient vector in a standard linear regression model.     

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.     

Total least squares and bootstrapping with applications in calibration

The solution to the errors-in-variables problem computed through total least squares is highly nonlinear. Because of this, many statistical procedures for constructing confidence intervals and testing hypotheses cannot be applied. One possible solution to this dilemma is bootstrapping. A nonparametric bootstrap technique could fail. Here, the proper nonparametric bootstrap procedure is provided and its correctness is proved. On the other hand, a residual bootstrap is not valid and suitable in this case. The results are illustrated through a simulation study. An application of this approach to calibration data is presented.     

On necessary and sufficient condition for superiority of ridge estimator over least squares estimator

The necessary and sufficient condition is obtained such that ridge estimator is better than the least squares estimator relative to the matrix mean square error.