Least squares variogram fitting by spatial subsampling

Summary. Least squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.     

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     

Estimation of linear models for field experiments

General mixed linear models for experiments conducted over a series of sltes and/or years are described. The ordinary least squares (OLS) estlmator is simple to compute, but is not the best unbiased estimator. Also, the usuaL formula for the varlance of the OLS estimator is not correct and seriously underestimates the true variance. The best linear unbiased estimator is the generalized least squares (GLS) estimator. However, t requires an inversion of the variance-covariance matrix V, whlch is usually of large dimension. Also, in practice, V is unknown.

We presented an estlmator [Vcirc] of the matrix V using the estimators of variance components [for sites, blocks (sites), etc.]. We also presented a simple transformation of the data, such that an ordinary least squares regression of the transformed data gives the estimated generalized least squares (EGLS) estimator. The standard errors obtained from the transformed regression serve as asymptotic standard errors of the EGLS estimators. We also established that the EGLS estlmator is unbiased.

An example of fitting a linear model to data for 18 sites (environments) located in Brazil is given. One of the site variables (soil test phosphorus) was measured by plot rather than by site and this established the need for a covariance model such as the one used rather than the usual analysis of variance model. It is for this variable that the resulting parameter estimates did not correspond well between the OLS and EGLS estimators. Regression statistics and the analysis of variance for the example are presented and summarized.     

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.     

Comparisons among regression estimators under the generalized mean square error criterion

It is not always prossible to establish a preference ordering among regression estimators in terms of the generalized mean square error criterion. In the paper, we determine when it is feasible to use this criteion to couduct comparisons among ordinary least squares, principal components, ridge regression, and shrunken least squares estimators.     

The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

It is well known that the ordinary least squares estimator of Xβ in the general linear model E _y = Xβ, cov y = σ² V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.     

Least squares estimation of nonhomogeneous poisson processes

We formulate and evaluate weighted least squares (WLS) and ordinary least squares (OLS) procedures for estimating the parametric mean-value function of a nonhomogeneous Poisson process. We focus the development on processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components. Unanticipated problems with the WLS procedure are explained by an analysis of the associated residuals. The OLS procedure is based on a square root transformation of the "detrended" event (arrival) times - that is, the fitted mean-value function evaluated at the observed event times; and under appropriate conditions, the corresponding residuals are proved to converge weakly to a normal distribution with mean 0 and variance 0.25. The results of a Monte Carlo study indicate the advantages of the OLS procedure with respect to estimation accuracy and computational efficiency.     

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.     

E-optimal block designs under heteroscedastic model

This paper mainly studies the E-optimality of block designs under a general heteroscedastic setting. The C-matrix of a block design under a heteroscedastic setting is obtained by using generalized least squares. Some bounds for the smallest positive eigenvalue of C-matrix are obtained in some general classes of connected designs. Use of these bounds is then made to obtain certain E-optimal block designs in various classes of connected block designs.     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

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。     

Confidence regions based on inefficient estimators

We consider an unbiased estimator of a function of mean value parameters, which is not efficient. This inefficient estimator is correlated with a residual vector. Thus, if a unit dispersion is unknown, it is impossible to determine the correct confidence region for a function of mean value parameters via a standard estimator of an unknown dispersion with the exception of the case when the ordinary least squares (OLS) estimator is considered in a model with a special covariance structure such that the OLS and the generalized least squares (GLS) estimator are the same, that is the OLS estimator is efficient. Two different estimators of a unit dispersion independent of an inefficient estimator are derived in a singular linear statistical model. Their quality was verified by simulations for several types of experimental designs. Two new estimators of the unit dispersion were compared with the standard estimators based on the GLS and the OLS estimators of the function of the mean value parameters. The OLS estimator was considered in the incorrect model with a different covariance matrix such that the originally inefficient estimator became efficient. The numerical examples led to a slightly surprising result which seems to be due to data behaviour. An example from geodetic practice is presented in the paper.     

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.     

Effects of non-normality and mild heteroscedasticity on estimators in regression

Several estimators are examined for the simple linear regression model under a controlled, experimental situation with multiple observations at each design point. The model is examined under normal and non-normal error distributions and mild heterogeneity of variances across the chosen design points. We consider the ordinary, generalized, and estimated generalized least squares estimators and several examples of M estimators. The asymptotic properties of the M estimator using the Huber ψ are presented under these conditions for the multiple regression model. A simulation study is also presented which indicates that the M estimator possesses strong robustness properties under the presence of both non-normality and mild heteroscedasticity o£ errors. Finally, the M estimates are compared to the least squares estimates in two examples.     

Two Kinds of Restricted Estimators in Singular Linear Regression

In this article, the parameter estimators in singular linear model with linear equality restrictions are considered. The restricted root estimator and the generalized restricted root estimator are proposed and some properties of the estimators are also studied. Furthermore, we compare them with the restricted unified least squares estimator and show their sufficient conditions under which their superior over the restricted unified least squares estimator in terms of mean squares error, and discuss the choice of the unknown parameters of the generalized restricted root estimator.     

Estimating partially linear panel data models with one-way error components

We consider the problem of estimating a partially linear panel data model whenthe error follows an one-way error components structure. We propose a feasiblesemiparametric generalized least squares (GLS) type estimator for estimating the coefficient of the linear component and show that it is asymptotically more efficient than a semiparametric ordinary least squares (OLS) type estimator. We also discussed the case when the regressor of the parametric component is correlated with the error, and propose an instrumental variable GLS-type semiparametric estimator.     

The unconditional distributions of the OLS,TSLS and LIML estimators in a simple structural equations model

The exact distributions of the standard estimators of the structural coe?cients in a linear structural equations model conditional on the exogenous variables have been shown to have some unexpected and quirky features. Since the argument for conditioning on exogenous (ancillary) variables has been weakened over the past 20 years by the discovery of an “ancillarity paradox,” it is natural to wonder whether such finite sample properties are in fact due to conditioning on the exogenous variables. This article studies the exact distributions of the ordinary least squares (OLS), two-stage least squares (TSLS), and limited information maximum likelihood (LIML) estimators of the structural coe?cients in a linear structural equation without conditioning on the exogenous variables.     

Robust extrapolation designs and weights for biased regression models with heteroscedastic errors

We consider the construction of designs for the extrapolation of regression responses, allowing both for possible heteroscedasticity in the errors and for imprecision in the specification of the response function. We find minimax designs and correspondingly optimal estimation weights in the context of the following problems: (1) for ordinary least squares estimation, determine a design to minimize the maximum value of the integrated mean squared prediction error (IMSPE), with the maximum being evaluated over both types of departure; (2) for weighted least squares estimation, determine both weights and a design to minimize the maximum IMSPE; (3) choose weights and design points to minimize the maximum IMSPE, subject to a side condition of unbiasedness. Solutions to (1) and (2) are given for multiple linear regression with no interactions, a spherical design space and an annular extrapolation space. For (3) the solution is given in complete generality; as one example we consider polynomial regression. Applications to a dose-response problem for bioassays are discussed. Numerical comparisons, including a simulation study, indicate that, as well as being easily implemented, the designs and weights for (3) perform as well as those for (1) and (2) and outperform some common competitors for moderate but undetectable amounts of model bias.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

Parameter estimation and inference with spatial lags and cointegration

This article studies dynamic panel data models in which the long run outcome for a particular cross-section is affected by a weighted average of the outcomes in the other cross-sections. We show that imposing such a structure implies a model with several cointegrating relationships that, unlike in the standard case, are nonlinear in the coe?cients to be estimated. Assuming that the weights are exogenously given, we extend the dynamic ordinary least squares methodology and provide a dynamic two-stage least squares estimator. We derive the large sample properties of our proposed estimator under a set of low-level assumptions. Then our methodology is applied to US financial market data, which consist of credit default swap spreads, as well as firm-specific and industry data. We construct the economic space using a “closeness” measure for firms based on input–output matrices. Our estimates show that this particular form of spatial correlation of credit default swap spreads is substantial and highly significant.