Stein rule prediction of the composite target function in a general linear regression model

This paper considers the problem of simultaneous prediction of the actual and average values of the dependent variable in a general linear regression model. Utilizing the philosophy of Stein rule procedure, a family of improved predictors for a linear function of the actual and expected value of the dependent variable for the forecast period has been proposed. An unbiased estimator for the mean squared error (MSE) matrix of the proposed family of predictors has been obtained and dominance of the family of Stein rule predictors over the best linear unbiased predictor (BLUP) has been established under a quadratic loss function.     

The Gauss—Markov Theorem and Random Regressors

In the standard linear regression model with independent, homoscedastic errors, the Gauss—Markov theorem asserts that = (X'X)^-1(X'y) is the best linear unbiased estimator of β and, furthermore, that is the best linear unbiased estimator of c'β for all p × 1 vectors c. In the corresponding random regressor model, X is a random sample of size n from a p-variate distribution. If attention is restricted to linear estimators of c'β that are conditionally unbiased, given X, the Gauss—Markov theorem applies. If, however, the estimator is required only to be unconditionally unbiased, the Gauss—Markov theorem may or may not hold, depending on what is known about the distribution of X. The results generalize to the case in which X is a random sample without replacement from a finite population.     

A comparison of estimators for the mean in the inverse gaussian distribution with a known coefficient of variation

In this paper, we discuss an estimation problem of the mean in the inverse Gaussian distribution with a known coefficient of variation. Two types of linear estimators for the mean, the linear minimum variance unbiased estimator and the linear minimum mean squared error estimator, are constructed by using the squared error loss function and their properties are examined. It is observed that, for small samples the performance of the proposed estimators is better than that of the maximum likelihood estimator, when the coefficient of variation is large.     

A Further Study of Predictions in Linear Mixed Models

This article is concerned with the prediction problems in linear mixed models (LMM). Both biased predictors and restricted predictors are introduced. It was found that the mean square error matrix (MSEM) of a predictor strongly depends on the MSEM of corresponding estimator of the fixed effects and precise formulas are obtained. As an application, we propose three new predictors to improve the best linear unbiased predictor (BLUP). The performance of the new predictors can be examined easily with the help of vast literature on the linear regression models (LM). We also illustrate our findings with a Monte Carlo simulation and a numerical example.     

The concept of risk unbiasedness in statistical prediction

This paper extends the concept of risk unbiasedness for applying to statistical prediction and nonstandard inference problems, by formalizing the idea that a risk unbiased predictor should be at least as close to the “true” predictant as to any “wrong” predictant, on the average. A novel aspect of our approach is measuring closeness between a predicted value and the predictant by a regret function, derived suitably from the given loss function. The general concept is more relevant than mean unbiasedness, especially for asymmetric loss functions. For squared error loss, we present a method for deriving best (minimum risk) risk unbiased predictors when the regression function is linear in a function of the parameters. We derive a Rao–Blackwell type result for a class of loss functions that includes squared error and LINEX losses as special cases. For location-scale families, we prove that if a unique best risk unbiased predictor exists, then it is equivariant. The concepts and results are illustrated with several examples. One interesting finding is that in some problems a best unbiased predictor does not exist, but a best risk unbiased predictor can be obtained. Thus, risk unbiasedness can be a useful tool for selecting a predictor.     

A Small Area Predictor under Area-Level Linear Mixed Models with Restrictions

A calibrated small area predictor based on an area-level linear mixed model with restrictions is proposed. It is showed that such restricted predictor, which guarantees the concordance between the small area estimates and a known estimate at the aggregate level, is the best linear unbiased predictor. The mean squared prediction error of the calibrated predictor is discussed. Further, a restricted predictor under a particular time-series and cross-sectional model is presented. Within a simulation study based on real data collected from a longitudinal survey conducted by a national statistical office, the proposed estimator is compared with other competitive restricted and non-restricted predictors.     

On the ABLUE of the normal mean from a censored sample

The asymptotically best linear unbiased estimate (ABLUE) of the normal mean is discussed. The estimate is based on k selected order statistics chosen from a singly or doubly censored large sample of size n(>k). The coefficients, the asymptotic relative efficiency of the estimate, and the optimum spacing of k real numbers between 0 and 1 which determines the optimum ranks of order statistics, are provided. A comparison between the ABLUE and the iterated maximum likelihood estimate is made.     

Fundamental Equations of BLUE and BLUP in the Multivariate Linear Model with Applications

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.     

Estimation in singular partitioned,reduced or transformed linear models

A singular partitioned linear model, i.e. the singular model comprising the main parameters and the nuisance parameters, can be reduced, or transformed to the form in which only linear functions concerning main parameters are involved. In the paper some properties of the best linear unbiased estimators of these functions following from these models are considered.     

On Necessary and Sufficient Conditions for Ordinary Least Squares Estimators to Be Best Linear Unbiased Estimators

Two often-quoted necessary and sufficient conditions for ordinary least squares estimators to be best linear unbiased estimators are described. Another necessary and sufficient condition is described, providing an additional tool for checking to see whether the covariance matrix of a given linear model is such that the ordinary least squares estimator is also the best linear unbiased estimator. The new condition is used to show that one of the two published conditions is only a sufficient condition.     

On equality of ordinary least squares estimator,best linear unbiased estimator and best linear unbiased predictor in the general linear model

The equality of ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) in the general linear model with new observations is investigated through matrix rank method, some new necessary and sufficient conditions are given.     

Exact linear inference and prediction for exponential distributions based on general progressively type-ii censored samples

In this paper, we make use of an algorithm of Huffer and Lin (2001) in order to develop exact interval estimation for the location and scale parameters of an exponential distribution based on general progressively Type-II censored samples. The exact prediction intervals for failure times of the items censored at the last observation are also presented for one-parameter and two-parameter exponential distributions. Finally, we give two examples to illustrate the methods of inference developed here.     

Design and analysis of computer experiments when the output is highly correlated over the input space

To build a predictor, the output of a deterministic computer model or “code” is often treated as a realization of a stochastic process indexed by the code's input variables. The authors consider an asymptotic form of the Gaussian correlation function for the stochastic process where the correlation tends to unity. They show that the limiting best linear unbiased predictor involves Lagrange interpolating polynomials; linear model terms are implicitly included. The authors then develop optimal designs based on minimizing the limiting integrated mean squared error of prediction. They show through several examples that these designs lead to good prediction accuracy.     

On Mean Squared Prediction Error Estimation in Small Area Estimation Problems

For a general linear mixed normal model, a new linearized weighted jackknife method is proposed to estimate the mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) of a general mixed effect. Different MSPE estimators are compared using a Monte Carlo simulation study.     

Linear predictors of future order statistics

In this paper we establish an optimal asymptotic linear predictor which does not involve the finite-sample variance-covariance structure. Extensions to the problem of finding the best linear unbiased and simple linear unbiased predictors for k samples are given. Moreover, we obtain alternative linear predictors by modifying the covariance matrix by either an identity matrix or a diagonal matrix. For normal, logistic and Rayleigh samples of size 10, the alternative linear predictors with these modifications have high efficiency when compared with the best linear unbiased predictor.     

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.      

Predictions and estimations under a group of linear models with random coefficients

This article investigates the problem of establishing best linear unbiased predictors and best linear unbiased estimators of all unknown parameters in a group of linear models with random coefficients and correlated covariance matrix. We shall derive a variety of fundamental statistical properties of the predictors and estimators by using some matrix analysis tools. In particular, we shall establish necessary and sufficient conditions for the predictors and estimators to be equivalent under single and combined equations in the group of models by using the method of matrix equations, matrix rank formulas, and partitioned matrix calculations.     

On the Loss Robustness of Least-Square Estimators

Abstract

The article revisits univariate and multivariate linear regression models. It is shown that least-square estimators (LSEs) are minimum risk estimators in general class of linear unbiased estimators under some general divergence loss. This amounts to the loss robustness of LSEs.     

Nonnegative Unbiased Estimation of Scale Parameters and Associated Quantiles Based on a Ranked Set Sample

In this article, we are interested in estimating the scale parameter in location and scale families. It is well known that the best linear unbiased estimator (BLUE) of scale parameter based on a simple random sample (SRS) is nonnegative. However, the BLUE of scale parameter based on a ranked set sample (RSS) can assume negative values. We suggest various modifications of BLUE of scale parameter based on RSS so that the resulting estimators are unbiased as well as nonnegative. Their performances in terms of relative efficiencies are compared and some recommendations are made for normal, logistic, double exponential, two-parameter exponential and Weibull distributions. We also briefly discuss an application of the proposed nonnegative BLUE of scale parameter for quantile estimation for the above populations.     

Optimal prediction in finite populations under matrix loss

Optimal prediction problems in finite population are investigated. Under matrix loss, we provide necessary and sufficient conditions for the linear predictor of a general linearly predictable variable to be the best linear unbiased predictor (BLUP). The essentially unique BLUP of a linearly predictable variable is obtained in the general superpopulation model. Surprisingly, the both BLUPs under matrix and quadratic loss functions are equivalent to each other. Next, we prove that the BLUP is admissible in the class of linear predictors. Conditions for optimality of the simple projection predictor (SPP) are given. Furthermore, the robust SPP and the robust BLUP are characterized on the misspecification of the covariance matrix.