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 the Use of Superpopulation Model in Estimation of Population Mean in Two-Occasion Rotation Patterns

The present work is an attempt to study the estimation of the population mean on the current occasion in two-occasion successive (rotation) sampling under a superpopulation model. Six different estimators are proposed for estimating the current population mean in two-occasion successive (rotation) sampling. Optimum replacement policies and performances of the proposed estimators have been discussed. Results are interpreted via empirical studies.     

Akaike Information Criterion for Selecting Variables in the Nested Error Regression Model

The Akaike Information Criterion (AIC) is developed for selecting the variables of the nested error regression model where an unobservable random effect is present. Using the idea of decomposing the likelihood into two parts of “within” and “between” analysis of variance, we derive the AIC when the number of groups is large and the ratio of the variances of the random effects and the random errors is an unknown parameter. The proposed AIC is compared, using simulation, with Mallows' C _p , Akaike's AIC, and Sugiura's exact AIC. Based on the rates of selecting the true model, it is shown that the proposed AIC performs better.     

On the Asymmetric Marcinkiewicz-Zygmund Strong Law of Large Numbers for Linear Random Fields

In this article, the asymmetric Marcinkiewicz-Zygmund strong law of large numbers for linear random field under negative association is obtained. Our result generalizes a result in Gut and Studtmüller (2009 Gut , A. , Studtmüller , U. ( 2009 ) An asymmetric Marcinkiewicz-Zygmund LLN for random fields . Statist. Probab. Lett. 79 : 1016 – 1020 .[Crossref], [Web of Science ®] , [Google Scholar]). An asymmetric Marcinkiewicz-Zygmund LLN for random fields to the linear random field by using the Beverige-Nelson decomposition.     

On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution

The Lomax (Pareto II) distribution has found wide application in a variety of fields. We analyze the second-order bias of the maximum likelihood estimators of its parameters for finite sample sizes, and show that this bias is positive. We derive an analytic bias correction which reduces the percentage bias of these estimators by one or two orders of magnitude, while simultaneously reducing relative mean squared error. Our simulations show that this performance is very similar to that of a parametric bootstrap correction based on a linear bias function. Three examples with actual data illustrate the application of our bias correction.     

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.     

Optimal Simultaneous Confidence Bands in Multiple Linear Regression with Predictor Variables Constrained in an Ellipsoidal Region

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model function, just as a confidence interval gives the plausible range of an unknown parameter. For a multiple linear regression model, confidence bands of different shapes, such as the hyperbolic band and the constant width band, can be constructed and the predictor variable region over which a confidence band is constructed can take various forms. One interesting but unsolved problem is to find the optimal (shape) confidence band over an ellipsoidal region χ_E under the Minimum Volume Confidence Set (MVCS) criterion of Liu and Hayter (2007 Liu, W., Hayter, A.J. (2007). Minimum area confidence set optimality for confidence bands in simple linear regression. J. Amer. Statist. Assoc. 102:181–190.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Liu et al. (2009 Liu, W., Bretz, F., Hayter, A.J., Wynn, H.P. (2009). Assessing non-superiority, non-inferiority or equivalence when comparing two regression models over a restricted covariate region. Biometrics 65:1279–1287.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This problem is challenging as it involves optimization over an unknown function that determines the shape of the confidence band over χ_E. As a step towards solving this difficult problem, in this paper, we introduce a family of confidence bands over χ_E, called the inner-hyperbolic bands, which includes the hyperbolic and constant-width bands as special cases. We then search for the optimal confidence band within this family under the MVCS criterion. The conclusion from this study is that the hyperbolic band is not optimal even within this family of inner-hyperbolic bands and so cannot be the overall optimal band. On the other hand, the constant width band can be optimal within the family of inner-hyperbolic bands when the region χ_E is small and so might be the overall optimal band.     

Some remarks on the traditional way of circular and diagonal circular systematic sampling schemes

ABSTRACT

In this paper, some deficiencies in traditional selection procedure of circular version of systematic sampling schemes are investigated and alternative methods are proposed. We also suggest some rules of thumb for coincidence of units in the sample. The end corrections proposed by Bellhouse and Rao (1975 Bellhouse, D.R., Rao, J.N.K. (1975). Systematic sampling in the presence of a trend. Biometrika. 62:694–697.[Crossref], [Web of Science ®] , [Google Scholar]) and Sampath and Varalakshmi (2008) for circular systematic sampling and diagonal circular systematic sampling, respectively, are also modified.     

Exponential Inequalities in Stochastic Inverse Problems Using an Iterative Method

We consider an iterative method in order to solve linear inverse problems. We establish exponential inequalities for the probability of the distance between the approximated solution and the exact one for a calibration problem. The approximate is given by an iterative method with Gaussian errors. We treat an operator equation of the form Ax = u, where A is a compact operator.