On prediction and mean squared error

Practical questions motivate the search for predictors either of an as yet unobserved random vector, or of a random function of a parameter. An extension of the classical UMVUE theory is presented to cover such situations. In includes a Rao-Blackwell-type theorem, a Cramer-Rao-type inequality, and necessary and sufficient conditions for a predictor to minimize the mean squared error uniformly in the parameter. Applications are considered to the problem of selected means, the species problem, and the examination of some u-v estimates of Robbins (1988).     

EMPIRICAL BAYES ESTIMATION FOR FIRST-ORDER AUTOREGRESSIVE PROCESSES

Large sample properties of an empirical Bayes estimate for a first order autoregressive process are obtained with respect to both the empirical Bayes and the frequentist frameworks.     

THE BOOTSTRAP RISK OF LINEAR EMPIRICAL BAYES ESTIMATES

This paper discusses the bootstrap risk of the linear empirical Bayes estimate of the form θ=Ǎ+B̌x, where x is the current observation, and Ǎ and B̌ are generally functions of the estimates of the prior parameters. The standard error of this risk is developed and ‘computations’ of both the bootstrap risk and its standard error are made.     

Population Trends in Metropolitan and Nonmetropolitan America: Selective Deconcentration and the Rural Rebound

This study uses revised annual population estimates that incorporate adjustments from the 2000 Census to backcast demographic change for U.S. counties during the 1990s. These data are supplemented with new post-censal population estimates for 2001–2003. We use these data to examine demographic trends in the late 1990s and first years of the new century. Our findings are consistent with a model suggesting that a selective deconcentration of the U.S. population is underway. Our findings also confirm the occurrence of the rural rebound in the first half of the 1990s and a waning of this rebound in the late 1990s. Post-censal data also suggest a modest upturn in nonmetropolitan population growth rates in 2001–2003.     

Testing a block exchangeable covariance matrix

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.     

Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring

In this paper we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution when it is known that data are hybrid Type I censored. The maximum likelihood and Bayes estimates are derived. In sequel interval estimates are also constructed. We further consider one- and two-sample prediction of future observations and also obtain prediction intervals. The performance of proposed methods of estimation and prediction is studied using simulations and an illustrative example is discussed in support of the suggested methods.     

Edgeworth Expansions for Semiparametric Averaged Derivatives

A valid Edgeworth expansion is established for the limit distribution of density‐weighted semiparametric averaged derivative estimates of single index models. The leading term that corrects the normal limit varies in magnitude, depending on the choice of bandwidth and kernel order. In general this term has order larger than the n^−1/2 that prevails in standard parametric problems, but we find circumstances in which it is O(n^−1/2), thereby extending the achievement of an n^−1/2 Berry‐Esseen bound in Robinson (1995a). A valid empirical Edgeworth expansion is also established. We also provide theoretical and empirical Edgeworth expansions for a studentized statistic, where some correction terms are different from those for the unstudentized case. We report a Monte Carlo study of finite sample performance.