Sample surveys are usually designed and analyzed to produce estimates for larger areas. However, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To overcome such difficulties,borrowing strength from related small areas via modeling becomes an appropriate approach. In line with this, we propose hierarchical models with power transformations for improving the precision of small area predictions. The proposed methods are applied to satellite data in conjunction with survey data to estimate mean acreage under a specified crop for counties in Iowa.

Summary

Motivated by the attractive features of robust priors, we develop Bayesian estimators for the parameters in a one-way ANOVA model using mixed priors, which are formed by incorporating at density into the usual conjugate priors to independently describe prior knowledge regarding the overall mean or regarding the factor effects. The effect of the independentt prior component is greatly different from that of the conjugate prior. The Bayesian estimators arising from such mixed priors are non-linear functions of the least squares estimators and adjust automatically to the value of the sum of squared errors. In this sense, they are adaptive and rather insensitive to extreme observations. The proposed estimators are clearly superior to the usual Bayesian estimators and to the traditional unbiased estimators, and may be practicable when the error terms are Cauchy distributed.

The so-called Lancaster probabilities on R² are a class of distributions satisfying an orthogonality condition involving orthogonal polynomials with respect to their marginal laws. They are characterized in the cases where the two identical margins are Gaussian, gamma (the latter are known results, but a new treatment is given), Poisson or negative binomial distributions. Some partial results are obtained in the cases of two different Poisson or negative binomial margins, and also in the case where one margin is gamma and the other margin is negative binomial.

Summary

Operational parameters are parameters that are defined in terms of the data that are being modelled. This paper shows under what conditions they can be used in place of the usual formal parameters and discusses the advantages of doing this. Example applications include the normal, exponential and uniform model, the multivariate-normal and other multivariate models, finitepopulation versions, and also several new models. Also presented is their relationship to de Finetti’s work on exchangeability and other symmetrybased approaches.

Cramer-Rao type integral inequalities are developed for loss functionsw(x) which are bounded below by functions of the typeg(x) =c|x|^l,l > 1. As applications, we obtain lower bounds of Hajek-LeCam type for locally asymptotic minimax error for such loss functions.

In two recent papers del Barrio et al. (1999) and del Barrio et al. (2000) consider a new class of goodness-of-fit statistics based on theL₂-Wasserstein distance. They derive the limiting distribution of these statistics and show that the normal distribution is the only location-scale family for which this limiting distribution has the “loss of degrees of freedom” property, due to the estimation of the unknown parameters. In this paper a weightedL₂-Wasserstein distance is considered and it is proven that these statistics retain the loss of degrees of freedom property for general classes of distributions if applied separately to the location family and to the scale family and if the “right” weight function is used. These weight functions are such that the corresponding minimum distance estimators for the location parameter and the scale parameter are asymptotically efficient. Examples are discussed for both location and scale families.

We provide a complete Bayesian method for analyzing a threshold autoregressive (TAR) model when the order of the model is unknown. Our approach is based on a version (Godsill (2001)) of the reversible jump algorithm of Green (1995), and the method for estimating marginal likelihood from the Metropolis-Hasting algorithm by Chib and Jeliazkov (2001). We illustrate our results with simulated data and the Wolfe’s sunspot data set.

In this paper an overview of the existing literature about bootstrapping for estimation and prediction in time series is presented. Some of the methods are detailed, organized according to the aim they are designed for (estimation or prediction) and to the fact that some parametric structure is assumed, or not, for the dependence. Finally, some new bootstrap (kernel based) method is presented for prediction when no parametric assumption is made for the dependence.

The estimation of the location and magnitude of the optimum has long been considered as an important problem in the realm of response surface methodology. In this paper, we consider the Bayes estimates in a single factor quadratic response function, after a reparametrization from the linear model, using noninformative priors. The usual constant noninformative prior for the reparametrized model does not yield a proper posterior, thus it is desirable to consider other noninformative priors such as the Jeffreys prior and reference priors. Comparisons will be made based on the resulting posterior means, variances and credible intervals by examples and simulations.

The posterior probabilities ofK given models when improper priors are used depend on the proportionality constants assigned to the prior densities corresponding to each of the models. It is shown that this assignment can be done using natural geometric priors in multiple regression problems if the normal distribution of the residual errors is truncated. This truncation is a realistic modification of the regression models, and since it will be made far away from the mean, it has no other effect beyond the determination of the proportionality constants, provided that the sample size is not too large. In the caseK=2, the posterior odds ratio is related to the usualF statistic in “classical” statistics. Assuming zero-one losses the optimal selection of a regression model is achieved by maximizing the posterior probability of a submodel. It is shown that the geometric criterion obtained in this way is asymptotically equivalent to Schwarz’s asymptotic Bayesian criterion, sometimes called the BIC criterion. An example of polynomial regression is used to provide numerical comparisons between the new geometric criterion, the BIC criterion and the Akaike information criterion.