J. Montanero

A note on kernel density estimation at a parametric rate

In the context of kernel density estimation, we give a characterization of the kernels for which the parametric mean integrated squared error (MISE) rate n −1 may be obtained, where n is the sample size. Also, for the cases where this rate is attainable, we give an asymptotic bandwidth choice that makes the kernel estimator consistent in mean integrated squared error at that rate and a numerical example showing the superior performance of the superkernel estimator when the bandwidth is properly chosen. †Research supported by Spanish Ministerio de Ciencia y Tecnología project MTM2005-06348.

On the Use of Bayes Factor in Frequentist Testing of a Precise Hypothesis

This article deals with Bayes factors as useful Bayesian tools in frequentist testing of a precise hypothesis. A result and several examples are included to justify the definition of Bayes factor for point null hypotheses, without merging the initial distribution with a degenerate distribution on the null hypothesis. Of special interest is the problem of testing a proportion (joint with a natural criterion to compare different tests), the possible presence of nuisance parameters, or the influence of Bayesian sufficiency on this problem. The problem of testing a precise hypothesis under a Bayesian perspective is also considered and two alternative methods to deal with are given.

Strong invariance under a Bayesian point of view

Sufficiency on the parameter space is used to extend to the Bayesian case a strong notion of invariance, obtaining the sufficiency of the almost-invariant [sigma]-field . We also give some results about the completeness of , giving in passing two partial answers to a question of J.K. Ghosh.

Partial sufficiency and density estimation

In this paper, we study the usefulness of the concept of partial sufficiency, in the sense of Fraser, when it is applied to the problem of nonparametric density estimation.

Exhaustivity, completeness and almost invariance

In this paper, the relationship between exhaustivity (or sufficiency in the sense of Blackwell) and invariance is studied. An analogous result of two classical theorems of Hall, Wijsman and Ghosh (1965) and Berk (1972) on the relationship between sufficiency and invariance is given making use of a concept introduced here under the name of boundedG-completeness; in particular, we get the same conclusion under the assumptions of stability and bounded completeness.