arXiv subjects
Publications and source records attributed to A. G. Nogales.
Under mild conditions, it is shown the strong consistency of the Bayes estimator of the density. Moreover, the Bayes risk (for some common loss functions) of the Bayes estimator of the density (i.e. the posterior predictive density) reaches zero when the sample size goes to $\infty$. In passing, a similar result is obtained for the estimation of the sampling distribution.
In this paper several related estimation problems are addressed from a Bayesian point of view and optimal estimators are obtained for each of them when some natural loss functions are considered. Namely, we are interested in estimating a regression curve. Simultaneously, the estimation problems of a conditional distribution function, or a conditional density, or even the conditional distribution itself, are considered. All these problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, parametric and non-parametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function comonly used in estimating a real function of the unknown parameter. The cornerstone of the mentioned Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate these results.
Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the $L^1$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed. Consistency of the estimator of the density is proved.
A known property of conditional expectation is extended to the framework of Markov kernels. Its meaning in terms of densities is provided. Some examples located in the field of clinical diagnosis are presented to delimit the main result of the paper.
Two known results on the relationship between conditional and unconditional independence are obtained as a consequence of the main result of this paper, a theorem that uses independence of Markov kernels to obtain a minimal condition which added to conditional independence implies independence. Some counterexamples and representation results are provided to clarify the concepts introduced and the propositions of the statement of the main theorem. Moreover, conditional independence and the mentioned results are extended to the framework of Markov kernels.
This work addresses the problem of estimating the parameters of the general half-normal distribution. Namely, the problem of determining the minimum risk equi\-va\-riant (MRE) estimators of the parameters is explored. Simulation studies are realized to compare the behavior of these estimators with maximum likelihood and unbiased estimators. A natural Monte Carlo method to compute conditional expectations is used to approximate the MRE estimation of the location parameter because its expression involves two conditional expectations not easily computables. The used Monte Carlo method is justified by a theorem of Besicovitch on differentiation of measures, and has been slightly modified to solve a sort of "curse of dimensionality" problem appearing in the estimation of this parameter. This method has been implicitly used in the last years in the context of ABC (approximate Bayesian computation) methods.
In the context of kernel density estimation, we give a characterization of the kernels for which the parametric mean integrated squared error 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.