Abstract
Sufficient conditions are derived for the asymptotic efficiency and equivalence of componentwise Bayesian and classical estimators of the infinite-dimensional parameters characterizing $l^{2}$ valued Poisson process, and Hilbert valued Gaussian random variable models. Conjugate families are considered for the Poisson and Gaussian univariate likelihoods, in the Bayesian estimation of the components of such infinite-dimensional parameters. In the estimation of the functional mean of a Hilbert valued Gaussian random variable, sufficient and necessary conditions, that ensure a better performance of the Bayes estimator with respect to the classical one, are also obtained for the finite-sample size case. A simulation study is carried out to provide additional information on the relative efficiency of Bayes and classical estimators in a high-dimensional framework.
Highlights
In the last two decades, there have been numerous contributions for the statistical inference in function spaces, motivated by the analysis of high-dimensional data
Recent developments in the context of directed acyclic graphs allow the efficient estimation of the adjacency matrix closely related with the influence matrix
Penalized likelihood estimation is considered in [19]. These results can be applied to the estimation of the autocorrelation operator eigenvalues of an autoregressive Hilbertian process
Summary
In the last two decades, there have been numerous contributions for the statistical inference in function spaces, motivated by the analysis of high-dimensional data (see, for example, [3, 6, 7, 11, 14, 15, 16], among others). The same statistics are computed for comparing classical and Bayesian estimators of the mean function of a Gaussian Hilbert valued random variable, when different rates of convergence of the eigenvalues of the covariance operator are considered, obtaining similar results in relation to the rate of convergence to zero of the empirical mean-square errors, when the sample size increases, as well as in relation to the empirical relative efficiency, which again is equal to one. A simulation study is undertaken in Section 6 providing additional information of the relative efficiency of Bayes and classical parameter estimators, when truncation is performed, and an increasing sequence of finite functional sample sizes is tested, for different rate of convergence to zero on the components of the infinite-dimensional parameters approximated
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