×

Incidental versus random nuisance parameters. (English) Zbl 0795.62029

Author’s summary: Let \(\{P_{\vartheta,\eta}\): \((\vartheta,\eta)\in \Theta\times H\}\), with \(\Theta\subset \mathbb{R}\) and \(H\) arbitrary, be a family of mutually absolutely continuous probability measures on a measurable space \((X,{\mathcal A})\). The problem is to estimate \(\vartheta\), based on a sample \((x_ 1,\dots,x_ n)\) from \(\times_ 1^ n P_{\vartheta, \eta_ \nu}\). If \((\eta_ 1,\dots, \eta_ n)\) are independently distributed according to some unknown prior distribution \(\Gamma\), then the distribution of \(n^{1/2} (\vartheta^{(n)}- \vartheta)\) under \(P^ n_{\vartheta,\Gamma}\) (\(P_{\vartheta,\Gamma}\) being the \(\Gamma\)-mixture of \(P_{\vartheta,\eta}\), \(\eta\in H\)) cannot be more concentrated asymptotically than a certain normal distribution with mean 0, say \(N_{(0, \sigma_ 0^ 2 (\vartheta,\Gamma))}\).
Folklore says that such a bound is also valid if \((\eta_ 1,\dots,\eta_ n)\) are just unknown values of the nuisance parameter: In this case, the distribution cannot be more concentrated asymptotically than \(N_{(0,\sigma_ 0^ 2 (\vartheta, E^{(n)}_{\eta_ 1,\dots,\eta_ n}))}\), where \(E^{(n)}_{(\eta_ 1, \dots,\eta_ n)}\) is the empirical distribution of \((\eta_ 1,\dots, \eta_ n)\). The purpose of the present paper is to discuss to which extent this conjecture is true.

MSC:

62G05 Nonparametric estimation
62F10 Point estimation
Full Text: DOI