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December, 1993 Incidental Versus Random Nuisance Parameters
J. Pfanzagl
Ann. Statist. 21(4): 1663-1691 (December, 1993). DOI: 10.1214/aos/1176349392

Abstract

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 $(X, \mathscr{A})$. The problem is to estimate $\vartheta$, based on a sample $(x_1, \cdots, x_n)$ from $\times^n_1 P_{\vartheta,\eta_\nu}$. If $(\eta_1, \cdots, \eta_n)$ 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 $N_{(0, \sigma^2_0(\vartheta,\Gamma))}$. Folklore says that such a bound is also valid if $(\eta_1, \cdots, \eta_n)$ are just unknown values of the nuisance parameter: In this case, the distribution cannot be more concentrated asymptotically than $N_{(0, \sigma^2_0(\vartheta,E^{(n)}_{(\eta_1, \cdots, \eta_n)}))}$, where $E^{(n)}_{(\eta_1, \cdots, \eta_n)}$ is the empirical distribution of $(\eta_1,\cdots, \eta_n)$. The purpose of the present paper is to discuss to which extent this conjecture is true. The results are summarized at the end of Sections 1 and 3.

Citation

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J. Pfanzagl. "Incidental Versus Random Nuisance Parameters." Ann. Statist. 21 (4) 1663 - 1691, December, 1993. https://doi.org/10.1214/aos/1176349392

Information

Published: December, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0795.62029
MathSciNet: MR1245763
Digital Object Identifier: 10.1214/aos/1176349392

Subjects:
Primary: 62G05
Secondary: 62G20

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.21 • No. 4 • December, 1993
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