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Inadmissible estimators of normal quantiles and two-sample problems with additional information. (English) Zbl 1326.62055

Fourdrinier, Dominique (ed.) et al., Contemporary developments in Bayesian analysis and statistical decision theory. A festschrift for William E. Strawderman. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-81-2). Institute of Mathematical Statistics Collections 8, 104-116 (2012).
Summary: We consider estimation problem of a normal quantile \(\mu+\eta\sigma\). For the scale invariant squared error loss and unrestricted values of the population mean and standard deviation \(\mu\) and \(\sigma\), J. V. Zidek [Ann. Math. Stat. 42, 1444–1447 (1971; Zbl 0225.62036)] established the inadmissibility of the MRE estimator for \(\eta\neq 0\). In this paper, we explore: (i) the impact of the loss with the study of scale invariant absolute value loss, and (ii) situations where there is a parameter space restriction of a lower bounded mean \(\mu\). We establish
(i)
the inadmissibility of the MRE estimator of \(\mu+\eta\sigma\); \(\eta\neq 0\) under scale invariant absolute value loss;
(ii)
the inadmissibility of the Generalized Bayes estimator of \(\mu+\eta\sigma\); \(\eta >0\); under scale invariant squared error loss, associated with the prior measure \(1_{(0,\infty)} (\mu)1_{(0,\infty)}(\sigma)\) which represents the truncation of the usual non-informative prior measure onto the restricted parameter space.
Both of these results are obtained through a conditional risk analysis and may be viewed as extensions of Zidek [loc. cit.]. Finally, we provide further applications to two-sample problems under the presence of the additional information of ordered means.
For the entire collection see [Zbl 1319.62003].

MSC:

62F10 Point estimation
62F30 Parametric inference under constraints
62C15 Admissibility in statistical decision theory

Citations:

Zbl 0225.62036
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