Comparing location experiments. (English) Zbl 0672.62008

Lehmann, E. L.

Ann. Stat. 16, No. 2, 521-533 (1988).

Motivated by the non-comparability of two experiments in the sense of the comparison based on all decision kernels and all loss functions available (Blackwell, LeCam et al) the author introduces the weaker comparison of more effectivity with respect to a subclass \({\mathcal C}\) of decision kernels which is large enough to include most of the kernels arising from interesting statistical problems, and small enough to exclude non- comparability. For the subclass \({\mathcal C}\) the class \({\mathcal M}\) of all kernels \(\delta\) is taken which are monotone in the sense that for any \(x<x'\), \(\delta (x,[a,\infty))>0\) implies \(\delta (x,[a,\infty))=1.\)
One of the innovative results presented by the author is the fact, that two experiments defined by families \((F_{\theta})\) and \((G_{\theta})\) of distribution functions admitting densities \(f_{\theta}\) and \(g_{\theta}\) with monotone likelihood ratios, are in the relationship of more effectivity w.r.t. \({\mathcal M}\) iff the function \(\theta \to G_{\theta}^{-1}(F_{\theta}(x))\) is nondecreasing for all x. The result is applied to location and scaling experiments. The conditions on F and G arising from effectivity lead to new tail-orderings and shapings of distributions.

Reviewer: H.Heyer

MSC:

62B15	Theory of statistical experiments
62C05	General considerations in statistical decision theory

Keywords:

location families; monotone decision procedures; spread ordering; non- comparability; decision kernels; loss functions; more effectivity; monotone likelihood ratios; tail-orderings