Estimating the ratio of two scale parameters: a simple approach

Abstract

We describe a simple approach for estimating the ratio ρ = σ 2 /σ 1 of the scale parameters of two populations from a decision theoretic point of view. We show that if the loss function satisfies a certain condition, then the estimation of ρ reduces to separately estimating σ 2 and 1/σ 1 . This implies that the standard estimator of ρ can be improved by just employing an improved estimator of σ 2 or 1/σ 1 . Moreover, in the case where the loss function is convex in some function of its argument, we prove that such improved estimators of ρ are further dominated by corresponding ones that use all the available data. Using this result, we construct new classes of double-adjustment improved estimators for several well-known convex as well as non-convex loss functions. In particular, Strawderman-type estimators of ρ in general models are given whereas Shinozaki-type estimators of the ratio of two normal variances are briefly treated.Keywords Decision theory · Improved estimation of a scale parameter · Improved estimation of ratio of scale parameters · Stein's and Brewster and Zidek's techniques · Strawderman's technique · Kubokawa's approach

General dominance properties of double shrinkage estimators for ratio of positive parameters

“…The dominance property given in Theorem 2.1 can be provided as a simple conclusion of Kubokawa and Srivastava (1996), Iliopoulos and Kourouklis (1999) and Bobotas, et al . (2012).…”

“…Thus, we could not answer whether the stronger dominance property holds or not for the quadratic loss. In stead ofρ * =θ * 2 /θ * 1 , Kubokawa (1994b) and Bobotas, et al . (2012) showed that the double shrinkage estimator of the formρ 2 .…”

“…This dominance result is also an application of Theorem 2.1. For the quadratic loss function, on the other hand, the double shrinkage estimator suggested by Kubokawa (1994b) and Bobotas, et al . (2012) is of the formρ…”

Εκτιμητές τύπου Strawderman για παραμέτρους κλίμακας

“…Τέλος, στην περίπτωση του προβλήµατος εκτίµησης του λόγου των παραµέτρων κλίµακας δύο εκθετικών κατανοµών, οι Madi and Tsui (1990b) , ενώ η Ενότητα 3.5 αποτελεί µέρος της εργασίας Bobotas, Iliopoulos and Kourouklis (2009).…”