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.
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Bobotas, P., Iliopoulos, G. & Kourouklis, S. Estimating the ratio of two scale parameters: a simple approach. Ann Inst Stat Math 64, 343–357 (2012). https://doi.org/10.1007/s10463-010-0308-3
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DOI: https://doi.org/10.1007/s10463-010-0308-3