Estimating the ratio of two scale parameters: a simple approach

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.