Estimating the ratio of two scale parameters: A simple approach
P. Bobotas, G. Iliopoulos and S. Kourouklis
In this paper we describe a simple approach for estimating the ratio
$\rho=\sigma_{2}/\sigma_{1}$ 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 $\rho$ reduces to separately estimating $\sigma_{2}$ and
$1/\sigma_{1}$. This implies that the standard (i.e., best
equivariant) estimator of $\rho$ can be improved by just employing
an improved estimator of $\sigma_{2}$ or $1/\sigma_{1}$. Moreover,
in the case where the loss function is convex in some function of
its argument, we prove that such improved estimators of $\rho$ 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 nonconvex loss functions. In particular, Strawderman-type
estimators of $\rho$ are given in general models having monotone
likelihood ratio properties. Moreover, in the case of two
normal populations, Shinozaki-type estimators of the ratio of the
variances are briefly treated.
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