Brown's loss function and estimation of the scale parameter and the entropy in scale mixture models with unknown location

arXiv:2610.13460v1 Announce Type: new Abstract: This paper deals with the problems of decision theoretic estimation of the scale parameter, $\sigma$, and the entropy (Shannon or Renyi) in general scale mixture models with unknown location. Employing Kubokawa's approach, under convexity of the loss $L(t)$ either in $t$ or in $lnt$, improved estimators of $\sigma$ are derived. If $L(t)$ satisfies an additional condition, the problem reduces to estimating $\sigma$ in the degenerate (scale mixture) model. As the entropy is a linear translation of $ln\sigma$, its estimation under a loss $L_0(u)$ amounts to estimating $\sigma$ under $L(t)=L_0(lnt)$, with the quadratic loss $L_0(u)=u^2$ and Brown's loss $L(t)=(lnt)^2$ being a typical case. Thus, in the case that $L_0(u)$ is convex improved estimators of the entropy are directly obtained from respective estimators of $\sigma$. The results are applied to scale mixtures of normal and exponential distributions, including the multivariate $t$ and multivariate Lomax distributions.