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

  • Stavros Kourouklis
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Abstract

This paper deals with the problems of decision theoretic estimation of the scale parameter, σ, 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 σ are derived. If L(t) satisfies an additional condition, the problem reduces to estimating σ in the degenerate (scale mixture) model. As the entropy is a linear translation of lnσ, its estimation under a loss L0(u) amounts to estimating σ under L(t)=L0(lnt), with the quadratic loss L0(u)=u2 and Brown's loss L(t)=(lnt)2 being a typical case. Thus, in the case that L0(u) is convex improved estimators of the entropy are directly obtained from respective estimators of σ. The results are applied to scale mixtures of normal and exponential distributions, including the multivariate t and multivariate Lomax distributions.

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Cite

BibTeXRIS
BibTeX
@misc{kourouklis2026browns,
  title = {{Brown's loss function and estimation of the scale parameter and the entropy in scale mixture models with unknown location}},
  author = {Stavros Kourouklis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13460},
  eprint = {2610.13460},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13460},
}
RIS
TY  - UNPB
TI  - Brown's loss function and estimation of the scale parameter and the entropy in scale mixture models with unknown location
AU  - Stavros Kourouklis
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13460
UR  - https://arxiv.org/abs/2610.13460
AB  - This paper deals with the problems of decision theoretic estimation of the scale parameter, σ, 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 σ are derived. If L(t) satisfies an additional condition, the problem reduces to estimating σ in the degenerate (scale mixture) model. As the entropy is a linear translation of lnσ, its estimation under a loss L_0(u) amounts to estimating σ 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 σ. The results are applied to scale mixtures of normal and exponential distributions, including the multivariate t and multivariate Lomax distributions.
ER  -