Showing papers by "Iain M. Johnstone published in 1984"
TL;DR: In this paper, a discrete univariate version of Brown's diffusion characterization of admissibility in estimation of a multivariate normal mean is proposed, where the objective is to decide whether a generalized Bayes estimator is admissible if and only if it is recurrent.
Abstract: Consider estimation of a Poisson mean $\lambda$ based on a single observation $x$, using estimator $d(x)$ and loss function $(d(x) - \lambda)^2/\lambda$. The goal is to decide (in)admissibility of $d(x)$. To every generalized Bayes estimator there corresponds a unique reversible birth and death process $\{X_t\}$ on $\mathbb{Z}_+$. Under side conditions $d(x)$ is admissible if and only if it is generalized Bayes and $\{X_t\}$ is recurrent. Explicit equivalent conditions exist in terms of difference equations and minimization problems. The theory is a discrete, univariate counterpart to Brown's (1971) diffusion characterization of admissibility in estimation of a multivariate normal mean. A companion paper discusses simultaneous estimation of several Poisson means.
29 citations
TL;DR: This paper showed that such measures do not ordinarily commute with affine coordinate reexpressions unless the distribution is centrosymmetric (location case) or elliptically contoured (dispersion case).
Abstract: Huber (1981b) and Li and Chen (1981) have proposed the use of projection pursuit techniques to construct and estimate measures of location and dispersion for multivariate distributions. In this paper we show that such measures do not ordinarily commute with affine coordinate reexpressions unless the distribution is centrosymmetric (location case) or elliptically contoured (dispersion case).
8 citations
TL;DR: In this article, a null recurrent diffusion on ℝp with generatorG=(1/r)∇·(r∇) for smooth positiver is constructed such that (Zt, Z′t)is transient in Ωp+1.
Abstract: LetZt be a null recurrent diffusion on ℝp with generatorG=(1/r)∇·(r∇) for smooth positiver. This note constructs an independent recurrent diffusionZ′ton ℝ1 such that (Zt, Z′t)is transient in ℝp+1. This resolves negatively an old question in simultaneous estimation: Is there an admissible but not Bayes estimatorσ(X) of the mean of a multivariate normal distribution for quadratic loss with the property: for every admissible σ′(X′), whereX′ is normal and independent ofX, (δ, δ′) remains admissible in the combined problem obtained by summing the component losses?
4 citations