×

Bayes rules for location problems. (English) Zbl 0582.62028

Statistical decision theory and related topics III, Proc. 3rd Purdue Symp., West Lafayette/Indiana 1981, Vol. 1, 315-327 (1982).
[For the entire collection see Zbl 0575.00019.]
The problem treated here is estimation of a location parameter when the error distribution is not completely known. Previous work from a Bayesian point of view is reviewed in Section 2. Some new nonparametric results are described.
For example: let \(X_ i=\theta +\epsilon_ i\), \(1\leq i\leq n\) be observed. Put prior distributions on \(\theta\) and \(\epsilon\) : \(\theta\) \(\sim \mu (d\theta)\) and \(\epsilon \sim D_{\alpha}\), where \(D_{\alpha}\) is the Dirichlet prior with parameter \(\alpha\). The posterior distribution is computed. The Bayes estimate of \(\theta\) under squared error loss is \[ (1)\quad {\hat \theta}=\int \theta \Pi^*\alpha '(X_ i-\theta)\mu (d\theta)/\int \Pi^*\alpha '(X_ i-\theta)\mu (d\theta), \] the product being over distinct \(X_ i\) only. Some frequentist properties of \({\hat \theta}\) are derived in Section 3. Roughly, if the parameter measure \(\alpha\) is long-tailed like a Cauchy distribution, then \({\hat \theta}\) is inconsistent, in the sense that there are random variables \(X_ i\), symmetric about 0, such that \({\hat \theta}\) oscillates indefinitely between two wrong answers as n tends to infinity. It is also shown that \({\hat \theta}\) is robust, in the sense of having a bounded influence curve, for \(\alpha\) having tails exponential or longer.
In Section 4 some extensions such as symmetrized Dirichlet priors are considered. The overall conclusions do not change: The estimate \({\hat \theta}\) given in (1) with \(\alpha\) ’ log convex, having exponential tails, is the only Bayes estimate which is both robust and consistent.

MSC:

62F15 Bayesian inference
62C10 Bayesian problems; characterization of Bayes procedures
62F35 Robustness and adaptive procedures (parametric inference)

Citations:

Zbl 0575.00019