Improving on the positive part of the UMVUE of a noncentrality parameter of a noncentral chi-square distribution
This paper gives explicit improvements to the positive part of the UMVUE of a noncentrality parameter of a noncentral \(\chi^2_n (\mu/2)\). Let \(Y \sim \chi^2_n (\mu/2)\) with \(n\) (known) degrees of freedom and unknown noncentrality parameter \(\mu\). We consider the problem of estimating \(\mu\) with loss \(= (\delta - \mu)^2\). The estimator \(Y + n\) is the generalized Bayes estimator of \(\mu\) with respect to a noninformative prior distribution and is dominated by \(Y - n\), the UMVUE of \(\mu\). Our results fall into two broad classes. The first set of results describes explicit estimators which change \((Y - n)_+\) only on the set \([n,n + 2]\). These results depend on some properties of central chi-square distributions which along with other preliminary results are given in Section 2. In Section 3, we will study alternative estimators of the form \[ \delta (a,g,Y) = (Y - n)_+ - ag(Y) I_{\{n \leq Y \leq n + 2\}}, \tag{1} \] where \(g (\cdot)\) is an even symmetric (``\(W\)-shaped'') piecewise linear function about \(Y = n + 1\) with \(g(n) = g(n + 2) = 0\), \(g'(n) < 0\) and \(|g'(Y) |\equiv 1\) a.e. on \([n,n + 2]\). To fully specify \(\delta (a,g,Y)\), it suffices to specify the constant \(a\) and a value \(y^*\) in \((n + 1,n + 2)\) such that \(g(y^*)\) attains its minimum. Values of \(a\) and \(y^*\) are given such that \(\delta (a,g,Y)\) dominates \((Y - n)_+\). In Section 4, we study estimators of the form (1) for certain nonpiecewise linear \(g\). It follows from the results of \textit{M. S. Chow} [Ann. Stat. 15, 800-804 (1987; Zbl 0627.62008)] that estimators of the form (1) cannot themselves be admissible because they are not generalized Bayes. Therefore we investigate a more general class of estimators in Section 5. This second class of estimators is of the form \[ \delta (Y) = (Y - n)_+ - ag (Y) I_{\{Y > q \}} - akh (Y) I_{\{0 \leq Y \leq q\}} \tag{2} \] with \(q > n\). We give conditions on \(a,k,q\), \(g (\cdot)\) such that (2) dominates \((Y - n)_+\). In this case, \(g (\cdot)\) will be ``\(W\)'' shaped on \([q, \infty)\) and \(h (\cdot)\) will be bounded continuous and nonpositive. We believe, but have not proved, that the admissible improvements can be found in this class.
scientific article; zbMATH DE number 769826