Estimating events
In the first part of this paper the author describes the problem of estimating an event with positive probability content based on a sample of n observations and gives the definition of optimal estimators of events. The most important result of this paper is application of this theory to the following problem: (i) Let \(\underset \tilde{} X=(x_ 1,...,x_ n)\) be a sample from the exponential \((\theta)\) model where \(\theta\) is unknown. How to estimate the \(\beta\)-quantile \(q(\theta)=-\theta.\ln(1-\beta)?\) (ii) Let \(\underset \tilde{} X=(\underset \tilde{} x_ 1,...,\underset \tilde{} x_ n)^ t\in R^{p\times n}\) be a sample from \(N_ p(\mu,\Sigma)\) distribution, where \(\mu \in R^ p\) and \(\Sigma \in R^{p\times p}\) positive definite are not known. Let \(Q(\mu,\Sigma)=\{y| (y-\mu)^ t\Sigma(y-\mu)\leq k_ 0\}\) where \(k_ 0=\chi^ 2_{1-\beta}(p)\). How to estimate \(Q(\mu,\Sigma)\)? Following two theorems are proved. Theorem 1. If \(\underset \tilde{} X=(x_ 1,...,x_ n)^ t\) is a sample from a distribution in the class \{exponential \((\theta)| \theta>0\}\) then a finite optimal equivariant estimator exists for the quantile - \(\theta.\ln(1-\beta)\) if and only if \(\chi^ 2_{0.5}(2n+2)>-2 \ln(1-\beta)\) and this is always true whenever \(\beta \leq.5\). When such an estimator exists it is of the form \[ \bar x_ n.(-n.\ln(1- \beta))/(\ln(1-\beta)+({1\over2}).\chi^ 2_{0.5}(2n+2)). \] Theorem 2. If \(\underset \tilde{} X=(\underset \tilde{} x_ 1,...,\underset \tilde{} x_ n)\) is a sample from a distribution in \(\{N_ p(\mu,\Sigma)| \mu \in R^ p\), \(\Sigma \in R^{p\times p} p.d.\}\) then an optimal estimator of the form \(C(\underset \tilde{} X)=\{\underset \tilde{} y| \underset \tilde{} y-\underset \tilde{} x)^ t.S^{-1}(x).(\underset \tilde{} y-\underset \tilde{} x)\leq k\}\) where \(k\in [0,\infty)\) exists for \(Q(\mu,\Sigma)\) if and only if \[ G_ p((n+1)k_ 0)>.5\text{ and }\sum^{\infty}_{m=0} b(m,(n/2),(1/n+1))G_{p+2m}((n+1)k_ 0)<0.5, \] where \(G_ p\) equals to the \(\chi^ 2_ p\) distribution function. Important part of the paper is concentrated in the comparison of this method (and results) with approaches proposed earlier for solving the same problem.
scientific article; zbMATH DE number 3862247