Stochastic estimation and testing
Several stochastic procedures (tests, estimates and confidence intervals) are introduced and the asymptotic theory of these procedures is developed. For example, let \(\{Q_{\theta}\), \(\theta\in \Theta \}\) be a family of distribution functions on \(R^ d\) with density f(\(\theta\),x), \(\Theta \subset R^ d\), \(X_ n=(x_ 1,...,x_ n)\) be an i.i.d. sample of size n, \(L_ n(\theta,X_ n)=\prod^{n}_{1}f(\theta,x_ i)\) be the likelihood function. Let \(S_ n=(s_ 1,...,s_{j_ n})\) be a certain random sample of \(j_ n\) elements in \(\Theta\). The stochastic maximum likelihood estimate (SMLE) \({\hat \theta}_ n={\hat \theta}_ n(X_ n,S_ n)\) is defined by the requirement \[ L_ n({\hat \theta}_ n,X_ n)=\max (L_ n(s_ j,X_ n),\quad 1\leq j\leq j_ n). \] Under some regularity assumptions, the SMLE is asymptotically (n\(\to \infty)\) normal and asymptotically efficient (like the usual MLE).
scientific article; zbMATH DE number 4050752
scientific article