Stochastic estimation and testing. (English) Zbl 0644.62028

Beran, R.; Millar, P. W.

Ann. Stat. 15, 1131-1154 (1987).

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).

Reviewer: V.Malinovskij

MSC:

62F12	Asymptotic properties of parametric estimators
62F05	Asymptotic properties of parametric tests
62E20	Asymptotic distribution theory in statistics

Keywords:

randomized tests; iterated bootstrap techniques; random sampling schemes; stochastic minimum distance tests; stochastic confidence bands; critical value; Monte Carlo; minimum distance estimate; goodness-of-fit test; likelihood ratio test; stochastic search; asymptotic normality; asymptotic efficiency; likelihood function; stochastic maximum likelihood estimate