Concavity and estimation. (English) Zbl 0699.62027

Haberman, Shelby J.

Ann. Stat. 17, No. 4, 1631-1661 (1989).

Given a “sample space” U (a separable complete metric space) and an auxiliary set \(V=R^ p\) for some finite positive integer p, an estimation function \(h: U\times V\to [-\infty,+\infty)\) is considered. The special requirement is that h(x,\(\cdot)\) is concave for each \(x\in U.\)
Let \(d(w,h,F)=E h(X,w)\), where X is an U-valued random element with distribution F. Given \(W\subset V\), the M-parameter M(W,h,F) is defined, roughly speaking, as a w maximizing d(w,h,F) with respect to \(w\in W\). To estimate M(W,h,F) the estimates \(M(W,h,F_ n)\) are used, where \(F_ n\) is the empirical distribution. Strong consistency and asymptotic normality of \(M(W,h,F_ n)\) is discussed.

Reviewer: R.Zielinski

MSC:

62F12	Asymptotic properties of parametric estimators
62E20	Asymptotic distribution theory in statistics
62F10	Point estimation

Keywords:

M-estimates; maximization of averages of independent identically; distributed random concave functions; maximum likelihood; estimation; empirical distribution; Strong consistency; asymptotic normality