On the rate of convergence for maximum likelihood estimates in a truncated case
Let \(X_ 1,...,X_ n\) be a random sample of size n from the probability density function f(x-\(\theta)\), \(x\in {\mathbb{R}}\), with location parameter \(\theta\in {\mathbb{R}}\). It is assumed that f is uniformly continuous and positive in (0,\(\infty)\). It is also assumed that \(f(x)\sim \alpha \beta x^{\alpha -1}\) as \(x\to 0+\), where \(1<\alpha <2\) and \(\beta >0\). If \({\hat \theta}{}_ n={\hat \theta}_ n(X_ 1,...,X_ n)\) is the maximum likelihood of \(\theta\), it is known [\textit{M. Woodroofe}, Ann. Stat. 2, 474-483 (1974; Zbl 0283.62030)] that \(\beta_ n^{1/\alpha}({\hat \theta}_ n-\theta)\) has a limiting distribution, as \(n\to \infty\), which is not normal. Let H be the distribution function of this limiting distribution. The purpose of the present paper is to study the rate of convergence of the distribution of \((\beta_ n)^{1/\alpha}({\hat \theta}_ n- \theta)\) to its limiting distribution H. Under certain conditions on f and the derivative (with respect to x) of \(g(x)=f(x-\theta)\) (where \(\theta\) is actually set equal to 0), the existence of maximum likelihood estimates (in a certain range) is ensured which are roots of the likelihood equation. Then the main result established is that \[ | P_{\theta}[b_ n({\hat \theta}_ n-\theta)\leq -t]-H(-t)| \leq Cn^{-(2-\alpha)/\alpha}(\log n)^{(4-\alpha)/2\alpha}\quad and \] \[ | P_{\theta}[b_ n({\hat \theta}_ n-\theta)\geq -t]-H(-t)| \leq Cn^{-(2-\alpha)/\alpha},\quad for\quad some\quad C>0,\quad for\quad all\quad \theta,\quad t\geq 0,\quad and\quad n\geq 1, \] provided \(\alpha +\lambda >2\). If \(\alpha +\lambda \leq 2\), then for every \(\lambda\) with \(0<\lambda <\gamma\), there exists \(C>0\) such that, for all \(\theta\), t and \(n\geq 1\), holds \(| P_{\theta}[b_ n({\hat \theta}_ n- \theta)\leq t]-H(t)| \leq Cn^{-\lambda /\alpha}.\) Here \(b_ n=(\beta \Lambda)^{1/\alpha}\) and the positive constant \(\lambda\) appears in the assumptions made on f and derivatives of g.
existence of maximum likelihood estimates
scientific article; zbMATH DE number 3898018