On the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate
Let \(X_ 1,X_ 2,..\). be an i.i.d. sequence of random variables with density function f on \({\mathbb{R}}\). Consider kernel density estimators \[ \hat f_ n(x,h)=(nh)^{-1}\sum^{n}_{j=1}K((x-X_ j)/h) \] and choose a data driven bandwidth \(\hat h_ n\) by maximizing the cross- validated likelihood product, as suggested by \textit{J. D. F. Habbema}, \textit{J. Hermans} and \textit{Vandenbroek} [A stepwise discriminant analysis program using density estimation. G. Bruckmann (ed.), Compstat 1974, 101- 110 (1975)] and \textit{R. P. W. Duin} [IEEE Trans. Comput. C-25, 1175-1179 (1976; Zbl 0359.93035)]. \textit{P. Hall} [Biometrica 69, 383-390 (1982; Zbl 0496.62038)] showed that even for densities with bounded support, \(\hat h_ n\) behaves suboptimal but leads to consistent estimates [see also \textit{Y.-S. Chow}, \textit{S. Geman} and \textit{L.-D. Wu}, Ann. Stat. 11, 25-38 (1983; Zbl 0509.62033)]. In this paper the case of densities with unbounded support is discussed. Under some minor assumptions the main result reads essentially as follows: \(\int | \hat f_ n-f| dx\to 0\) in prob. (a.s.) iff \(X_{1,n}\) and \(X_{n,n}\) (the extremes of the sample) are stable (strongly stable), i.e. there exist sequences \(\alpha_ n\), \(\beta_ n\) s.t. \(X_{1,n}-\alpha_ n\to 0\) in prob. (a.s.) and \(X_{n,n}-\beta_ n\to 0\) in prob. (a.s.). But even then the rate of convergence may be slower than \(n^{-\epsilon}\) for any \(\epsilon >0\), as shown for certain densities in the domain of attraction of Gumbel's law.
scientific article; zbMATH DE number 4149370