Universal Consistency of Delta Estimators

This paper considers delta estimators of the Radon-Nikodym derivative of a probability function with respect to a a-finite measure. We provide sufficient conditions for universal consistency, which are checked for some wide classes of non-parametric estimators. Let P be a probability measure in the Borel space (]l~d,~d), absolutely continuous with respect to the a-finite measure # and f = dP/d# be the corresponding Radon-Nikodym derivative, which is assumed to belong to the space Lp(]~d,~d,p), with 1 < p < c~. Usually, the Lebesgue measure A is considered, and f = dP/dA is the associated probability density function (pdf). Given a random sample {Xi}i~l from P, a delta estimator of f is defined as, n 1 E Kin. (x; Xi), i=1 where mn= re(n) is called a smoothing sequence, and {Km, }hen a generalized kernel sequence. The sequence {mn}neN is not necessarily a sequence of numbers, it may be a sequence of positive definite matrices ordered by decreasing norm, in the usual kernel estimator of a multivariate density; or the order of a polynomial, in the Fourier series estimator. We consider that the smoothing sequence {ran}heN belongs to some directed set ~. We say that the set ][ is directed if it is a non empty set endowed with a partial preorder <, such that if Vml,m2 E ][, 3m3 E IT such that ml _~ m3 and m2 _~ m3. We also assume that {mn}nCN diverges in ][ as n ~ ~, i.e., VM E IT, 3riM C I~ such that mn ~_ M, Vn ~ nM. The class of delta estimators was introduced by Whittle (1958), encompassing most of the existing nonparametric estimators. Terrell (1984) and Terrell and Scott (1992) have shown that all nonparametric density estimators which are continuous and differ-entiable functionals of the empirical distribution function, can be interpreted as delta estimators, at least asymptotically.