Uniform consistency of a histogram density estimator and modal estimation

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Abstract
Let be an ordered sample of n independent observations, X1,X2,…,Xn of a random variable X with distribution function F(x) and density f(x) continuous on its support set . As a nonparametric histogram estimator of the density function f(x), consider an estimator fn (x) of the form: where {An (x)} is a suitably chosen sequence of non-negative integer-valued indexing random variables; and {kn} is also an appropriately defined sequence of positive integers which depends only on the sample size n . J. Van Ryzin (1973) has given conditions under which the above estimators are pointwise consistent. In this paper we establish conditions under which such a histogram density estimator is uniformly consistent almost surely. When the density has a unique mode, the results are used to obtain a strongly consistent estimator of the mode similar to that of Venter (1967).

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