More on Estimation of Distributions Using Orthogonal Expansions

Let f(x) be a continuous, strictly positive probability density function over an interval [a, b], and let F(x) be its associated cumulative distribution function. Suppose ϕ0(x), ϕ1(x), ϕ2(x), … form a complete orthonormal basis for L 2[a, b], the set of square-integrable functions on [a, b]. Suppose, further, that over [a, b] and that log f(x) has a similar expansion. Given a random sample X 1, X 2, …, Xn from the distribution, some aspects of the estimation of f(x) and F(x) are investigated.