Estimation of Distributions Using Orthogonal Expansions

Let f (x) be a continuous, strictly positive probability density function over an interval a, b and F (x) its associated cdf. Suppose \ᵢ (x) \^₈=₀ is a complete orthonormal basis for L₂ a, b and that f (x) and f (x) have orthogonal series expansions, in the ᵢ's, over a, b. Estimators for f (x) and F (x) are chosen from the canonical exponential family of distributions generated by \ᵢ (x) \^₈=₀, and convergence theorems are presented for these estimators in the special case of Legendre polynomials over -1, 1.