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 \φᵢ(x)\^∞ᵢ₌₀ is a complete orthonormal basis for L₂ a, b and that $f(x)$ and log f(x) have orthogonal series expansions, in the φᵢ's, over a, b. Estimators for $f(x)$ and $F(x)$ are chosen from the canonical exponential family of distributions generated by \φᵢ(x)\^∞ᵢ₌₀, and convergence theorems are presented for these estimators in the special case of Legendre polynomials over -1, 1.