Estimation of Distributions Using Orthogonal Expansions

Abstract

Let $f(x)$ be a continuous, strictly positive probability density function over an interval $\lbrack a, b\rbrack$ and $F(x)$ its associated $\operatorname{cdf}$. Suppose $\{\phi_i(x)\}^\infty_{i=0}$ is a complete orthonormal basis for $L_2\lbrack a, b\rbrack$ and that $f(x)$ and $\log f(x)$ have orthogonal series expansions, in the $\phi_i$'s, over $\lbrack a, b\rbrack$. Estimators for $f(x)$ and $F(x)$ are chosen from the canonical exponential family of distributions generated by $\{\phi_i(x)\}^\infty_{i=0}$, and convergence theorems are presented for these estimators in the special case of Legendre polynomials over $\lbrack -1, 1\rbrack$.