Open Access
May, 1974 Estimation of Distributions Using Orthogonal Expansions
Bradford R. Crain
Ann. Statist. 2(3): 454-463 (May, 1974). DOI: 10.1214/aos/1176342706


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 $\lbrack -1, 1\rbrack$.


Download Citation

Bradford R. Crain. "Estimation of Distributions Using Orthogonal Expansions." Ann. Statist. 2 (3) 454 - 463, May, 1974.


Published: May, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0283.62042
MathSciNet: MR362678
Digital Object Identifier: 10.1214/aos/1176342706

Primary: 62G05
Secondary: 41A10 , 42A08 , 62G99

Keywords: cumulative distribution functions , densities , estimation of densities , estimation of distributions , exponential families , restricted maximum likelihood estimation

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 3 • May, 1974
Back to Top