Estimation of Distributions Using Orthogonal Expansions

Crain, Bradford R.

Ann. Statist., Tome 2 (1974) no. 1, p. 454-463 / Harvested from Project Euclid

Résumé

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 $\lbrack -1, 1\rbrack$.

Détail

Publié le : 1974-05-14
Classification:  Densities,  estimation of densities,  cumulative distribution functions,  estimation of distributions,  restricted maximum likelihood estimation,  exponential families,  62G05,  62G99,  41A10,  42A08