The Annals of Statistics

Estimation of Distributions Using Orthogonal Expansions

Bradford R. Crain

Full-text: Open access

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 $\lbrack -1, 1\rbrack$.

Article information

Source
Ann. Statist., Volume 2, Number 3 (1974), 454-463.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342706

Digital Object Identifier
doi:10.1214/aos/1176342706

Mathematical Reviews number (MathSciNet)
MR362678

Zentralblatt MATH identifier
0283.62042

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G99: None of the above, but in this section 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 42A08

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

Citation

Crain, Bradford R. Estimation of Distributions Using Orthogonal Expansions. Ann. Statist. 2 (1974), no. 3, 454--463. doi:10.1214/aos/1176342706. https://projecteuclid.org/euclid.aos/1176342706


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