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December, 1991 Asymptotics for Doubly Flexible Logspline Response Models
Charles J. Stone
Ann. Statist. 19(4): 1832-1854 (December, 1991). DOI: 10.1214/aos/1176348373


Consider a $\mathscr{Y}$-valued response variable having a density function $f(\cdot\mid x)$ that depends on an $\mathscr{X}$-valued input variable $x.$ It is assumed that $\mathscr{X}$ and $\mathscr{Y}$ are compact intervals and that $f(\cdot\mid\cdot)$ is continuous and positive on $\mathscr{X} \times \mathscr{Y}.$ Let $F(\cdot\mid x)$ denote the distribution function of $f(\cdot\mid x)$ and let $Q(\cdot\mid x)$ denote its quantile function. A finite-parameter exponential family model based on tensor-product $B$-splines is constructed. Maximum likelihood estimation of the parameters of the model based on independent observations of the response variable at fixed settings of the input variable yields estimates of $f(\cdot \mid \cdot), F(\cdot \mid \cdot)$ and $Q(\cdot \mid \cdot).$ Under mild conditions, if the number of parameters suitably tends to infinity as $n \rightarrow \infty,$ these estimates have optimal rates of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated.


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Charles J. Stone. "Asymptotics for Doubly Flexible Logspline Response Models." Ann. Statist. 19 (4) 1832 - 1854, December, 1991.


Published: December, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0785.62034
MathSciNet: MR1135151
Digital Object Identifier: 10.1214/aos/1176348373

Primary: 62G05
Secondary: 62F12

Keywords: B-splines , exponential families , Input-response model , maximum likelihood , rates of convergence

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • December, 1991
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