The Annals of Statistics

Adaptive demixing in Poisson mixture models

Nicolas W. Hengartner

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Abstract

Let $X_1, X_2, \dots, X_n$ be an i.i.d. sample from the Poisson mixture distribution $p(x) = (1/x!) \int_0^{\infty} s^x e^{-s}f(s) ds$. Rates of convergence in mean integrated squared error (MISE) of orthogonal series estimators for the mixing density f supported on $[a, b]$ are studied. For the Hölder class of densities whose rth derivative is Lipschitz $\alpha$, the MISE converges at the rate $(\log n/ \log \log n)^{-2(r +\alpha)}$. For Sobolev classes of densities whose rth derivative is square integrable, the MISE converges at the rate $(\log n/ \log \log n)^{-2r}$. The estimator is adaptive over both these classes.

For the Sobolev class, a lower bound on the minimax rate of convergence is $(\log n/ \log \log n)^{-2r}$, and so the orthogonal polynomial estimator is rate optimal.

Article information

Source
Ann. Statist., Volume 25, Number 3 (1997), 917-928.

Dates
First available in Project Euclid: 20 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1069362730

Digital Object Identifier
doi:10.1214/aos/1069362730

Mathematical Reviews number (MathSciNet)
MR1447733

Zentralblatt MATH identifier
0876.62042

Subjects
Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Keywords
Poisson mixtures demixing optimal rates of convergence orthonormal polynomial estimator adaptive estimation

Citation

Hengartner, Nicolas W. Adaptive demixing in Poisson mixture models. Ann. Statist. 25 (1997), no. 3, 917--928. doi:10.1214/aos/1069362730. https://projecteuclid.org/euclid.aos/1069362730


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