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June 1997 Adaptive demixing in Poisson mixture models
Nicolas W. Hengartner
Ann. Statist. 25(3): 917-928 (June 1997). DOI: 10.1214/aos/1069362730


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.


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Nicolas W. Hengartner. "Adaptive demixing in Poisson mixture models." Ann. Statist. 25 (3) 917 - 928, June 1997.


Published: June 1997
First available in Project Euclid: 20 November 2003

zbMATH: 0876.62042
MathSciNet: MR1447733
Digital Object Identifier: 10.1214/aos/1069362730

Primary: 62G07
Secondary: 62G20

Keywords: adaptive estimation , demixing , Optimal rates of convergence , orthonormal polynomial estimator , Poisson mixtures

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • June 1997
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