## The Annals of Statistics

### Adaptive demixing in Poisson mixture models

Nicolas W. Hengartner

#### 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

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

#### 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