Adaptive demixing in Poisson mixture models. (English) Zbl 0876.62042

Hengartner, Nicolas W.

Ann. Stat. 25, No. 3, 917-928 (1997).

Summary: Let \(X_1,X_2, \dots, X_n\) be an i.i.d. sample from the Poisson mixture distribution \[ p(x)= (1/x!) \int^\infty_0 s^xe^{-s} f(s)ds. \] 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 \(r\)th derivative is Lipschitz \(\alpha\), the MISE converges at the rate \((\log n/ \log\log n)^{-2 (r+\alpha)}\). For Sobolev classes of densities whose \(r\)th 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.

MSC:

62G20	Asymptotic properties of nonparametric inference
62G07	Density estimation

Keywords:

Poisson mixtures; demixing; optimal rates of convergence; orthonormal polynomial estimator; adaptive estimation; Hölder class; Sobolev classes; MISE