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.