On estimating mixing densities in discrete exponential family models. (English) Zbl 0841.62027

Zhang, Cun-Hui

Ann. Stat. 23, No. 3, 929-945 (1995).

Summary: This paper concerns estimating a mixing density function \(g\) and its derivatives based on i.i.d. observations from \(f(x)= \int f(x\mid\theta) g(\theta) d\theta\), where \(f(x\mid\theta)\) is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence.
If \(f(x\mid\theta_0) \geq \varepsilon^{x+1}\) \(\forall x\), for some positive numbers \(\theta_0\) and \(\varepsilon\), then our estimators achieve the optimal rate of convergence \((\log n)^{- \alpha+ m}\) for estimating the \(m\) th derivative of \(g\) under a Lipschitz condition of order \(\alpha> m\). The optimal rate of convergence is almost achieved when \((x! )^\beta f(x\mid \theta_0)\geq \varepsilon^{x+ 1}\). Estimation of the mixing distribution function is also considered.

MSC:

62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

Fourier transformation; mixing density function; derivatives; i.i.d. observations; exponential family of density functions; counting measure; kernel estimators; upper bounds; rate of convergence; lower bounds; optimal rate of convergence