Projection estimates for the density of a priori distributions and for functionals of it. (Russian) Zbl 0588.62053

Penskaya, M. Ya.

Teor. Veroyatn. Mat. Stat. 31, 99-110 (1984).

This paper deals with the construction of projection estimates for the density of a priori distributions and for functionals of it. The author used Chensov’s ideas (which are used in other statistical problems) to study the ideal behaviour of the obtained estimators.
The author considered N independent observations \((X_ i,\theta_ i),\quad i=\overline{1,N}\), with known first component \(x_ i\) and unknown \(\theta_ i\) with prior distribution g(\(\theta)\) and conditional distribution P(X\(| \theta)\), to get estimations for the unknown \(g(\theta)\in L^ 2(r)\). Assuming \(\{\phi_ j(\theta)\}\) to be an orthonormal base in F, A a bounded linear operator of F to E, \(g\in F\), \(P\in E\), \(Ag=P\), the unknown density function g(\(\theta)\) can be estimated as a part of the orthonormal series \(g_ n(\theta)=\sum^{n}_{j=0}c_ j\phi_ j(\theta)\). The possibility of constructing unbiased estimates \(C_{JN}\) for the coefficients \(C_ j\) is given by \(C_{jN}=N^{-1}\sum^{N}_{h=1}\Psi_ j(X_ h).\)
The author also gets the mean square error for the preceding estimates. By choosing n, N large enough, he constructs estimators of the unknown density. As examples he studies Laplace, Poisson and exponential distributions.

Reviewer: A.Hosni

MSC:

62G05

Nonparametric estimation

Keywords:

density estimation; projection estimates; density of a priori distributions; functionals; prior distribution; conditional distribution; orthonormal base; bounded linear operator; orthonormal series; unbiased estimates; mean square error; preceding estimates