Efficient estimation of linear functionals of a probability measure \(P\) with known marginal distributions. (English) Zbl 0742.62034

Bickel, Peter J.; Ritov, Ya’acov; Wellner, Jon A.

Ann. Stat. 19, No. 3, 1316-1346 (1991).

The authors consider the semiparametric model of the estimation of functionals \[ \theta(h)=\int h dP, \] where \(P\) is a distribution of random variables \((X,Y)\) on a product space \(X\times Y\) with known marginal distributions \(P_ X\) and \(P_ Y\), and \(h\) is a fixed function from \(X\times Y\) in \(R\). The proposed estimator \(\theta_ n\) is based on partitions of both \(X\) and \(Y\) and modified minimum chi-square estimates. The influence function of the estimator is characterized by the so-called ACE equations arising from the alternating conditional expectations algorithm for calculating projections on a certain subspace of \(L_ 2(P)\) with sum space structure. The subspaces are nonorthogonal and so explicit formulas of the projections are not available. But showing that the influence function lies in the tangent space of the model the asymptotic efficiency of \(\theta_ n\) is stated.

Reviewer: H.Liero (Berlin)

MSC:

62G05	Nonparametric estimation
60F05	Central limit and other weak theorems
62G30	Order statistics; empirical distribution functions
60G44	Martingales with continuous parameter

Keywords:

modified minimum chi square; alternating projections; asymptotic normality; semiparametric model; estimation of functionals; known marginal distributions; modified minimum chi-square estimates; influence function; ACE equations; alternating conditional expectations algorithm; tangent space; asymptotic efficiency