Efficient estimation in local parametric regression analysis

Abstract

We consider a random vector , where the predictor variable X is d-dimensional and the response variable Y realizes in IR. We assume that for each x in some given subset S of IRd the conditional distribution P(Y ∊.|X = x) of Y, given that X = x, belongs to some k-parametric exponential family Q_v(.|x). The exponential family Q_v(.|x) is for each x controlled by the same but unknown parameter v ∊θ ⊂ IR^m ¶Based on n independent copies of (X, F), we derive asymptotically efficient estimators of the underlying parameter v₀ in case of a single x=x₀ i.e., S={X₀}, as well as in the case, where S has a nonzero probability of occurence P{X ∊ S} > 0. The estimators are defined as solutions of systems of equations, which coincide in the case of local linear models with the normal equations of regression analysis. ¶Our approach uses local asymptotic normality of thinned empirical point processes, where the thinning function comes from a fuzzy set approach in the case of a single value S ={x₀}. This requires results from the theory of rare events. Efficiency is then established in the class of all regular estimates in the Hajek-LeCam sense.

Keywords:

• Conditional distributions
• regression analysis
• fuzzy sets
• efficient estimators
• LAN
• thinned empirical point processes
• exponential families
• regular estimators
• normal equations