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Efficient estimation in local parametric regression analysis
Ist Teil von
Communications in statistics. Theory and methods, 2000-01, Vol.29 (11), p.2363-2389
Ort / Verlag
Philadelphia, PA: Marcel Dekker, Inc
Erscheinungsjahr
2000
Link zum Volltext
Quelle
Taylor & Francis
Beschreibungen/Notizen
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 ε.|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 εθ ⊂ IR
m
¶Based on n independent copies of (X, F), we derive asymptotically efficient estimators of the underlying parameter v
0
in case of a single x=x
0
i.e., S={X
0
}, as well as in the case, where S has a nonzero probability of occurence P{X ε 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 ={x
0
}. 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.