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A NOTE ON SMOOTHED ESTIMATING FUNCTIONS

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A. THAVANESWARAN^{1} AND JAGBIR SINGH^{2}

^{1} *Department of Statistics, University of Manitoba, Winnipeg, Canada R3T 2N2*

^{2} *Department of Statistics, Temple University, Philadelphia, PA 19122, U.S.A.*
(Received December 6, 1991; revised December 21, 1992)

**Abstract.**
The kernel estimate of regression function in
likelihood based models has been studied in Staniswalis (1989,
*J. Amer. Statist. Assoc.*, **84**, 276-283). The
notion of optimal estimation for the nonparametric kernel estimation
of semimartingale intensity *alpha*(*t*) is proposed. The goal is to
arrive at a nonparametric estimate ^{^}*theta*_{0} of
*theta*_{0} = *alpha*(*t*_{0}) for a fixed point *t*_{0}\in [0, 1]. We consider
the estimator that is a solution of the smoothed optimal estimating
equation *S*_{t0, theta0} = \int^{1}_{0} *w*((*t*_{0}-*s*)/*b*)*dG*^{0}_{s} = 0 where
*G*^{0}_{t} = \int^{t}_{0} *a*^{0}_{s, theta0} *dM*_{s, theta0} is the optimal
estimating function as in Thavaneswaran and Thompson (1986,
*J. Appl. Probab.*, **23**, 409-417).

*Key words and phrases*:
Censored observations,
semimartingales, optimal estimation, smoothing.

**Source**
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