A NOTE ON SMOOTHED ESTIMATING FUNCTIONS

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

1Department of Statistics, University of Manitoba, Winnipeg, Canada R3T 2N2 2Department 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 α(t) is proposed. The goal is to arrive at a nonparametric estimate θ̂_{0} of θ_{0} = α(t_{0}) for a fixed point t_{0} ∈ [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation S_{t_{0},θ_{0}} = ∫_{0}^{1} w((t_{0} −s)/b)dG_{s}^{0} = 0 where G_{t}^{0} = ∫_{0}^{t} a_{s,θ_{0}}^{0}dM_{s,θ_{0}} 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.