(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 ^theta0 of theta0 = alpha(t0) for a fixed point t0\in [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation St0, theta0 = \int10 w((t0-s)/b)dG0s = 0 where G0t = \intt0 a0s, theta0 dMs, 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.
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