ASYMPTOTIC PROPERTIES OF SELF-CONSISTENT ESTIMATORS WITH MIXED INTERVAL-CENSORED DATA

QIQING YU, GEORGE Y. C. WONG AND LINXIONG LI

1Department of Mathematical Sciences, SUNY, Binghamton, NY 13902-6000, U.S.A.

2Strang Cancer Prevention Center, Cornell University Medical College, 428 E 72nd Street, NY 10021, U.S.A.

3Department of Mathematics, University of New Orleans, Lake Front, New Orleans, LA 70148, U.S.A.

(Received September 6, 1999; revised April 24, 2000)

Abstract. Mixed interval-censored (MIC) data consist of n intervals with endpoints L_{i} and R_{i}, i = 1, …, n. At least one of them is a singleton set and one is a finite non-singleton interval. The survival time X_{i} is only known to lie between L_{i} and R_{i}, i = 1, 2, …, n. Peto (1973, Applied Statistics, 22, 86–91) and Turnbull (1976, J. Roy. Statist. Soc. Ser. B, 38, 290–295) obtained, respectively, the generalized MLE (GMLE) and the self-consistent estimator (SCE) of the distribution function of X with MIC data. In this paper, we introduce a model for MIC data and establish strong consistency, asymptotic normality and asymptotic efficiency of the SCE and GMLE with MIC data under this model with mild conditions.

Key words and phrases: Asymptotic normality, generalized maximum likelihood estimator, mixture distribution, strong consistency.