AISM 53, 469-486

© 2001 ISM

## Asymptotic properties of self-consistent estimators with mixed
interval-censored data

### Qiqing Yu^{1}, George Y.C. Wong^{2} and Linxiong Li^{3}

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

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

^{3}Department 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 end-points $L_i$ and $R_i$, $i=1, \ldots, n$.
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 $R_i$, $i=1,2,\ldots,n$.
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.

**Source**
(TeX , DVI )