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LIMIT THEOREMS FOR THE MAXIMUM LIKELIHOOD

ESTIMATE UNDER GENERAL MULTIPLY

TYPE II CENSORING

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FANHUI KONG^{ 1} AND HELIANG
FEI^{ 2}

^{1} *Department of Mathematics and Statistics, University of
Maryland Baltimore County,*

Baltimore, MD 21228, U.S.A.

^{2} *Department of Mathematics, Shanghai Normal University,
Shanghai, 200234, China*
(Received February 7, 1995; revised August 7, 1995)

**Abstract.**
Assume *n* items are put on a life-time test,
however for various reasons we have only observed the *r*_{1}-th,
....., *r*_{k}-th failure times *x*_{r1,n}, ....., *x*_{rk,n}
with 0 __<__ *x*_{r1,n} __<__ ··· __<__ *x*_{rk,n} < \infty. This is a
multiply Type II censored sample. A special case where each
*x*_{ri,n} goes to a particular percentile of the population has
been studied by various authors. But for the general situation where
the number of gaps as well as the number of unobserved values in some
gaps goes to \infty, the asymptotic properties of MLE are still not
clear. In this paper, we derive the conditions under which the maximum
likelihood estimate of *theta* is consistent, asymptotically normal
and efficient. As examples, we show that Weibull distribution, Gamma
and Logistic distributions all satisfy these conditions.

*Key words and phrases*:
Maximum likelihood estimation,
multiply Type II censoring, law of large numbers, central limit
theorem, order statistic.

**Source**
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