AISM 54, 641-658

© 2002 ISM

## Modified maximum likelihood estimators based on ranked set samples

### Gang Zheng^{1} and Mohammad F. Al-Saleh^{2}

^{1}Office of Biostatistics Research, National Heart,
Lung, and Blood Institute, II Rockledge Centre, 6701 Rockledge Drive,
Bethesda, MD 20892-7938, U.S.A., e-mail: zhengg@nhlbi.nih.gov

^{2}Department of Mathematics and Statistics, Sultan
Qaboos University, P.O. Box 36 Al-Khodh, Postal Code 123, Sultanate of Oman, e-mail: malsaleh@squ.edu.om

(Received September 13, 2000; revised March 26, 2001)

Abstract.
The maximum likelihood estimator (MLE) using a ranked set sample
(RSS) usually has no closed expression because the maximum likelihood equation
involves both hazard and inverse hazard functions, and may no longer be
efficient when the judgment ranking is imperfect. In this paper, we consider a
modified MLE (MMLE) using RSS for general parameters, which has the same
expression as the MLE using a simple random sample (SRS), except that the
SRS in
the MLE is replaced by the RSS. The results show that, for the location
parameter, the MMLE is always more efficient than the MLE using SRS, and for
the scale parameter, the MMLE is at least as efficient as the MLE using SRS,
when the same sample size is used. Under the perfect judgment ranking,
numerical examples also show that the MMLE has good efficiency relative to the
MLE based on RSS. When the judgment error is present, we conduct simulations to
show that the MMLE is more robust than the MLE using RSS.

Key words and phrases:
Asymptotic relative efficiency, estimating equation, judgment error, modified maximum likelihood equation, order statistics, ranked set sampling, robustness.

**Source**
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