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ESTIMATION OF A SMOOTH QUANTILE FUNCTION

UNDER THE PROPORTIONAL HAZARDS MODEL

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J. K. GHORAI

*Department of Mathematical Sciences, The University of Wisconsin-Milwaukee,*

P.O. Box 413, Milwaukee, WI 53201, U.S.A.
(Received April 16, 1990; revised September 20, 1990)

**Abstract.**
The problem of estimating a smooth
quantile function, *Q*(·), at a fixed point *p*,
0 < *p* < 1, is treated under a nonparametric smoothness
condition on *Q*. The asymptotic relative deficiency of the
sample quantile based on the maximum likelihood estimate of
the survival function under the proportional hazards model
with respect to kernel type estimators of the quantile is
evaluated. The comparison is based on the mean square
errors of the estimators. It is shown that the relative
deficiency tends to infinity as the sample size, *n*, tends
to infinity.

*Key words and phrases*:
Relative deficiency,
mean square error, kernel type estimators, quantile
function, right censored data, proportional hazards model.

**Source**
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