We consider quantile estimation under a two-sample semi-parametric model in
which the log ratio of two unknown density functions has a known parametri
c form. This two-sample semi-parametric model, arising naturally from case-
control studies and logistic discriminant analysis, can be regarded as a bi
ased sampling model. A new quantile estimator is constructed on the basis o
f the maximum semi-parametric likelihood estimator of the underlying distri
bution function. It is shown that the proposed quantile estimator is asympt
otically normally distributed with smaller asymptotic variance than that of
the standard quantile estimator. Also presented are some results on simula
tion and from the analysis of a real data set.