  This paper focuses on drawing statistical inference based on a novel variant
of maxima or minima nomination sampling (NS) designs. These sampling designs
are useful for obtaining more representative sample units from the tails of the
population distribution using the available auxiliary ranking information.
However, one common difficulty in performing NS in practice is that the
researcher cannot obtain a nominated sample unless he/she uniquely determines
the sample unit with the highest or the lowest rank in each set. To overcome
this problem, a variant of NS which is called partial nomination sampling is
proposed in which the researcher is allowed to declare that two or more units
are tied in the ranks whenever he/she cannot find with high confidence the
sample unit with the highest or the lowest rank with high confidence. Based on
this sampling design, two asymptotically unbiased estimators are developed for
the cumulative distribution function, which are obtained using maximum
likelihood and moment-based approaches, and their asymptotic normality is
proved. Several numerical studies have shown that the developed estimators have
higher relative efficiencies than their counterpart in simple random sampling
in analyzing either the upper or the lower tail of the parent distribution. The
procedures that we are developed are then implemented on a real dataset from
the Third National Health and Nutrition Examination Survey (NHANES III) to
estimate the prevalence of osteoporosis among adult women aged 50 and over. It
is shown that in some certain circumstances, the techniques that we have
developed require only one-third of the sample size needed in SRS to achieve
the desired precision. This results in a considerable reduction of time and
cost compared to the standard SRS method.
