Statistical Inference from Partially Nominated Sets: An Application to Estimating the Prevalence of Osteoporosis

Abstract: 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.