Quantile regression with nominated samples: An application to a bone mineral density study

This paper studies quantile regression analysis with maxima or minima nomination sampling designs. These designs are often used to obtain more representative samples from the tails of the underlying distribution using the easy to access rank information during the sampling process. We propose new loss functions to incorporate the rank information of nominated samples in the estimation process. Also, we provide an alternative approach that translates estimation problems with nominated samples to corresponding problems under simple random sampling (SRS). Strategies are given to choose proper nomination sampling designs for a given population quantile. Numerical studies show that quantile regression models with maxima (or minima) nominated samples have higher relative efficiencies compared with their counterparts under SRS for analyzing the upper (or lower) tail quantiles of the distribution of the response variable. Results are then implemented on a large cohort study in the Canadian province of Manitoba to analyze quantiles of bone mineral density using available covariates. We show that in some cases, methods based on nomination sampling designs require about one-tenth of the sample used in SRS to estimate the lower or upper tail conditional quantiles with comparable mean squared errors. This is a dramatic reduction in time and cost compared with the usual SRS approach.