Quantile and quantile-function estimations under density ratio model

1. Introduction. Forestry plays a major role in the Canadian economy; maintaining the high quality of wood products is vital economically and socially. We are designing an effective long-term monitoring plan for the quality of forestry products in Canada. Two important quality indices for a piece of lumber are the modulus of elasticity (MOE) and the modulus of rupture (MOR), its strength in terms of elasticity and toughness. The reliability of lumber-based structures may depend heavily on the lower population quantiles of these indices. However, it is costly, time consuming and laborious to obtain these quality measurements. Therefore, efficient estimates of the population quantiles and their functions are important. The estimation of quantiles based on a single random sample is a wellresearched topic. Empirical quantiles have been shown to admit a Bahadur representation [Bahadur (1966); Kiefer (1967); Serfling (1980)], making it simple to study the joint limiting distributions of any number of sample quantiles and their smooth functions. In the presence of auxiliary information, the empirical likelihood [EL; Owen (1988, 2001)] can be utilized to improve efficiency. The Bahadur

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