Recovery of quantile and quantile density function using the frequency moments

The problem of recovering quantiles and quantile density functions of a positive random variable via the values of frequency moments is studied. The uniform upper bounds of the proposed approximations are derived. Several simple examples and corresponding plots illustrate the behavior of the recovered approximations. Some applications of the constructions are discussed as well. Namely, using the empirical counterparts of the constructions yield the estimates of the quantiles, and the quantile density functions. By means of simulations, the average errors in terms of L2-norm are evaluated to justify the consistency of the estimate of the quantile density function. As an application of the constructions, the question of estimating the so-called expected shortfall measure in risk models is also studied.