Quantiles

The main results of this chapter come from Rychlik [90]. The bounds for quantiles of general distributions were obtained by Moriguti [58]. Those for symmetric and symmetric unimodal distributions may also be concluded from the Chebyshev and Gauss inequalities, respectively. Vysochanskii and Petunin [102] presented a refinement of the Gauss inequality for unimodal distributions. Further generalizations can be found in Dharmadhikari and Joag-dev [25, Section 1.5]. We also notice that the Markov inequality yields $${F^{ - 1}}(p) \leqslant \frac{{\mu F}}{{1 - p}}$$ for quantiles of nonnegative random variables. Another implication of the Markov inequality is the second moment bound $${F^{ - 1}}(p) \leqslant \frac{{mF}}{{{{(1 - p)}^{1/2}}}}$$.