Abstract
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
for quantiles of nonnegative random variables. Another implication of the Markov inequality is the second moment bound
.
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© 2001 Springer-Verlag New York, Inc.
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Rychlik, T. (2001). Quantiles. In: Projecting Statistical Functionals. Lecture Notes in Statistics, vol 160. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2094-7_3
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DOI: https://doi.org/10.1007/978-1-4612-2094-7_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95239-0
Online ISBN: 978-1-4612-2094-7
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