Generalized Bootstrap Confidence Intervals for High Quantiles

Authored by: Bin Wang , Satya N. Mishra , Madhuri S. Mulekar , Nutan S. Mishra , Kun Huang

Handbook of Fitting Statistical Distributions with R

Print publication date:  October  2010
Online publication date:  April  2016

Print ISBN: 9781584887119
eBook ISBN: 9781584887126
Adobe ISBN:

10.1201/b10159-28

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Abstract

Let X = {X 1, X 2, …, Xn } be a random sample from a probability distribution F(x). For any 0 < p < 1, let ?p be the p-th quantile of level p of F(x), which is any root of the equation F(?p )= p. When the quantile level p is high, ?p is called a high quantile. High quantile estimation has many practical applications. In real life, it is often necessary to evaluate the risk of large but possibly rare losses such as earthquakes and hurricanes and the safety of nuclear power stations. In foodborne disease control, risk assessors are usually more interested in high quantiles of contaminant exposure distributions rather than the averages of the distributions (Albert and Gauchi (2002)); accurate estimation of the far right tail of the future lifetime distribution of the annuitants is crucial in setting adequate insurance premiums in life annuities (Einmahl, Fils-Villetard, and Guillou (2008)). In a value-at-risk context, precisely predicting the probability of an extreme movement in the value of a portfolio is essential for both risk management and regulatory purpose (Gençay, Selçuk, and Ulugülya?ci (2003)). Applications of high quantile estimation include customer wallet estimation (Perlich, Rosset, Lawrence, and Zadrozny (2007)), network performance evaluation (Choi, Moon, Cruz, Zhang, and Diot (2006)), and many others. For patients with life-threatening diseases, an estimate of the mean or median lifetime of patients with the disease might be too optimistic for patients with very bad conditions and too discouraging for otherwise very healthy patients. People may be more interested in the answer to a question such as, “How long can the top 10% of very healthy men/women survive AIDS?”

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