On optimising the estimation of high quantiles of a probability distribution

One of the major aims of one-dimensional extreme-value theory is to estimate quantiles outside the sample or at the boundary of the sample. The underlying idea of any method to do this is to estimate a quantile well inside the sample but near the boundary and then to shift it somehow to the right place. The choice of this “anchor quantile” plays a major role in the accuracy of the method. We present a bootstrap method to achieve the optimal choice of sample fraction in the estimation of either high quantile or endpoint estimation which extends earlier results by Hall and Weissman (1997) in the case of high quantile estimation. We give detailed results for the estimators used by Dekkers et al. (1989). An alternative way of attacking problems like this one is given in a paper by Drees and Kaufmann (1998).

[1]  Jón Dańıelsson,et al.  Tail Index and Quantile Estimation with Very High Frequency Data , 1997 .

[2]  Dennis W. Jansen,et al.  On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective , 1989 .

[3]  Richard L. Smith.  Threshold Methods for Sample Extremes , 1984 .

[4]  U. Stadtmüller,et al.  Generalized regular variation of second order , 1996, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[5]  Laurens de Haan,et al.  Fighting the arch–enemy with mathematics‘ , 1990 .

[6]  I. Weissman.  Estimation of Parameters and Large Quantiles Based on the k Largest Observations , 1978 .

[7]  L. Haan,et al.  A moment estimator for the index of an extreme-value distribution , 1989 .

[8]  H. Joe.  Estimation of quantiles of the maximum of N observations , 1987 .

[9]  Holger Drees,et al.  On Smooth Statistical Tail Functionals , 1998 .

[10]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[11]  Patrick Billingsley,et al.  Weak convergence of measures - applications in probability , 1971, CBMS-NSF regional conference series in applied mathematics.

[12]  Dennis D. Boos,et al.  Using extreme value theory to estimate large percentiles , 1984 .

[13]  L. de Haan,et al.  On the maximal life span of humans. , 1994, Mathematical population studies.

[14]  Peter Hall,et al.  On the estimation of extreme tail probabilities , 1997 .

[15]  H. Bergström,et al.  Weak convergence of measures , 1982 .

[16]  Edgar Kaufmann,et al.  Selecting the optimal sample fraction in univariate extreme value estimation , 1998 .

[17]  L. Haan,et al.  Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation , 2000 .

[18]  J. Einmahl.  The empirical distribution function as a tail estimator. , 1990 .