Estimating probabilities for normal extremes

作者：Peter A. Hall · 发表于：Advances in Applied Probability · 年份：1980 · DOI：10.2307/1426608 · 被引用次数：41 · 研究领域：Credit Risk and Financial Regulations、Financial Risk and Volatility Modeling

Let Xnn denote the largest of n independent N(0, 1) variables. Several methods of estimating P(Xnn ≦ x) are considered. It is shown that X2nn, when normalized in an optimal way, converges to the extreme value distribution at a rate of only 1/(log n)2, and that if 0 < t ≠ 2 then |Xnn|t converges at a rate of 1/log n. Therefore it is not feasible to use the extreme value distribution to estimate probabilities for normal extremes unless the sample size is extremely large. An alternative approach is presented, which gives very good estimates of P(Xnn ≦ x) for n ≧ 10. The case of rth extremes is also considered.