Exercise Solution 11.1

  1. The largest and second largest realizations in Exhibit 11.1 are 21.9963 and 18.7076. Their mean, 20.35, is our estimate for the .90-quantile.
  2. The sample mean and standard deviation, calculated with sample mean estimator [4.4] and the square root of sample variance estimator [4.5], are 13.5853 and 4.1802. Treating the underlying distribution as normal, we note that the .90-quantile of a normal distribution occurs 1.282 standard deviations above its mean, and estimate the .90-quantile as


  3. With the actual quantile equalling 18.55, we see that the second approach yielded the more accurate estimate. If you repeat this experiment multiple times, each time generating your own χ2(12,0) pseudorandom variates, you will find this, quite consistently, to be the case.