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

   [s1]

   
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.