MAST20005/MAST90058: Week 10 Lab
Goals: (i) Properties of order statistics; (ii) Confidence intervals for quantiles; (iii) An intro-
duction to the bootstrap and an example of its use.
1 Order statistics
Let X(1) < X(2) < X(3) be order statistics of a random sample X1, X2, X3 from the the uniform distribution Unif(✓ 0.5, ✓ + 0.5). Three possible estimators for the median are the sample mean W1 = X ̄, the sample median W2 = X(2), and the midrange W3 = (X(1) + X(3))/2.
1. The theoretical pdf for the smallest order statistic, X(1), is g1(x)=3(1 F(x))2f(x)=3(1 x)2, ✓ 0.5
We discussed this in the lectures early on. Two estimators we proposed were T1 = X ̄ 1 andT2 =X(1) n1. Using✓=3andasamplesizeofn=10,usesimulationstoshow that both of these are unbiased and that T2 has clearly smaller variance.
3. Consider the scenario in Section 1.
(a) Consider the estimator W4 = X(3) 0.5. Use simulations to show that it is biased. (b) Determine a value of a that makes W5 = X(3) a an unbiased estimator.
(c) Use simulations to compare the variance of W5 to that of W1, W2 and W3.
4. Calculate a 95% confidence interval for the simulated coverage estimate in Section 2.
Repeat for 1000 simulations.
5. Do question 5 from the tutorial problems. Also, find an approximate 95% confidence interval for the first quartile.
6. Consider the following random sample on X: 0.252, 0.287, 0.537, 0.511, 0.054,
0.022, 0.142, 0.021, 0.155, 0.241
Calculate the statistic T = 1/X ̄. Suppose this is an estimator for some underlying parameter ✓. Calculate a 95% confidence interval for ✓ using the percentile bootstrap.
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