7.11 Consider a one-sided Kolmogorov–Smirnov test for the null hypothesis (7.9), where F0 is a discrete distribution with the following jumps:

x 1 2 3 4 5 6 F0(x) 0.033 0.600 0.833 0.933 0.961 1.000

[the exercise is based on an example given in Wood and Altavela (1978)].

The alternative is H

−

1 given below (7.9), so the statistic D

−

n of (7.12) is considered.

(i) Show that for any λ > 0, lim n→∞

P(

√

nD

−

n > λ) = 1 − P(Z1 ≤ λ, . . . , Z5 ≤ λ), (7.44)

where (Z1, . . . , Z5) has a multivariate normal distribution with means 0 and covariances given by cov(Zi,Zj ) = F0(i) ∧ F0(j ) − F0(i)F0(j ), 1 ≤ i, j ≤ 5.

(ii) The observed value of

√

nD

−

n in Wood and Altavela (1978) was 1.095.

For each sample size n, where n = 30, 100, and 200, generate 10,000 random vectors (Z1, . . . , Z5)

 as above and evaluate the right side of (7.44) with λ = 1.095 by Monte-Carlo method.