Arnab Chakraborty's classical nonparametric statistics notes

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Last updated on Fri May 21 11:52:17 IST 2010.

We have same set up as before. We want to test

H₀: F = G Vs H₁: F ≠ G

H₀: F = G Vs H₁: F G

Let F_m and G_n denote the empirical distributions of the X's and the Y's , respectively. Define

D = sup_x | F_m(x)-G_n(x) |
D₊ = sup_x (F_m(x)-G_n(x))
D = sup_x (G_n(x)-F_m(x))

Exercise 8.1: Show that
D₊ = max{0, max_i(i/m - G_n(X_(i)))}
Here X_(i) is the i-th order statistic of the X's.

Exercise 8.2: State and prove a similar result for D_-

Exercise 8.3: Show that
D = max{ D_-, D₊}

Exercise 8.4: We want want to reject H₀ for large values of the test statistic. Which test statistic should we use for which H₁?

The actual distributions of D, D_-and D₊ are complicated. Here are the asymptotic distributions under H₀.

Theorem As min(m,n) → ∞,

P(sqrt(mn/(m+n)) D ≤ x) → H(x), under H₀,

where

H(x) = 1- 2 ∑ _j (-1)^j-1 exp(-2j² x²) if x ≥ 0

0 else

Theorem As min(m,n) → ∞,

P(sqrt(mn/(m+n)) D₊ ≤ x) → H(x), under H₀,

where

H(x) = 1- exp(-2x²) if x ≥ 0

0 else

The very same limit works for D_- as well.