Arnab Chakraborty's classical nonparametric statistics notes

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Last updated on Fri May 21 11:52:18 IST 2010.

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Two sample inference

Mann Whitney U-test

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

Define

D_ij = I{X_i < Y_j}, for all i,j.

Then Mann Whitney U-test uses the test statistic

U = ∑ _i ∑ _j D_ij

Exercise 9.1: Consider the three alternative hypothesies listed above. When should we reject H₀ in favour of each of them, for large or small U?

Let

θ = E(D_ij)
θ₁ = E(D_ijD_ik)
θ₂ = E(D_ijD_kj)

Exercise 9.2: Compute θ under H₀.

Exercise 9.3: Show that if H₀ is true then
θ₁ = θ₂ = 1/3.

Exercise 9.4: Compute E(U) in general, as well as under H₀.

Exercise 9.5: Show that

cov(D_ijD_kl) = 0 if i ≠ k and j ≠ l

θ-θ² if i=k and j=l

θ₁-θ² if i=k and j ≠ l

θ₂-θ² if i ≠ k and j=l

Exercise 9.6: Use the last exercise to show that
Var(U) = mn(θ-θ²) +mn(m-1)(θ₂-θ²) +mn(n-1)(θ₁-θ²),
in general. Check that this reduces to
mn(m+n+1)/12
under H₀.

Exercise 9.7: Argue that U is distribution-free under H₀ by showing that
P(U = u) = r_mn(u)/ ^m+nC_m
under H₀, where
r_mn(u) = number of arrangements of m X's and n Y's such that the number of times an X precedes an Y is u.

Exercise 9.8: Show that
r_mn(u) = r_mn(mn-u)
Hnece conclude that undr H₀, U is distributed symmetrically around mn/2.

Theorem For large m,n,

(U-E(U))/sqrt(Var(U)) ~ AN(0,1).

Proof: Shall do later using a theorem for U-statistics.

An estimation problem

We can use the above idea to get a confidence interval for a two sample location problem. We have the same set up as above. But now we also assume that G is a location shift of F, i.e.,

G(x) = F(x- θ) for all x,

where θ is unknown.

Exercise 9.9: Which of the following two statements is true under this model?
Y₁-θ, ..., Y_n-θ are iid F
or
Y₁+θ, ..., Y_n+θ are iid F ?

Define

U(θ) = ∑ _i ∑ _j I{Zij > 0)

Exercise 9.10: Show that U(θ) has distribution free of θ. In particular, show that its distribution is same as the null distribution of the Mann-Whitney U statistic under H₀.

Suppose that we are given &alpha, and we want to get a (1-&alpha)-level CI for θ. Since U(θ) has a completely known distribution, hence we can find integer c such that

P(c ≤ U(θ) ≤ mn-c) = 1-&alpha.

Due to the discrete nature of U(θ), the equality may not hold exactly. Next, order the mn differences, Z_ij's, as

W₁ < ... < W_mn

The inequalities are strict with probability 1, because of the continuity assumption on F.

Exercise 9.11: Show that
{c ≤ U(θ) ≤ mn-c} iff {W_c < θ < W_mn-c+1}
Hnece argue that (W_c, W_mn-c+1) is a (1-&alpha)-level CI for θ.
See the hint for ONE.7.

Hodges-Lehmann approach

Recall that we had described the HL approach for one sample problem as a method to obtain point estimates from a test. We can apply this aproach to Mann-Whitney U-test to get a two sample HL estimate for θ as follows.

Notice that U(θ) is distributed symmetrically around mn/2. The HL approach suggests choosing

in a way so that U(

) is as close to mn/2 as possible.

Exercise 9.12: Show that
= median of the Z_ij's.