Arnab Chakraborty's classical nonparametric statistics notes

cwave.eu5.org
Also see: http://www.angelfire.com/dragon/letstry
cwave04 at yahoo dot com

My Guestbook

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

Home >

... Statistics >

... ... Classical nonparametrics

Two sample inference

Rank statistics

So far we have seen a number of two sample tests. These as well as many others all use test statistics of a general form called linear rank statistics. We discuss this below.

Our set up is as before:

X₁,...,X_m iid F
Y₁,...,Y_n iid G
X's and Y's are mutually independent
F, G both continuous and unknown.

Pool the samples together as

X₁,...,X_m,Y₁,...,Y_n

Let the rank vector of the pooled data be

R₁,...,R_m,R_m+1,..., R_m+n

Thus,

R_i = rank of X_i in the pooled data for 1 ≤ i ≤ m

and

R_m+j = rank of Y_j in the pooled data for 1 ≤ j ≤ n.

Example: In a controlled experiment to measure the effectiveness of a treatment suppose that the control group yields the following measurements:
X₁ = 2, X₂ = 4, X₃ = 9,
and the treatment group produces
Y₁ = 5, Y₂ = 8.
Then here are the pooled data and the rank vector:

pooled data: 2 4 9 5 8 rank vector: 1 2 5 3 4

Definition A statistic that is a function of the rank vector is called a rank statistic.

Definition Let A be any known (m+n) by (m+n) matrix with (i,j)-th element denoted by a(i,j). The

∑ a(i,R_i)

is called a linear rank statistic.

How to check that a given rank statistic is indeed a linear rank statistic? The following theorem helps you to do this.

Theorem Suppose that T(R) is a rank statistics, where R denotes the rank vector, ie, permutations of 1,...,N, say. Let S be any rank vector, and i,j be in {1,...,N}. Define S(i,j) as the permutation obtained from S by swapping i and j. Then T is a linear rank statistic iff

T(S)-T(S(i,j)) is free of S_k for all k ≠ i,j.

Proof: Direct argument.

Exercise 10.1: Is the Mann-Whitney U-statistic a rank statistic? Is it linear? If so, find A.
[Hint: Explicitly write down the U-statistic in terms of the ranks.]

Exercise 10.2: Do the same for the 2-stample KS test statistic, D₊
Use the theorem above.

Exercise 10.3: Do the same for the Wald-Wolfowitz run test.
Use the theorem above.

Definition A statistic of the form

∑ c_i a(R_i)

is called a simple, linear rank statistic, where c_i's are any constants and a(.) is any function.

Exercise 10.4: To justify the terminology used we need to show that a simple linear rank statistic is indeed a linear rank statistic as we have defined earlier. In particular, show that a simple linear rank statistic corresponds to a A matrix of the form
A = u v'
where u and v are column vectors.

Computing moments of simple linear rank statistics

Let

S = ∑ c_i a(R_i)

We shall compute E(S) and Var(S) under H₀: F = G.

Exercise 10.5: Show that under H₀,
each R_i ~ discrete uniform {1,...,m+n}
and
(R₁,...,R_m+n)) ~ discrete uniform over all (m+n)! possible permutations of {1,...,m+n}

Exercise 10.6: Use the last exercise to show that
E(S) = (m+n)cbar * abar
where cbar and abar of averages of c_i's and a(i)'s respectively.

Exercise 10.7: Show that
cov(a(R_i),a(R_j))= -σ²_a/(m+n)

Exercise 10.8: Show that
Var(S) = (m+n-1) σ²_a σ²_c,
where σ²_c is the variance of the c_i's. The denominator in σ²_c is (m+n-1). Similarly for σ²_a.