Arnab Chakraborty's classical nonparametric statistics notes

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Last updated on Wed May 19 16:12:02 IST 2010.

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U-statistics

Introduction

Many of the statistics that we have dealt with so far in this course are related to a special family called U-statistics. We discuss general properties of this family now.

Definition Let h: R^m → R be some function. Then U-statitistic with kernel h, based on a sample

X₁,...,X_n (where n ≥ m)

is defined as

U = ∑ ^' h(x_i₁,..., x_{i_m})/ ⁿP_m,

where ∑ ^' is over all i₁,...,i_m such that

1 ≤ i₁ ≠ ... ≠ i_m ≤ n

Example: If we take h(x) = x (for m=1) then the corresponding U statistics is the sample mean.

Exercise .1: Suppose that we take m=2, and the kernel
h(x,y) = x² - xy
What is the U statistic? It should be something familiar.

Suppose that X₁,...,X_n are iid F, and θ is some real-valued parameter of interest..

Definition We call θ (unbiasedly) estimable if there is m ≥ 1 and a function h(X₁,...,X_m) such that

E(h(X₁,...,X_m) = θ for all θ.

The smallest value of m for which there exists such an h is called the degree of θ.

Exercise .2: If E(h(X₁,...,X_m) = θ, then show that E(U) = θ, as well. Hence conclude that every estimable parameter θ has an unbiased estimator which is a U-statistic.

Definition A function h:R^m → R is called symmetric if for all permutations p of {1,...,m} we have

h(X₁,...,X_m) = h(X_p(1),...,X_p(m))

Example: h(x,y,z) = xy+xz+yz is a symmetric function. But h(x,y) = x²y is not. h(x,y)=xy is symmetric, but h(x,y,z)=xy is not!

Exercise .3: If E(h(X₁,...,X_m) = θ, then show that there is a symmetric function
g:R^m → R
such that E(g(X₁,...,X_m) = θ.

Exercise .4: Show that for any estimable θ there is a U statistic with symmetric kernel that is an unbiased estimator for θ.

Note that if h is symmetric then the U-statistics based on h is same as

U = ∑ ^' h(x_i₁,..., x_{i_m})/ ⁿC_m,

where ∑ ^' is over all i₁,...,i_m such that

1 ≤ i₁ < ... < i_m ≤ n

Exercise .5: Obtain unbiased symmetric kernel U-statistic estimators for the 3rd central moment.
First try to do it for product of raw moments.

Exercise .6: Show that the sign test statistic is a constant multiple of a U-statistic.

Exercise .7: Show that the Wilcoxon signed rank statistic is a linear combination of two U-statistics. In particular, it has the form
n U₁ + ⁿC₂ U₂.

Two sample U-statistics

This is defined in a way similar to the above one sample case. Here we have a kernel

h:R^r x R^s → R

Based on a two sample dataset

X₁,...,X_m, Y₁,...,Y_n,

(where m ≥ r and n ≥ s) we define the two-sample U-statistic with kernel h as

U = ∑ ^' h(X_i₁,..., X_{i_m}, Y_j₁,...,Y_{j_n})/ ( ^mC_r ⁿC_s)

Exercise .8: Consider the parameter
θ = P(X < Y).
Find a kernel to estimate it unbiasedly. Find out the corresponding U-statistic. Show that it is a multiple of the Mann-Whitney U-test statistic.