My previous two blog posts revolved around derivation of the limiting distribution of U-statistics for one sample and multiple independent samples.
For derivation of the limiting distribution of a U-statistic for a single sample, check out Getting to know U: the asymptotic distribution of a single U-statistic.
For derivation of the limiting distribution of a U-statistic for multiple independent samples, check out Much Two U About Nothing: Extension of U-statistics to multiple independent samples.
The notation within these derivations can get quite complicated and it may be a bit unclear as to how to actually derive components of the limiting distribution.
In this blog post, I provide two examples of both common one-sample U-statistics (Variance, Kendall’s Tau) and two-sample U-statistics (Difference of two means, Wilcoxon Mann-Whitney rank-sum statistic) and derive their limiting distribution using our previously developed theory.
Asymptotic distribution of U-statistics
One sample
For a single sample, , the U-statistic is given by
where is a symmetric kernel of degree
.
For a review of what it means for to be symmetric, check out U-, V-, and Dupree Statistics.
In the examples covered by this blog post, , so we can re-write
as,
Alternatively, this is equivalent to,
The limiting variance of is given by,
where
or equivalently,
Note that when ,
.
For , these expressions reduce to
where
and
The limiting distribution of for
is then,
For derivation of the limiting distribution of a U-statistic for a single sample, check out Getting to know U: the asymptotic distribution of a single U-statistic.
Two independent samples
For two independent samples denoted and
, the two-sample U-statistic is given by
where is a kernel that is independently symmetric within the two blocks
and
.
In the examples covered by this blog post, , reducing the U-statistic to,
The limiting variance of is given by,
where
and
Equivalently,
and
For , these expressions reduce to
where
and
The limiting distribution of for
and
is then,
For derivation of the limiting distribution of a U-statistic for multiple independent samples, check out Much Two U About Nothing: Extension of U-statistics to multiple independent samples.
Continue reading One, Two, U: Examples of common one- and two-sample U-statistics