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