U-statistic

A statistical measure used in non-parametric hypothesis testing

U-statistics are a class of statistics that are especially useful in non-parametric hypothesis testing. They were introduced by Wassily Hoeffding in 1948 and are used to estimate parameters of a population. U-statistics are defined as the average value of a kernel function applied to all possible subsets of a given size from a sample.

Definition

A U-statistic is defined for a sample of size n and a kernel function h of m arguments, where m \leq n. The U-statistic is given by:

\( U_n = \frac{1}{\binom{n}{m}} \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} h(X_{i_1}, X_{i_2}, \ldots, X_{i_m}) \)

where \( \binom{n}{m} \) is the binomial coefficient, representing the number of ways to choose m elements from n elements.

Properties

U-statistics have several important properties:

• Unbiasedness: U-statistics are unbiased estimators of the parameter they are estimating.
• Asymptotic normality: Under certain conditions, U-statistics are asymptotically normally distributed as the sample size n goes to infinity.
• Minimum variance: Among all unbiased estimators that are symmetric functions of the sample, U-statistics have the minimum variance.

Examples

Some common examples of U-statistics include:

• The sample mean, which is a U-statistic with a kernel function that is simply the identity function.
• The sample variance, which can be expressed as a U-statistic with a kernel function that computes the squared difference between pairs of observations.
• The Mann-Whitney U test, which uses a U-statistic to test for differences between two independent samples.

Applications

U-statistics are widely used in non-parametric statistics, particularly in hypothesis testing and estimation. They are used in:

• Non-parametric tests such as the Wilcoxon signed-rank test and the Kruskal-Wallis test.
• Estimation of parameters in survival analysis and reliability engineering.
• Bootstrap methods for estimating the distribution of a statistic.

Also see

• V-statistic
• Order statistic
• Rank statistic
• Kernel density estimation

References

• Hoeffding, W. (1948). "A Class of Statistics with Asymptotically Normal Distribution." Annals of Mathematical Statistics, 19(3), 293-325.
• Lehmann, E. L. (1999). "Elements of Large-Sample Theory." Springer.