The term “U statistic” can have several meanings:

- Unbiased (U) statistics.
- Mann Whitney U Statistic.
- U statistic in L-estimators.
- Theil’s U.

## 1. Unbiased (U) statistics

Unbiased “U” statistics are a way to construct unbiased estimators to study large sample behavior; the “U” in *U-statistics* stands for “unbiased”.

Constructing U-statistics involves use of a *kernel*— a symmetric real valued function h(x,y). An (Unbiased) U statistic is any statistic that can be placed into the following form:

Where:

- (n 2) is the binomial coefficient,
- {u
_{t}} are independent and identically distributed variables. - Σ = summation notation (“add up”).

Although they look complicated, a U statistic is really just a type of generalized average (Abraham et. al, 2013).

## 2. Mann Whitney U Statistic

The result of performing a Mann Whitney U Test is a U Statistic. For details on that particular statistic, see: Mann Whitney U Test.

## 3. L-Estimators

In L-estimation, Kaigh (1983) proposed a type of non-parametric U statistic as a quantile estimator.

## 4. Theil’s U Statistic

Theil proposed **two** U statistics, used in finance. The first (U_{1}) is a measure of forecast *accuracy *(Theil, 1958, pp 31-42); The second (U_{2}) is a measure of forecast *quality *(Theil, 1966, chapter 2).

Wharton University’s J. Scott Armstrong gleaned the following formulae from **Theil’s original works;** the same formula for U_{2} appeared in Brown and Rozeff (1978):

**Where:**

- A = change in actual earnings,
- P = predicted change in earnings.

**If you perform a Google search for Theil’s U, you’ll find an interesting assortment of different formulae, each purporting to be “Theil’s U”. **According to Armstrong, the existence of multiple formulas “…has caused some confusion.” For example, the occasional author refers to the Uncertainty Coefficient (unrelated to the U statistic) as Theil’s U, because Theil originally worked on its derivation. The end result is that if you need to use Theil’s U for some reason, the best idea is to refer to Theil’s original formulae, shown above. That said, if you’re reading about Theil’s U in the literature, you may want to make sure that the author is using the correct formula.

## References

Abraham, N. et. al (2013). Measures of Complexity and Chaos. Springer Science & Business Media.

Armstrong, J. (No Title). Retrieved December 8, 2017 from: http://armstrong.wharton.upenn.edu/dictionary/definitions/theil%27s%20u.html

Brown, L. & Rozeff, M. (1978). The superiority of analyst forecasts as measures of expectations: Evidence from earnings. In The Journal of Finance, Vol. XXXIII, March. No. 1.

Kaigh, W.D. (1983), “Quantile Interval Estimation,” Communications in Statistics, Part A – Theory and Methods, 12,2427-2443.

Theil, H. (1958), Economic Forecasts and Policy. Amsterdam: North Holland.

Thiel, H. (1966), Applied Economic Forecasting. Chicago: Rand McNally.