# Exploring the power curve of the Lepage test

Previously, I already discussed the Cucconi test. In this post, I continue the topic of nonparametric tests and check out the Lepage test.

### The Lepage test

Let $$\mathbf{x} = (x_1, x_2, \ldots, x_n)$$ and $$\mathbf{y} = (y_1, y_2, \ldots, y_m)$$ be random samples. Let $$N=n+m$$. Let $$\mathbf{V} = (V_1, V_2, \ldots, V_N)$$ be a boolean vector of zeros and ones, where $$V_i = 1$$ if the $$i^\textrm{th}$$ order statistic of the pooled sample $$(\mathbf{x}, \mathbf{y})$$ is come from $$\mathbf{x}$$.

With this notation, we can define the Wilcoxon statistic $$T_1$$ and Ansari-Bradley statistic $$T_2$$ as follows:

$T_1 = \sum_{i=1}^N i \cdot V_i,$

$T_2 = \frac{1}{2} n (N + 1) - \sum_{i=1}^N \left| i - \frac{1}{2}(N+1) \right| \cdot V_i.$

Let

$\mu_1 = \frac{1}{2} n (N + 1),$

$\sigma_1^2 = \frac{1}{12} nm (N + 1),$

$\mu_2 = \begin{cases} \frac{1}{4} n(N + 2), & \;\textrm{if}\;N\;\textrm{is even,}\\ \frac{1}{4} n(N + 1)^2/N, & \;\textrm{if}\;N\;\textrm{is odd,} \end{cases}$

$\sigma_2^2 = \begin{cases} nm(N^2-4) / (48(N-1)) & \;\textrm{if}\;N\;\textrm{is even,}\\ nm(N+1)(N^2+3) / (48N^2) & \;\textrm{if}\;N\;\textrm{is odd.} \end{cases}$

The Lepage statistic is the squared Euclidean distance of standardized Wilcoxon and Ansari–Bradley statistics:

$T = \left( \frac{T_1-\mu_1}{\sigma_1} \right)^2 + \left( \frac{T_2-\mu_2}{\sigma_2} \right)^2$

### Power curve

Here is the power curve by effect size for the two-sided Student’s t-test, Mann-Whitney U test, Cucconi test, and Lepage test under normality ($$n=\{ 5, 7 \}$$, $$\alpha = 0.05$$):

As we can see, in this simulation study, the Cucconi test and the Lepage test have approximately the same statistical power.

• [Lepage1971]
Lepage, Yves. “A combination of Wilcoxon’s and Ansari-Bradley’s statistics.” Biometrika 58, no. 1 (1971): 213-217.
DOI: 10.2307/2334333