Mathematical Science Statistics & Exploratory Data Analysis Discrete & Continuous Univariate Distributions
Let (X1, Y1), …, (X4, Y4) be a random sample from a continuous bivariate distribution function FX, Y with marginal distributions of X and Y being FX and FY respectively. In order to test the null hypothesis H0: 'X and Y are independent' against the alternative H1: 'X and Y are positively associated', consider the Kendall sample correlation statistic
K = \(\rm\displaystyle\sum_{i = 1}^3 \sum_{j = i+1}^4\) ψ((Xi, Yi), (Xj, Yj))
where
ψ((a, b), (c, d)) = \(\begin{cases}1, & \text { if } \rm (d-b)(c-a)>0 . \\ -1, & \text { if }\rm (d-b)(c-a)<0 .\end{cases}\)
Assuming no ties, which of the following are true?
1
The test that rejects H0 for K ≥ 4 has size 1/4.
2
The test that rejects H0 for K ≥ 4 has size 1/6.
3
PH0(K = 4) = 3/24
4
PH0(K = 6) = 1/12