Mathematical Science Statistics & Exploratory Data Analysis Discrete & Continuous Univariate Distributions
Let\( (A_1, B_1), …, (A_4, B_4) \)be a random sample from a continuous bivariate distribution function \(F_{A, B} \)with marginal distributions of A and B being \(F_A \)and\( F_B\) respectively. In order to test the null hypothesis \(H_0\)\(H_0\): 'A and B are independent' against the alternative \(H_1\): 'A and B are negatively associated', consider the Kendall sample correlation statistic
K = \(\rm\displaystyle\sum_{i = 1}^3 \sum_{j = i+1}^4\) ϕ((Ai, Bi), (Aj, Bj))
where ϕ((e, f), (g, h)) = \(\begin{cases}1, & \text { if } \rm(h−f)(g−e)>0. \\ -1, & \text { if }\rm (h−f)(g−e)<0.\end{cases}\)
then which of the following are true?
1
If K>0, reject the null hypothesis, H0
2
If K<0, do not reject the null hypothesis, H0
3
If K>0, do not reject the null hypothesis, H0
4
If K<0, reject the null hypothesis, H0