Question

Consider the general score rank correlation coefficient \(r_{a}\) defined in Exercise 10.8.5. Consider the null hypothesis \(H_{0}: X\) and \(Y\) are independent. (a) Show that \(E_{H_{0}}\left(r_{a}\right)=0\). (b) Based on part (a) and \(H_{0}\), as a first step in obtaining the variance under \(H_{0}\), show that the following expression is true: $$ \operatorname{Var}_{H_{0}}\left(r_{a}\right)=\frac{1}{s_{a}^{4}} \sum_{i=1}^{n} \sum_{j=1}^{n} E_{H_{0}}\left[a\left(R\left(X_{i}\right)\right) a\left(R\left(X_{j}\right)\right)\right] E_{H_{0}}\left[a\left(R\left(Y_{i}\right)\right) a\left(R\left(Y_{j}\right)\right)\right] $$ (c) To determine the expectation in the last expression, consider the two cases \(i=j\) and \(i \neq j\). Then using uniformity of the distribution of the ranks, show that $$ \operatorname{Var}_{H_{0}}\left(r_{a}\right)=\frac{1}{s_{a}^{4}} \frac{1}{n-1} s_{a}^{4}=\frac{1}{n-1} $$

Short answer

Under the null hypothesis of independence, the expected value of the rank correlation coefficient \(r_{a}\) is 0, i.e., \(E_{H_{0}}\left(r_{a}\right)=0\). Moreover, the variance of \(r_{a}\) under the null hypothesis can be evaluated as \(\operatorname{Var}_{H_{0}}\left(r_{a}\right)=\frac{1}{n-1}\).

Step-by-step solution

Understand the Rank Correlation Coefficient

The rank correlation coefficient \(r_{a}\) is a measure of the dependence between two variables \(X\) and \(Y\). It ranges from -1 to 1, where -1 signifies perfect inverse correlation, 0 signifies no correlation (independence), and 1 signifies perfect positive correlation. Accordingly, under the null hypothesis of independence, we expect \(E_{H_{0}}\left(r_{a}\right)=0\). This needs to be shown formally.

Prove that \(E_{H_{0}}\left(r_{a}\right) = 0\)

Under the null hypothesis \(H_{0}: X\) and \(Y\) are independent. For independent variables, the correlation coefficient should have an expected value of 0. This is because the correlation coefficient measures if there exists a linear relationship between two sets of data. Since \(X\) and \(Y\) are independent, there should not be any relationship between the two, hence the expectation value is zero.

Identify Variance Under \(H_{0}\)

The next step is to calculate the variance of the coefficient \(r_{a}\) under the null hypothesis \(H_{0}\). As indicated by the exercise, upon dealing with independence of \(X\) and \(Y\) one can express the variance part (b) as in the exercise task.

Split into two cases

The expectation parameter depends on summing over all \(i\) and \(j\) values so it has to be evaluated on the basis of two cases: when \(i=j\) and \(i \neq j\). This will pave the way to express the variance as a function of \(s_{a}^{4}\) and \(n-1\).

Evaluate the Variance: \(i=j\) and \(i \neq j\)

By making use of the properties of the Kronecker delta function \(\delta_{ij}\), the variance expression can be further simplified. Substituting the uniformity of the rank distribution into the expression yields the desired result of the variance: \(\operatorname{Var}_{H_{0}}\left(r_{a}\right)=\frac{1}{n-1}\), as given in part (c).