Rank correlation

Athletes whose ranks when lined up by height were $1,2, \ldots , 13$ showed the following long-jump results (in m): 6,94; 7,12; 7,01; 6,98; 7,24; 7,42; 7,13; 6,95; 7,34; 7,82; 7,23; 7,05; 7,15. Is there a dependence between an athlete's height and the length of his jump? Take the significance level to be 0,05.

Prove equalities (9.8).

Prove that the rank correlation coefficient $\widehat{\rho }_{X Y}(n)$ defined by formula (9.12) can be transformed to the form (9.13).

Prove that $\mathbb {E}\left[\hat{\tau }_{X Y}\right] = \tau_{X Y}$.

There are 12 disks of the same size whose coloring differs in tone — from light blue to dark blue. To assess a fashion designer's ability to distinguish color shades, he is asked to arrange the disks in order of increasing color intensity. The designer established the following order of the disks: $1,4,7,2,3,5$, $8,12,10,6,11,9$. The objective assessment of color intensity, obtained by a colorimetric test: $1,2, \ldots , 12$. Based on these data, characterize the designer's ability to distinguish color shades.

A panel of six judges evaluates the performances of five figure skaters who reached the final of a competition. The results are given in Table 9.14.

Table 9.14

Compute Kendall's coefficient of concordance. Can one assume, at the significance level 0,05, that this panel of judges has a common system of preferences?

GDP growth rates of Russia and Ukraine over 2006 - 2012, in percent (total and per capita, relative to 2005):

Test the hypothesis that these samples are independent at significance level $0.05$ (choose the most convenient and applicable test in this case).

A study was conducted at the institute to detect a relationship between a student's academic performance and the average number of doughnuts they eat per day.

Test the hypothesis that academic performance and the number of doughnuts eaten are independent at significance level $0.05$ (choose the most convenient and applicable test in this case).

Prove that, assuming the hypothesis of independence of the samples holds, $\mathrm{D} \rho_{S}=\frac{1}{n-1}$, where $\rho_{S}$ is the Spearman correlation coefficient.

Using the statistics $\left\{ T_{i}\right\}$ introduced in problem $13.2$, prove that

Given samples $X=\left(X_{1}, \ldots , X_{n}\right), Y=\left(Y_{1}, \ldots , Y_{n}\right)$. Consider an arbitrary function $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$. For any $1 \leq i<j \leq n$ set $c_{i j}(X)=f\left(X_{i}, X_{j}\right)$. The generalized correlation coefficient is the following statistic:

Prove that this is indeed a sample correlation coefficient, i.e., that $\left|\hat{r}\right| \leq 1$ and the values $\pm 1$ are attained, and also that under the hypothesis of independence of the samples $\mathbb {E}\left[\hat{r}\right]=0$

Find the Pearson, Spearman, and Kendall correlation coefficients using the generalized correlation coefficient.

The Fechner sign correlation coefficient is defined by the formula

where $C$ is the number of pairs for which the signs of the deviations of the values from their means agree, and $H$ is the number of pairs for which the signs of the deviations do not agree. Find the distribution of $I$ under the hypothesis of independence of the samples, if the samples are drawn from the standard normal distribution.

Can we, at significance level $0.05$, consider that the sequence of numbers

is a realization of a random vector all of whose 10 components are independent identically distributed random variables? Base your conclusion on correlation coefficients (choose the most convenient and applicable test in this case).