Math Problems Vault

Problem 2.13.1

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).

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Problem 2.13.2

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.

Donuts	9	1	3	7	3	5	1	0	60	63	2	9	0	0
Grade	4	9	8	7	3	8	7	5	8	3	6	5	7	8

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).

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Problem 2.13.3

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.

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Problem 2.13.4

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

\begin{align} \rho _{S}=1-\frac{12}{n^{3}-n} \sum _{i<j}(j-i) \; \mathbb {1}_{T_{i}>T_{j}}, \\ \tau =1-\frac{4}{n^{2}-n} \sum _{i<j} \; \mathbb {1}_{T_{i}>T_{j}}. \end{align}

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Problem 2.13.5

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:

\hat{r}=\frac{\sum _{i<j} c_{i j}(X) c_{i j}(Y)}{\sqrt{\sum _{i<j} c_{i j}(X)^{2} \sum _{i<j} c_{i j}(Y)^{2}}}

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 $\mathbb {E}\left[\hat{r}\right]=0$

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Problem 2.13.6

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

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Problem 2.13.7

The Fechner sign correlation coefficient is defined by the formula

I = \frac{C-H}{C + H},

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.

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Problem 2.13.8

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

1.05,1.12,1.37,1.50,1.51,1.73,1.85,1.98,2.03,2.17

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).

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