3. You are studying the relationship between two variables, $X_1$ and $X_2$, in your data set.
The data set contains 655 pairs of observations, with the following summary statistics:
Variable Min. Q1 Median Mean Q3 Max.
First 0.5498 0.8894 0.9788 0.9722 1.0555 1.4318
Second 2.632 3.219 3.335 3.327 3.432 4.153
To explore the dependence structure, your assistant plotted the following graph:
(a) Describe clearly the steps your assistant took in plotting the above graph, and suggest two copula families that may be used to model this data set. Justify your suggestion.
[6 marks]
The data set is then transformed to have Uniform(0, 1) margins. An Archimedean copula with a single parameter $\alpha > 0$ and generator function
$\psi(t) = -\ln[1-(1-t)^\alpha]$ (1)
is fitted to the data set, giving a fitted parameter value $\hat{\alpha} = 1.68$.
(b) Show that $\psi^{-1}(t) = 1 - (1-e^{-t})^{1/\alpha}$.
[2 marks]
(c) Let $(U, V)$ be a random pair with Uniform(0, 1) margins and the dependence structure described by this Archimedean copula with the fitted parameter value. Calculate $P(U > 0.9, V > 0.9)$.
[5 marks]
(d) Using the formulas
$\lambda_L = 2 \lim_{t \to \infty} \frac{\psi^{-1}(2t)}{\psi^{-1}(t)}; \lambda_U = 2 - 2 \lim_{t \to 0^+} \frac{\psi^{-1}(2t)}{\psi^{-1}(t)}$
calculate the fitted upper and lower tail dependence coefficients.
[6 marks]
(e) The full data set contains three more variables, $X_3, X_4$ and $X_5$. Seeing that the fitted copula describes the dependence structure well, your assistant suggests that you fit the full data set using the 5-dimensional generalization of the Archimedean copula with the same generator function (1). Do you agree with your assistant's suggestion? Explain your answer.
[4 marks]
[Total: 23 marks]
