Chapter 24 ERA 3.3 Modeling and Dependency: Correlations and Copulas - G. Venter

\(\star\) Correlation

Pearson’s correlation:

• Formula (24.1) and its properties

• Outliers will have disproportionate weight

Kendall’s \(\tau\): depends on the rank

• Formula (24.2) for discrete and continuous (24.3) & (24.4)

Copulas

Limitation of joint distribution and advantages for using copulas

Joint distribution plots where the best is to plot the percentile for the marginal distribution

Using copula and Sklar’s Theorem 24.1

• Joint density function express with copula (24.5)

Properties of the main copulas in table 24.1 and the partial perfect correlation copula

• Simulation for each of the copula Frank, Gumble, HRT is same as Frank, and Normal

  • Gumbel’s formula (24.15) & (24.16)

Tail Concentration Functions

Left tail and right tail concentration function for each copulas 24.1

Methods for selecting Copulas

• Plot the percentile plot

• Empirical tail concentration function

• Multivariate Copulas: Normal and t-copula and their properties

• Fitting copulas to data

  • Using the \(J(z)\) and \(\chi(z)\)

  • Graph the possible \(J(z)\) with empirical \(J(z)\) to see which fits best; Similarly for \(\chi(z)\)