# MAT8886 copulas and extremes

### Maximum likelihood estimates for multivariate distributions

September 22, 2012 |

Consider our loss-ALAE dataset, and - as in Frees & Valdez (1998) - let us fit a parametric model, in order to price a reinsurance treaty. The dataset is the following, __ library(evd) __ data(lossalae) __ Z=lossalae __ X=Z[,1];Y=Z[,2] ... [Read more...]

### Interactive 3d plot, in R

September 20, 2012 |

Following the course of this afternoon, I will just upload some codes to make interactive 3d plots, in R. __ library(rgl) __ library(evd); __ data(lossalae) __ U=rank(lossalae[,1]+rnorm(nrow(lossalae), + mean=0,sd=.001))/(nrow(lossalae)+1) ... [Read more...]

### (nonparametric) Copula density estimation

September 20, 2012 |

Today, we will go further on the inference of copula functions. Some codes (and references) can be found on a previous post, on nonparametric estimators of copula densities (among other related things).  Consider (as before) the loss-ALAE data... [Read more...]

### Copulas and tail dependence, part 3

September 18, 2012 |

We have seen extreme value copulas in the section where we did consider general families of copulas. In the bivariate case, an extreme value can be writtenwhere is Pickands dependence function, which is a convex function satisfyingObserve that in ... [Read more...]

### Copulas and tail dependence, part 2

September 18, 2012 |

An alternative to describe tail dependence can be found in the Ledford & Tawn (1996) for instance. The intuition behind can be found in Fischer & Klein (2007)). Assume that and have the same distribution. Now, if we assume that those vari... [Read more...]

### Copulas and tail dependence, part 1

September 17, 2012 |

As mentioned in the course last week Venter (2003) suggested nice functions to illustrate tail dependence (see also some slides used in Berlin a few years ago). Joe (1990)'s lambda Joe (1990) suggested a (strong) tail dependence index. For lower t... [Read more...]

### Kendall’s function for copulas

September 12, 2012 |

As mentioned in the course on copulas, a nice tool to describe dependence it Kendall's cumulative function. Given a random pair with distribution  , define random variable . Then Kendall's cumulative function is Genest and Rivest (1993) intr... [Read more...]

### Association and concordance measures

September 12, 2012 |

Following the course, in order to define assocation measures (from Kruskal (1958)) or concordance measures (from Scarsini (1984)), define a concordance function as follows: let be a random pair with copula , and with copula . Then define the so-... [Read more...]

### MAT8886 reducing dimension using factors

February 16, 2012 |

First, let us recall a standard result from linear algebra: "real symmetric matrices are diagonalizable by orthogonal matrices". Thus, any variance-covariance matrix can be written since a variance-covariance matrix is also definite positive. In ... [Read more...]

### MAT8886 the Dirichlet distribution

February 15, 2012 |

In the course, still introducing some concept of dependent distributions, we will talk about the Dirichlet distribution (which is a distribution over the simplex of ). Let denote the Gamma distribution with density (on ) Let denote independent... [Read more...]

### MAT8886 elliptically contoured distributions

February 15, 2012 |

### MAT8886 exchangeability, credit risk and risk measures

February 10, 2012 |

Exchangeability is an extremely concept, since (most of the time) analytical expressions can be derived. But it can also be used to observe some unexpected behaviors, that we will discuss later on with a more general setting. For instance, in a old... [Read more...]

### MAT8886 a short word on profile likelihood

February 7, 2012 |

Profile likelihood is an interesting theory to visualize and compute confidence interval for estimators (see e.g. Venzon & Moolgavkar (1988)). As we will use is, we will plot But more generally, it is possible to consider where . Then (... [Read more...]

### MAT886 mean excess function (and reinsurance)

February 1, 2012 |

Tomorrow, in the course on extreme value, we will focus on applications. We will discuss reinsurance pricing. Consider a random variable , a threshold and define the mean excess function. This function is known in life insurance as the average ... [Read more...]

### MAT8886 Extremes and sums (of i.i.d. random variables)

January 20, 2012 |

Yesterday, we have discussed briefly sums and maximas of i.i.d. random variables using the concept of subexponential distributions. Today, we will introduce the concept of regular variation: a positive function is said to be regularly varying (at i... [Read more...]

### MAT8886 Fisher-Tippett theorem and limiting distribution for the maximum

January 12, 2012 |

Tomorrow, we will discuss Fisher-Tippett theorem. The idea is that there are only three possible limiting distributions for normalized versions of the maxima of i.i.d. samples . For bounded distribution, consider e.g. the uniform distribution on t... [Read more...]