Department seminar by Harry Joe

Please Note: This seminar will be given online.

Estimation of copula-based tail quantities

Let $F$ be a $d$-dimensional distribution, say of risk variables. With a random sample from $F$.  Under weak assumptions on the form of the tail of the copula $C$ of $F$, based on a random sample from $F$, we estimate quantities such as (a) extreme joint tail probabilities, (b) multivariate return curves/surfaces, (c) tail dependence parameters, (d) tail order, (e) tail-weighted dependence measures.

The main theory is based on (i) tail expansions of the distribution $D()$ of directional maxima or minima of random vectors in the copula scale and (ii) tail expansions of an integral of $D()$.  Item (ii) comes from investigating a tail-weighted dependence measure that arises from estimating the extremal index for multivariate extreme value data.  The estimation methods for extreme joint tail probabilities consist of likelihood-based threshold methods (for observations of appropriate maxima/minima that lie beyond a threshold).  Examples will be used for illustration of the main ideas.