Estimation of Stochastic Dependence via Kendall's Tau

Start: 10/17/2013 - 4:00pm
End  : 10/17/2013 - 5:00pm

Statistics/OR/Math Finance Seminar

Uwe Schmock, TU, Vienna


We discuss the performance of different estimators of dependence
measures, concentrating on the linear correlation coefficient and the
rank-based measure Kendall's tau. As the estimator of Kendall's tau
is a U-statistic, it is asymptotically normal. We calculate the
asymptotic variance explicitly for the Farlie-Gumbel-Morgenstern
copula, the Clayton copula, the Ali-Mikhail-Haq copula and the
Marshall-Olkin copula. For the Clayton copula, the result is
expressed using a generalized hypergeometric function, which can be
reduced to a more elementary form for specific parameters values of
the copula.

For elliptical distributions there is a unique connection between the
linear correlation coefficient and Kendall's tau. This offers two
ways for estimation: using the standard estimator of the linear
correlation directly or estimating Kendall's tau and transforming it
into a linear correlation estimate. Since both estimators are
asymptotically normal under appropriate moment conditions, we use the
asymptotic variance to compare their performance. For the
uncorrelated t-distribution, using the fact that it is a variance
mixture of normal distributions, we show that the asymptotic variance
equals an integral involving the square of the arctangent and a
hypergeometric function. For all integer-valued degrees of freedom,
involved and tricky integrations lead to a surprisingly simple closed
form for the asymptotic variance. It turns out that especially for
small degrees of freedom the alternative estimation via Kendall's tau
performs much better than the standard estimator. (This is joint work
with Barbara Dengler.)

Roberts South 105, CMC