Statistics Seminar: Philip Kokic -- Multivariate quantiles, expectiles and M-quantiles

Philip Kokic Mathematics, Informatics and Statistics CSIRO 

Location: Carslaw 173 

Time: 2pm Friday, August 6, 2010 

Title: Multivariate quantiles, expectiles and M-quantiles 

Abstract: Ordinary quantiles and expectiles are special cases of univariate M-quantiles,
which can be defined either in terms of the minimum of a specific loss function or the
solution to the corresponding estimating equation.  In this presentation we show how
M-quantiles can be extended to a multivariate situation by generalising the estimating
equation approach.  Some of the features and properties of Multivariate M-quantiles are
explored using a variety of simulated data sets.  In particular, it is illustrated how,
to a certain extent, they are able to capture non-linear relationships between
variables.  For data of more than 1 dimension they are unique and can also be computed
efficiently even for high dimensional data.  Unlike alternatives based on multivariate
density estimates they do not suffer from ’the curse of dimensionality’.  

Multivariate outliers present some interesting statistical issues; for example, an
outlier may or may not become an influential observation depending on the type of
analysis performed.  In this presentation we also describe how M-quantiles need to be
modified for multivariate outlier detection.  To this end a semi-parametric form of
multivariate expectiles is described, and the approach is illustrated using UK annual
business survey data.