Weak limits for exploratory plots in the analysis of extremes

Exploratory data analysis is often used to test the goodness-of-fit of sample observations to specific target distributions. A few such graphical tools have been extensively used to detect subexponential or heavy-tailed behavior in observed data. In this paper we discuss asymptotic limit behavior of two such plotting tools: the quantile-quantile plot and the mean excess plot. The weak consistency of these plots to fixed limit sets in an appropriate topology of $\mathbb{R}^2$ has been shown in Das and Resnick (Stoch. Models 24 (2008) 103-132) and Ghosh and Resnick (Stochastic Process. Appl. 120 (2010) 1492-1517). In this paper we find asymptotic distributional limits for these plots when the underlying distributions have regularly varying right-tails. As an application we construct confidence bounds around the plots which enable us to statistically test whether the underlying distribution is heavy-tailed or not.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ401 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

QQ plots, Random sets and data from a heavy tailed distribution

The QQ plot is a commonly used technique for informally deciding whether a univariate random sample of size n comes from a specified distribution F. The QQ plot graphs the sample quantiles against the theoretical quantiles of F and then a visual check is made to see whether or not the points are close to a straight line. For a location and scale family of distributions, the intercept and slope of the straight line provide estimates for the shift and scale parameters of the distribution respectively. Here we consider the set S_n of points forming the QQ plot as a random closed set in R^2. We show that under certain regularity conditions on the distribution F, S_n converges in probability to a closed, non-random set. In the heavy tailed case where 1-F is a regularly varying function, a similar result can be shown but a modification is necessary to provide a statistically sensible result since typically F is not completely known.Comment: 19 pages, 2 figure