Mathematics – Statistics Theory
Scientific paper
2010-08-16
Mathematics
Statistics Theory
30 pages, 8 figures
Scientific paper
Exploratory plotting tools have been devised aplenty in order to diagnose the goodness-of-fit of data sets to a hypothesized distribution. Some of them have found extensive use in diverse areas of finance, telecommunication, environmental science, etc. in order to detect sub-exponential or heavy-tailed behavior in observed data. In this paper we concentrate on two such plotting methodologies: the Quantile-Quantile plots for heavy-tails and the Mean Excess plots. Under the assumption of heavy-tailed behavior of the underlying sample the convergence in probability of these plots to a fixed set in a suitable topology of closed sets of $\R^2$ has been studied in \cite{das:resnick:2008} and \cite{ghosh:resnick:2009}. These results give theoretical justifications for using the plots to test the null hypothesis that the underlying distribution is heavy-tailed by checking if the observed plot is ``close'' to the limit under the null hypothesis. In practice though one set of observations would lead to only one plot of each kind. This is often not good enough to verify proximity of the plot to the fixed set of interest under the null hypothesis of heavy-tails. An appropriate confidence bound around the plots covering or not covering the fixed set would stand as a better indicator for such a purpose. In this paper we provide weak limits for Quantile-Quantile plots for heavy-tails and Mean Excess plots when the underlying distribution of the sample is heavy-tailed. As an application we are able to construct confidence bounds around the plots which enables us to check whether the underlying distribution is heavy-tailed or not.
Das Bikramjit
Ghosh Souvik
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