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boost::accumulators::impl::peaks_over_threshold_impl — Peaks over Threshold Method for Quantile and Tail Mean Estimation.

// In header: <boost/accumulators/statistics/peaks_over_threshold.hpp>template<typenameSample,typenameLeftRight>structpeaks_over_threshold_impl {// typestypedefnumeric::functional::average< Sample, std::size_t >::result_type float_type;typedefboost::tuple< float_type, float_type, float_type > result_type;typedefmpl::int_< is_same< LeftRight, left >::value?-1:1 > sign;// construct/copy/destructtemplate<typenameArgs> peaks_over_threshold_impl(Argsconst&);// public member functionstemplate<typenameArgs>voidoperator()(Argsconst&) ;template<typenameArgs> result_type result(Argsconst&)const; };

According to the theorem of Pickands-Balkema-de Haan, the distribution function of the excesses over some sufficiently high threshold of a distribution function may be approximated by a generalized Pareto distribution

with suitable parameters and that can be estimated, e.g., with the method of moments, cf. Hosking and Wallis (1987),

and being the empirical mean and variance of the samples over the threshold . Equivalently, the distribution function of the exceedances can be approximated by . Since for the distribution function can be written as

and the probability can be approximated by the empirical distribution function evaluated at , an estimator of is given by

It can be shown that is a generalized Pareto distribution with and . By inverting , one obtains an estimator for the -quantile,

and similarly an estimator for the (coherent) tail mean,

cf. McNeil and Frey (2000).

Note that in case extreme values of the left tail are fitted, the distribution is mirrored with respect to the axis such that the left tail can be treated as a right tail. The computed fit parameters thus define the Pareto distribution that fits the mirrored left tail. When quantities like a quantile or a tail mean are computed using the fit parameters obtained from the mirrored data, the result is mirrored back, yielding the correct result.

For further details, see

J. R. M. Hosking and J. R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, Volume 29, 1987, p. 339-349

A. J. McNeil and R. Frey, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach, Journal of Empirical Finance, Volume 7, 2000, p. 271-300