Threshold Selection in Extreme Value Analysis ∗ Frederico Caeiro D.M. and C.M.A., Universidade Nova de Lisboa and M. Ivette Gomes D.E.I.O. and C.E.A.U.L., Universidade de Lisboa September 1, 2014 Abstract. The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the extreme value index (EVI). Whenever we are interested in large values, such estimation is usually performed on the basis of the largest k + 1 order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive EVI-estimation. Such a choice can be either heuristic or based on sample paths stability or on the minimization of a mean squared error estimate as a function of k. Some of these procedures will be reviewed. Despite of the fact that the methods provided can be applied, with adequate modifications, to any real EVI and not only to the adaptive EVI-estimation but also to the adaptive estimation of other relevant right-tail parameters, we shall illus- trate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI. Keywords and phrases. Bootstrap methodology, heuristic methods, opti- mal sample fraction, sample-paths stability, semi-parametric estimation, statistical theory of extremes 1 Introduction As usual, let us consider that we have access to a sample (X 1 ,...,X n ) of independent, identically distributed (IID) random variables (RVs), or possi- bly weakly dependent and stationary RVs from an underlying population with unknown cumulative distribution function (CDF) F (·). Let us further use the notation (X 1:n ≤···≤ X n:n ) for the associated sample of ascending order statistics (OSs). * Research partially supported by National Funds through FCT —Funda¸c˜ ao para a Ciˆ encia e a Tecnologia, projects PEst-OE/MAT/UI0006/2014 (CEAUL) and PEst- OE/MAT/UI0297/2014 (CMA/UNL). 1