Cabral, I., Caeiro F., & Gomes I. M.
(2016).
Reduced bias Hill estimators.
International Conference of Computational Methods in Sciences and Engineering 2016, ICCMSE 2016. 1790, , 2016/12/6: American Institute of Physics Inc.
AbstractFor heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.For heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.
Caeiro, F., Marques F. J., Mateus A., & Atal S.
(2016).
A note on the Jackson exponentiality test.
International Conference of Computational Methods in Sciences and Engineering 2016, ICCMSE 2016. 1790, , 2016/12/6: American Institute of Physics Inc.
AbstractIn this paper we revisit the Jackson exponentiality test. We study and provide functions in R language to compute theoretical moments, the distribution function and quantiles of the statistic test. Approximations to the exact distribution function and quantiles are also provided and their precision discussed. In addition, we provide an application of the Jackson test to real data.In this paper we revisit the Jackson exponentiality test. We study and provide functions in R language to compute theoretical moments, the distribution function and quantiles of the statistic test. Approximations to the exact distribution function and quantiles are also provided and their precision discussed. In addition, we provide an application of the Jackson test to real data.
Caeiro, F., & Gomes I. M.
(2015).
Revisiting the maximum likelihood estimation of a positive extreme value index.
Journal Of Statistical Theory And PracticeJournal Of Statistical Theory And Practice. 9(1), 200 - 218., 2015/1/13
AbstractIn this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.
Caeiro, F., & Gomes I. M.
(2016).
Threshold Selection in Extreme Value Analysis.
Extreme Value Modeling and Risk Analysis. 69 - 86., 2016/1/7: Chapman and Hall/CRC 2007
AbstractThe 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 estimateas a function of k. Some of these procedures will be reviewed. Despite of thefact 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 illustrate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI.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 estimateas a function of k. Some of these procedures will be reviewed. Despite of thefact 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 illustrate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI.