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Efficient estimators and LAN in canonical bivariate POT models

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  • Falk, Michael
  • Reiss, Rolf-Dieter

Abstract

Bivariate generalized Pareto distributions (GPs) with uniform margins are introduced and elementary properties such as peaks-over-threshold (POT) stability are discussed. A unified parameterization with parameter [theta][set membership, variant][0,1] of the GPs is provided by their canonical parameterization. We derive efficient estimators of [theta] and of the dependence function of the GP in various models and establish local asymptotic normality (LAN) of the loglikelihood function of a 2x2 table sorting of the observations. From this result we can deduce that the estimator of [theta] suggested by Falk and Reiss (2001, Statist. Probab. Lett. 52, 233-242) is not efficient, whereas a modification actually is.

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  • Falk, Michael & Reiss, Rolf-Dieter, 2003. "Efficient estimators and LAN in canonical bivariate POT models," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 190-207, January.
    • Handle: RePEc:eee:jmvana:v:84:y:2003:i:1:p:190-207
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    1. Drees, Holger & Huang, Xin, 1998. "Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 25-47, January.
    2. M. Falk, 1983. "Relative efficiency and deficiency of kernel type estimators of smooth distribution functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 37(2), pages 73-83, June.
    3. Falk, Michael & Reiss, Rolf-Dieter, 2001. "Estimation of canonical dependence parameters in a class of bivariate peaks-over-threshold models," Statistics & Probability Letters, Elsevier, vol. 52(3), pages 233-242, April.
    4. Lu, Jye-Chyi & Bhattacharyya, Gouri K., 1991. "Inference procedures for bivariate exponential model of Gumbel," Statistics & Probability Letters, Elsevier, vol. 12(1), pages 37-50, July.
    5. E. Kaufmann & R. Reiss, 1993. "Strong convergence of multivariate point processes of exceedances," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(3), pages 433-444, September.
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    1. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On the distribution of Pickands coordinates in bivariate EV and GP models," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 267-295, April.
    2. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On Pickands coordinates in arbitrary dimensions," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 426-453, February.

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