Estimating extreme bivariate quantile regions

• Published in: Extremes (Boston) Vol. 16; no. 2; pp. 121 - 145
• Main Authors: Einmahl, John H. J., de Haan, Laurens, Krajina, Andrea
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.06.2013; Springer Nature B.V
• Subjects: Airlines; Aviation; Civil Engineering; Complement; Consistency; Density; Economics; Environmental Management; Estimators; Euclidean space; Finance; Hydrogeology; Insurance; Insurance claims; Management; Mathematics and Statistics; Monitoring; Probability; Quality Control; Quantiles; Reliability; Risk; Safety and Risk; Simulation; Statistical analysis; Statistics; Statistics for Business; Studies; United States > US; 62G05; Level set; Extreme value; Semiparametric estimation; 62G20; 60G70; Multivariate quantile; 62G32; Tail dependence; Rare event; Density contour
• ISSN: 1386-1999; 1572-915X
• DOI: 10.1007/s10687-012-0156-z

When simultaneously monitoring two possibly dependent, positive risks one is often interested in quantile regions with very small probability p . These extreme quantile regions contain hardly any or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we need to deal with probabilities p  < 1/ n , with n the sample size. We consider quantile regions of the form {( x , y ) ∈ (0, ∞ ) 2 : f ( x , y ) ≤  β }, where f , the joint density, is decreasing in both coordinates. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. A detailed simulation study shows the very good statistical performance of the estimated quantile regions. We also apply the method to find extreme risk regions for bivariate insurance claims.