Journal of the American Statistical Association ISSN: 0162-1459 (Print) 1537-274X (Online) Journal homepage: http://www.tandfonline.com/loi/uasa20 Fixed-k Asymptotic Inference About Tail Properties Ulrich K. Müller & Yulong Wang To cite this article: Ulrich K. Müller & Yulong Wang (2017) Fixed-k Asymptotic Inference About Tail Properties, Journal of the American Statistical Association, 112:519, 1334-1343, DOI: 10.1080/01621459.2016.1215990 To link to this article: https://doi.org/10.1080/01621459.2016.1215990 Accepted author version posted online: 12 Aug 2016. Published online: 13 Jun 2017. Submit your article to this journal Article views: 206 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=uasa20 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION , VOL. , NO. , –, Theory and Methods https://doi.org/./.. Fixed-k Asymptotic Inference About Tail Properties Ulrich K. Müller and Yulong Wang Department of Economics, Princeton University, Princeton, NJ ABSTRACT ARTICLE HISTORY We consider inference about tail properties of a distribution from an iid sample, based on extreme value the- Received February ory. All of the numerous previous suggestions rely on asymptotics where eventually, an infinite number of Accepted July observations from the tail behave as predicted by extreme value theory, enabling the consistent estimation KEYWORDS of the key tail index, and the construction of confidence intervals using the delta method or other classic Extreme quantiles; tail approaches. In small samples, however, extreme value theory might well provide good approximations for conditional expectations; only a relatively small number of tail observations. To accommodate this concern, we develop asymptoti- extreme value distribution cally valid confidence intervals for high quantile and tail conditional expectations that only require extreme value theory to hold for the largest k observations, for a given and fixed k. Small-sample simulations show that these “fixed-k” intervals have excellent small-sample coverage properties, and we illustrate their use with mainland U.S. hurricane data. In addition, we provide an analytical result about the additional asymp- totic robustness of the fixed-k approach compared to kn →∞inference. 1. Introduction of ξˆ, and thus undermines the distribution theory that justifies Tail properties of distributions have been an important and the confidence interval construction. On the other hand, picking ongoing empirical and theoretical issue. A large literature is kn large amounts to imposing the generalized Pareto approxima- devoted to exploiting implications of extreme value theory and tion on a relatively large fraction of the distribution, especially tail regularity conditions for purposes of estimation and infer- if n is only moderately large. This can be a poor approximation, ence: see Embrechts, Klüppelberg, and Mikosch (1997), Coles even for underlying F in the domain of attraction of an extreme (2001), Reiss and Thomas (2007), Resnick (2007), de Haan and value distribution, leading to substantial bias and undersized Ferreira (2007), Beirlant, Caeiro, and Gomes (2012) and Gomes confidence intervals. As such, for some combinations of n and and Guillou (2015) for reviews and references. As demonstrated F,nochoiceofkn leads to satisfactory inference derived under →∞ by Balkema and de Haan (1974)andPickands(1975), a distribu- kn asymptotics. tion F is in the domain of attraction of an extreme value distribu- This article develops an alternative asymptotic embedding to tion with tail index ξ if and only if its tail is well approximated by address this issue. In particular, we derive asymptotically valid a generalized Pareto distribution with shape parameter ξ.This inference under the sole assumption that extreme value theory regularity assumption about the tail implies that common tail applies to the k largest observations, for given and fixed k.This properties of interest, such as tail probabilities, high quantiles, or should yield good small-sample approximations whenever (a lit- / tail conditional expectations are functions of ξ and the location tle more than) a fraction k n of the tail of F is well approximated and scale of the generalized Pareto distribution, at least approx- by a generalized Pareto distribution. These “fixed-k”asymp- imately. Corresponding estimators of tail properties can hence toticsreflectthesmall-samplenatureoftheinferenceproblem be constructed by plugging in estimators of these three param- in the sense that only relatively few observations are assumed eters. The literature contains numerous suggestions along those to stem from the nicely behaved tail, while previous approaches lines, reviewed in the surveys mentioned above. crucially exploit that the number of such observations is (even- A common feature of all of these approaches is that they tually) large. Our approach is close in spirit to recently devel- oped alternative asymptotic embeddings in other contexts, such model an increasing number kn →∞, kn/n → 0oftailobser- vation in a sample of size n as stemming from the approximate as the “fixed-b” asymptotics considered by Kiefer and Vogelsang generalized Pareto tail. Indeed, under additional regularity con- (2005) in the context of heteroskedasticity and autocorrelation ditions, tail index estimators ξˆ cantypicallybeshowntobecon- robust inference, or the small bandwidth asymptotics in non- sistent and asymptotically Gaussian, enabling the construction parametric kernel estimation studied by Cattaneo, Crump and of confidence intervals about tail properties using variants of the Jansson (2010, 2014). More specifically, we focus in this article on the construc- delta method or similar approaches. For these asymptotics to − / provide a good approximation, however, k has to satisfy a del- tion of confidence intervals for the 1 h n quantile, for given n h, and corresponding tail conditional expectation from an iid icatebalance:ontheonehand,pickingkn small invalidates the asymptotic approximation of consistency and/or Gaussianity sample. By restricting attention to scale and location equivariant CONTACT Ulrich K. Müller [email protected] Department of Economics, Princeton University, Princeton, NJ . © American Statistical Association JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 1335 X intervals, the asymptotic problem under fixed- k extreme value for any fixed k, where the joint p.d.f. of is given by ( ,..., ) = ( ) k ( )/ ( ) ≤ theory becomes a reasonably transparent parametric small- fX|ξ x1 xk Gξ xk i=1 gξ xi Gξ xi on xk xk−1 sample problem: Given a single k-dimensional draw from the ≤···≤x1 with gξ (x) = dGξ (x)/dx,andzerootherwise. joint extreme value distribution (which is indexed only by the We seek to construct an inference method that yields valid scalar ξ), construct an equivariant confidence interval for a spe- asymptotic inference whenever (3)holds,forsomegivenfinite cific deterministic function of ξ and h. One straightforward con- k, to better approximate the small-sample effect of having a struction is obtained by inverting a generalized likelihood ratio finite number of order statistics that are reasonably modeled as statistic. A more systematic approach along the lines of Elliott, stemming from the generalized Pareto tail. Thus, the effective Müller, and Watson (2015) and Müller and Norets (2016) mini- data becomes Yk = (Yn:n,...,Yn:n−k+1 ) ,andtheobjectiveisto ξ mizes weighted expected length (over )subjecttothecoverage construct a confidence set S(Yk) ⊂ R such that P(U (F, n/h) ∈ constraint. S(Yk)) ≥ 1 − α,atleastasn →∞. Section 2 contains the details of the corresponding deriva- By definition of U (F, n/h), P(Yn:n ≤ U (F, n/h)) = (1 − n −h tions. Monte Carlo simulations in Section 3 show that these new h/n) → e . Thus, under (3), (U (F, n/h) − bn)/an converges −h confidence intervals have excellent small-sample coverage and to the e quantile of X1, denoted q(ξ, h) in the following. length properties for moderately large k and n compared to pre- A straightforward calculation shows q(ξ, h) = (h−ξ − 1)/ξ vious confidence interval constructions. We illustrate the appli- for ξ = 0andq(0, h) =−log(h).Ifan and bn were known, cation of the new intervals with data on mainland U.S. Hurricane the asymptotic problem would hence become inference about damage in Section 4.Finally,inSection 5,weconcludewitha q(ξ, h) based on the k × 1vectorofobservationsX.Butan theorem that provides an analytical sense in which the fixed-k and bn are not known, and they depend on tail properties of F approach leads to more robust large sample inference compared themselves. To make further progress, we hence impose loca- →∞ to potentially more informative kn approaches. tion and scale equivariance on the confidence set S. Specifi- cally, we impose that for any constants a > 0andb, S(aYk + ) = (Y ) + (Y ) + ={ ( − )/ ∈ (Y )} 2. Derivation of Fixed-k Inference b aS k b,whereaS k b y : y b a S k . Under this equivariance constraint, we can write 2.1. High Quantile ( , / ) − Y − ( ( , / ) ∈ (Y )) = U F n h bn ∈ k bn , ,..., P U F n h S k P S We observe a random sample Y1 Y2 Yn from some popula- an an ( , ) tion with cumulative distribution function F.WriteU F p for → Pξ q(ξ, h) ∈ S(X) , the 1 − 1/p quantile of F.Weinitiallyconsiderinferenceabout ( , / ) the quantile U F n h for given h. The objective is to construct where the notation Pξ indicatesthattheprobabilityoftheevent