Estimation of Extreme Conditional Quantiles Through Power Transformation

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1 This article was downloaded by: [North Carolina State University] On: 26 January 2015, At: 18:20 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of the American Statistical Association Publication details, including instructions for authors and subscription information: Estimation of Extreme Conditional Quantiles Through Power Transformation Huixia Judy Wang a & Deyuan Li b a Department of Statistics, North Carolina State University, Raleigh, NC, b Department of Statistics, Fudan University, Shanghai, , China Accepted author version posted online: 16 Jul 2013.Published online: 27 Sep To cite this article: Huixia Judy Wang & Deyuan Li (2013) Estimation of Extreme Conditional Quantiles Through Power Transformation, Journal of the American Statistical Association, 108:503, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Supplementary materials for this article are available online. Please go to Estimation of Extreme Conditional Quantiles Through Power Transformation Huixia Judy WANG and Deyuan LI The estimation of extreme conditional quantiles is an important issue in numerous disciplines. Quantile regression (QR) provides a natural way to capture the covariate effects at different tails of the response distribution. However, without any distributional assumptions, estimation from conventional QR is often unstable at the tails, especially for heavy-tailed distributions due to data sparsity. In this article, we develop a new three-stage estimation procedure that integrates QR and extreme value theory by estimating intermediate conditional quantiles using QR and extrapolating these estimates to tails based on extreme value theory. Using the power-transformed QR, the proposed method allows more flexibility than existing methods that rely on the linearity of quantiles on the original scale, while extending the applicability of parametric models to borrow information across covariates without resorting to nonparametric smoothing. In addition, we propose a test procedure to assess the commonality of extreme value index, which could be useful for obtaining more efficient estimation by sharing information across covariates. We establish the asymptotic properties of the proposed method and demonstrate its value through simulation study and the analysis of a medical cost data. Supplementary materials for this article are available online. KEY WORDS: Box Cox power transformation; Extreme value; Heavy-tailed distribution; High quantile; Quantile regression. 1. INTRODUCTION Events such as high medical cost, heavy rainfall, and large financial loss are rare but have significant consequences. For such rare events, researchers are interested in estimating the extreme quantiles of the response of interest rather than its central summaries such as the mean or median. In real practice, the variable of interest may depend on some predicting variables at the tails; see Jagger and Elsner (2008), Friederichs and Hense (2007), and Engle and Manganelli (2004) for applications in different areas. To incorporate the covariate information in the analysis, we focus on the estimation of extreme conditional quantiles for heavy-tailed distributions, which are encountered in many applications, such as financial returns and insurance claims. Existing literature on the estimation of extreme conditional quantiles can be roughly divided into four classes. One class of work models extremes by fitting a fully parametric model, such as generalized extreme value distribution or generalized Pareto distribution (GPD), where the location, shape and scale parameters are allowed to depend on covariates either parametrically or nonparametrically (Davison and Ramesh 2000; Hall and Tajvidi 2000; Beirlant and Goegebeur 2003, 2004; Chavez-Demoulin and Davison 2005; Wang and Tsai 2009). These methods either require maximal data or involve choices of covariate-dependent tuning parameters (e.g., threshold values). The second class of work extends extreme value theory for univariate data to regression setup through local estimation (Gardes and Girard 2010; Gardes, Girard, and Lekina 2010; Daouia et al. 2011). These methods estimate the extreme quantiles of the response variable Y given covariates X = x by using observations in a small neighborhood of x, thus their finite sample behavior heavily depends on the richness of data in the neighborhood. Huixia Judy Wang is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC ( hwang3@ncsu.edu). Deyuan Li is Associate Professor, Department of Statistics, Fudan University, Shanghai , China ( deyuanli@fudan.edu.cn). The research is partially supported by the NSF Award DMS , NSF CAREER Award DMS , and NNSFC grants and Quantile regression (QR; Koenker 2005) provides a direct semiparametric technique for modeling the conditional quantiles of a response variable given covariates. The third class of work applies QR directly to estimate conditional quantiles at tails; see for instance Bremnes (2004), Friederichs and Hense (2007), and Jagger and Elsner (2008). However, since QR does not make any distributional assumptions, it is difficult to make inference in data-sparse regions, such as the extreme tails. The fourth class of work integrates QR with extreme value theory by assuming some reasonable assumptions on tail behaviors. For instance, Chernozhukov and Du (2006) and Wang, Li, and He (2012) developed different estimation methods, where conventional QR is used to estimate the intermediate conditional quantiles, which are then extrapolated to the far tails based on some reasonable assumptions on tail behaviors. In both articles, the conditional quantiles of Y are assumed to be linear in x at tails. Such linearity assumption is restrictive and may fail in real applications. In addition, both articles assumed that the conditional distribution of Y is tail equivalent across covariate values x. However, in studies of rare events, the shape of the tail distribution may depend on the covariates. Various examples with covariate-dependent shape parameters can be found in Beirlant, de Wet, and Goegebeur (2004), Wang and Tsai (2009), Gardes, Girard, and Lekina (2010), and Das et al. (2010), to name a few. In this article, we develop a flexible integration approach by relaxing the assumptions of linear quantile functions of Y and tail equivalency across x required in Chernozhukov and Du (2006) and Wang, Li, and He (2012). We first show that for the conditional quantiles of Y to be linear in x, tail equivalency is a necessary assumption. However, we further demonstrate that for some cases with covariate-dependent extreme value index (EVI), a measurement of the heaviness of the tail distribution, quantiles of Y may still be linear in x after some appropriate transformation such as log transformation. This motivates us to consider a power-transformed QR model, which assumes that 2013 American Statistical Association Journal of the American Statistical Association September 2013, Vol. 108, No. 503, Theory and Methods DOI: /

3 Wang and Li: Estimation of Extreme Conditional Quantiles Through Power Transformation 1063 the conditional quantiles of the transformed Y are linear in x at right tails. Based on the power-transformation model, we propose a new three-stage procedure for estimating the extreme conditional quantiles of Y. The method works by first finding an appropriate power transformation to ensure the linearity of quantile functions of the transformed Y at tails, then estimating the intermediate conditional quantiles of Y on the original scale using conventional QR and the equivariance property of quantiles to monotone transformations, and finally extrapolating these estimates to extreme tails. This new method is more flexible compared to those in Chernozhukov and Du (2006) and Wang, Li, and He (2012). On the other hand, compared to nonparametric approaches (Beirlant, de Wet, and Goegebeur 2004; Gardes and Girard 2010; Gardes, Girard, and Lekina 2010; Gardes, Guillou, and Schorgen 2012), the parametric modeling of intermediate conditional quantile functions allows us to borrow information across covariates to obtain more stable estimation of extreme quantiles. The current article has the following major distinctions from Wang, Li, and He (2012), which also studied estimating extreme conditional quantiles through extrapolation. First, the method in Wang, Li, and He (2012) requires the conditional quantiles of Y to be linear in the covariates. This assumption is often restrictive for heavy-tailed distributions, and the estimator of Wang, Li, and He (2012) may perform badly when the linearity assumption is violated; see Section 4 for numerical evidence. In contrast, by considering a power-transformed QR model, our newly proposed estimator is more flexible and can accommodate a wider class of conditional quantile functions. Second, we formally study the relation between the linearity of quantile functions and tail equivalency, which is the first attempt in the literature. Our results suggest that to accommodate covariatedependent EVI, the linear QR model assumed in Wang, Li, and He (2012) cannot hold so the theory developed there cannot be applied. However, we can still employ linear QR on the transformed, such as log scale while allowing for EVI to be covariate dependent. Third, in this article we develop a test procedure to assess if the EVI depends on covariates. If the test suggests that the EVI is constant across x or in a subregion of x, we can pool information across covariates to estimate the common EVI and thus improve the estimation efficiency. The rest of the article is organized as follows. In Section 2,we present the motivation for the power-transformed QR model and describe the proposed three-stage estimating procedure. In Section 3, we discuss the theoretical properties of the proposed estimator, and present the proposed test procedure for assessing the constancy of EVI. In Section 4, we conduct a simulation study to assess the finite sample performance of the proposed method. In Section 5, the value of the proposed method is demonstrated by the analysis of a medical cost data where high costs are of interests. Section 6 concludes the article with a discussion. All the technical details are provided in the supplementary material. 2. PROPOSED METHOD 2.1 Notations Let Y be the response variable of interest, and X = (X 1,...,X p ) T be a p-dimensional covariate vector with the first element X 1 = 1 corresponding to the intercept. Suppose we observe a random sample {(y i, x i ),i = 1,...,n} of the random vector (Y, X). Let Q Y (τ x) = F 1 Y (τ x) := inf{y : F Y (y x) τ} denote the τth conditional quantile of Y given X = x, where F Y ( x) is the conditional cumulative distribution function of Y. We are interested in estimating the extremely high conditional quantiles Q Y (τ n X = x), where as n, τ n 1 at any arbitrary rate. In this article, we focus on heavy-tailed distributions, for which the estimation of high quantiles is especially challenging due to data sparseness in the tail area. Specifically, we assume that F Y ( x) is in the maximum domain of attraction of an extreme value distribution G γ (x), denoted by F Y ( x) D(G γ (x) ), where γ (x) > 0istheEVI. That is, suppose that Z 1,...,Z m is a random sample of size m from the conditional distribution F Y ( x) with a given x, there exist constants a m (x) > 0 and b m (x) R such that ( ) max1 i m Z i b m (x) P z G γ (x) (z) a m (x) = exp { (1 + γ (x)z) 1/γ (x)} (2.1) as m,for1+ γ (x)z >0. The maximum domain of attraction assumption is a standard assumption in the extreme value literature. A wide range of heavy-tailed distributions satisfy this assumption. 2.2 Power-Transformed Linear QR Model Conventional methods in QR literature often estimate Q Y (τ x) by fitting the linear QR model Q Y (τ x i ) = x T i β(τ), (2.2) where 0 τ 1 is the quantile level of interest; see for instance Portnoy and Jurecčková (1999), Tang and Leng (2011), and Yang and He (2012). However, such direct quantile estimations are often inaccurate at tails due to data sparseness, and the linearity assumption in model (2.2) is restrictive and may fail in practice. Furthermore, for the study of extreme quantiles, Chernozhukov (2005), Chernozhukov and Du (2006), and Wang, Li, and He (2012) assumed that the conditional distribution F Y ( x) are tail equivalent across covariate values x, that is, the EVI γ (x) is a constant across x. In studies of rare events, the EVI γ (x) may depend on x (Beirlant, de Wet, and Goegebeur 2004; Wang and Tsai 2009; Gardes, Girard, and Lekina 2010). However, the following proposition shows that in situations where the EVI γ (x) varies with x, it is rarely the case that the conditional high quantiles of Y are still linear in x. Proposition 2.1. Assume the linear QR Model (2.2) holds for τ [1 ε, 1], where ε is a small positive constant. Then γ (x) is a constant. Proposition 2.1 suggests that to accommodate covariatedependent EVI, one has to consider nonparametric QR. However, it is known that nonparametric quantile estimation is very unstable at tails for small sample sizes especially when the dimension of x is high. We hope to employ parametric modeling to borrow information across covariates to obtain more stable estimation of extreme quantiles. The following example shows that for some cases where the EVI is covariate dependent, the

4 1064 Journal of the American Statistical Association, September 2013 quantiles of Y may still be linear in x after some appropriate transformation. Example 1. Consider the following data-generating process: Y X = x Pareto { γ (x) }, γ(x) = x T β, (2.3) 1/γ (x) that is, P (Y y X = x) = 1 y for y 1. Under Model (2.3), we have Q Y (τ x) = (1 τ) γ (x), and Q log Y (τ x) = x T β(τ), where β(τ) = βlog(1 τ). To achieve more accuracy and model flexibility, we assume the following power-transformed QR model Q λ (Y )(τ x i ) = x T i θ(τ),τ [1 ε, 1], (2.4) where λ ( ) denotes the family of power transformations (Box and Cox 1964) with { y λ 1 if λ 0 λ (y) = λ log(y) if λ = 0. Model (2.4) states that the upper quantiles of the powertransformed Y are linear in the covariates. By the appealing equivariance property of quantiles to monotone transformations, the τth conditional quantile of Y given x i is simply 1 λ {xt i θ(τ)}, the inverse power transformation of the linear index x T i θ(τ). In this article, we consider power transformation for easy interpretation and model parsimony. 2.3 Three-Stage Estimation Method We propose a three-stage procedure for estimating the high conditional quantiles of Y, Q Y (τ n x). The method works by first estimating the intermediate conditional quantiles of Y through linear QR with a power transformation on Y, and then extrapolating these estimates to extreme tails based on reasonable assumptions of the tail behaviors. In the first stage, we adopt the method proposed in Mu and He (2007) and Yin, Zeng, and Li (2008), and estimate the power transformation parameter λ by n λ = argmin {R n (x i,λ;1 ε)} 2, (2.5) λ R i=1 where R n (t,λ; τ) = 1 n I(x j t) [ τ I{ λ (y j ) x T j n θ(τ; λ) 0} ], j=1 n { θ(τ; λ) = argmin ρ τ λ (y i ) x T i b}, b R p i=1 and ρ τ (u) = u{τ I(u <0)}. Here I( ) is the indicator function, and for vectors x and t R p, x t means that each component of x is less than or equal to the corresponding component of t. The estimation is obtained by focusing on a single quantile level τ = 1 ε with ε asmall positive constant. Alternatively, we could examine a set of upper quantile levels τ [1 ε, 1) to estimate the common λ. In the second stage, we estimate the conditional quantiles of Y by first fitting the Model (2.4) on the transformed responses, λ(y i ), and then transforming the conditional quantile estimates back to the original scale. Let τ j = j/(n + 1) for j = 1, 2,...,m, where m = n [n η ]forsome0<η<1 and [u] denotes the integer part of u. Throughout our empirical studies we let η = 0.1. For j = 1,...,m, we estimate Q Y (τ j x)by Q Y (τ j x) = 1 λ {xt θ(τ j ; λ)}. (2.6) In practice, due to estimation error or lack of data in regions of interest, the estimated conditional quantiles from QR may not be monotonically increasing in τ. To account for this issue, we employ the quantile rearrangement procedure proposed in Chernozhukov, Fernández-Val, and Galichon (2010), which constructs monotone quantile curves by sorting or monotone rearranging { Q Y (τ j x),j = 1,...,m}. The third stage involves extrapolating the intermediate quantile estimates to the extreme tails. Specifically, for τ n 1, we estimate Q Y (τ n x) by an adaptation of Weissman s estimator to the conditional case (Weissman 1978; Daouia et al. 2011), that is, ( 1 ) γ (x) τn k Q Y (τ n x) = Q Y (τ n k x), (2.7) 1 τ n where k = k n and k/n 0, and γ (x) = 1 k [n η ] k j=[n η ] log Q Y (τ n j x) Q Y (τ n k x), (2.8) with n being some small positive constant. Here γ (x) can be viewed as the Hill s (1975) estimator based on the pseudo upper order statistics of samples from F Y ( x). In Equation (2.8), the truncation of the sum at j [n η ] is included to ensure that n(1 τ n j ) for [n η ] j k, which is needed to establish the Bahadur representation of the estimated quantile coefficients θ(τ) and consequently to establish the asymptotic normality of γ (x). In our proposed Stage 3, extrapolation of extreme conditional quantiles is carried out on the original scale of Y. Alternatively, one could also consider extrapolation on the transformed scale, which can then be transformed back to the original scale. Note that for heavy-tailed distributions, the EVI of the transformed scale is γ (x) = λγ (x) R with the sign determined by λ; see Teugels and Vanroelen (2004). When λ>0, under power transformation, these two approaches are equivalent if both are based on the Hill s estimator. On the other hand, if λ 0, Hill s estimator cannot be used to estimate γ (x). Instead, one could use some other estimators (e.g., the moment estimator in de Haan and Ferreira 2006) to estimate γ (x), extrapolate the extreme conditional quantiles based on the transformed scale, and then transform it back to the original scale. However, our numerical studies (not reported due to space limit) showed that the alternative method based on moment estimator and extrapolation on the transformed scale often gives less efficient estimation than the proposed method. The following two examples demonstrate the possibility of the power-transformed QR model (2.4)forλ = 0 and λ<0, and the connections between the EVI of Y x and that of λ (Y ) x. Examples with λ>0 can be found in Section 4. Example 2. (λ = 0) Consider the random variable W such that for τ close to 1, F 1 W (τ x) = xt β (x T σ )log(1 τ) + d(x T σ )ρ 1 (1 τ) ρ,

5 Wang and Li: Estimation of Extreme Conditional Quantiles Through Power Transformation 1065 where d 0, ρ<0, and x T σ > 0. Since F 1 W (τ x)/ τ > 0for τ close to 1, the extreme high quantiles of W are well defined. It is easy to verify that W x D(G 0 ). Let Y = e W. By Lemma 5(iii) in the supplementary material, we can easily obtain that ( y ) [ 1/γ (x) 1 F Y (y x) = 1 + d ( y ) ] ρ/γ(x) {1 + o(1)}, c ρ c y, (2.9) where c = c(x) = exp(x T β) > 0 and γ (x) = x T σ. Therefore, this is a case where the EVI of Y depends on x but the EVI of the transformed Y is a constant, and the conditional quantiles of the log transformed Y are linear in x. Example 3. (λ <0) Consider the transformation model ( = x T β + 1 λ + xt β λ (Y ) = Y λ 1 λ ) ɛ, (2.10) where x T β>0and ɛ Uniform(0, 1). It is easy to show that the EVI of λ (Y )isγ (x) = 1. Note that given x, Y λ = (1 λx T β)(1 ɛ) U(0, 1 λx T β). Therefore, P (Y >t) = P (Y λ <t λ ) = t λ /(1 λx T β). That is, Y x follows a GPD and the EVI of Y x is 1/λ. 3. THEORETICAL RESULTS 3.1 Second-Order Condition We first introduce a second-order condition for the distribution F D(G γ ) with γ R. Define U(t) = F 1 (1 1/t). Then F D(G γ ) implies that there exists a function a 1 ( ) such that for x>0, as t, U(tx) U(t) a 1 (t) xγ 1. γ In most extreme value literatures (de Haan and Ferreira 2006), to obtain the asymptotic results, the following second-order condition is assumed, that is, U(tx) U(t) a 1 (t) a 2 (t) xγ 1 γ 1 ϱ ( x γ +ϱ ) 1 γ + ϱ xϱ 1. (3.1) ϱ Here a 2 (t) 0 as t and a 2 (t) RV (ϱ) with ϱ 0, which means a 2 (t) is a regularly varying function with index ϱ, that is, lim t a 2 (tx)/a 2 (t) = x ϱ for all x>0. In the following, we provide more explanation of (3.1) for three cases: γ>0, γ = 0, and γ<0, separately. Case 1. γ>0. The second-order condition (3.1) is equivalent to that there exists a function A with lim t A(t) = 0 and A(t) RV (ϱ), ϱ 0 such that for all x>0, A(t) 1 { U(tx) U(t) xγ } x γ xϱ 1, as t. (3.2) ϱ If γ>0and ϱ<0, then (3.2) is also equivalent to [ U(t) = ct γ 1 + A(t) ] {1 + o(1)}, ast, ϱ with some constant c>0, in which case we say that U satisfies the second-order condition (3.2) indexed by (c, γ, ϱ, A). Case 2. γ = 0. By de Haan and Ferreira (2006, p. 388), it follows that there exists some nonzero constant c such that U(t) c log t RV (ϱ). If U(t) as t, then c>0. For ϱ<0, Equation (3.1) implies U(t) = c log t + A(t) {1 + o(1)}, (3.3) ϱ where A(t) RV (ϱ). Case 3. γ<0. Then U( ) = sup{t : F (t) < 1} < and (3.1) is equivalent to that there exists a function A( ) with lim t A(t) = 0 and A(t) RV (ϱ), ϱ 0 such that for all x>0, { } U( ) U(tx) A(t) 1 U( ) U(t) xγ 1 ϱ xγ (x ϱ 1), as t. (3.4) Similarly, (3.4) with γ<0and ϱ<0isequivalent to [ U( ) U(t) = ct γ 1 + A(t) ] {1 + o(1)}, as t, ϱ (3.5) with some constant c>0, in which case we say U satisfies the second-order condition (3.4) indexed by (c, γ, ϱ, A). 3.2 Assumptions and Theoretical Results Throughout the article, we assume that the conditional distributions of Y x and λ (Y ) x belong to the maximum domain of attraction with extreme value indices γ (x) and γ (x), respectively. Teugels and Vanroelen (2004) showed that under the Box Cox transformation, γ (x) = λγ (x). For heavy-tailed distributions with γ (x) > 0, the sign of γ (x) is determined by the sign of λ. Denote a(t) b(t) ifa(t)/b(t) 1ast.We make the following additional assumptions. B1. The variable X has a compact support X, and E(XX T ) is positive definite. B2. There exists an auxiliary line x x T θ(r) with0< r<1 and θ(r) a bounded vector such that for Y = λ (Y ) x T θ(r) and some distribution function F 0 D(G γ 0 ) with a density at tails and γ0 R, (i) if γ0 0 and sup{t : F 0(t) < 1} =, 1 F Y (t x) K(x){1 F 0 (t)} 1 ={1 F 0 (t)} δ K(x){1 + o(1)}, (3.6) uniformly in x X as t, where K( ) > 0 and K( ) R are continuous and bounded functions on X, and δ>0isaconstant. (ii) if γ0 < 0, 1 F Y (v t x) K(x){1 F 0 (v 0 t)} 1 ={1 F 0 (v 0 t)} δ K(x){1 + o(1)}, (3.7) uniformly in x X as t 0, where v = sup{t : F Y (t x) < 1} <, v 0 = sup{t : F 0 (t) < 1} <, K( ) > 0, and K( ) R are continuous and bounded functions on X, and δ>0isaconstant. B3. τ F 1 Y (τ x) τ F 1 0 {τ/k(x)} uniformly in x X as τ 1. B4. Let U 0 (t) = F 1 0 (1 1/t). Assume that U 0 ( ) = for γ0 0 and U 0( ) < for γ0 < 0, and that

6 1066 Journal of the American Statistical Association, September 2013 (i) if γ 0 > 0, U 0 satisfies the second-order condition (3.2) with γ 0 > 0, ϱ<0, and A(t) = γ 0 dtϱ with d 0. (ii) if γ 0 = 0, U 0 satisfies that U 0 (t) = γ (x)logt + A(t) {1 + o(1)} (3.8) ϱ with γ (x) > 0, ϱ<0, and A(t) = γ (x)dt ϱ with d 0. (iii) if γ0 < 0, U 0 satisfies the second-order condition (3.4) with γ0 < 0, ϱ<0, and A(t) = γ 0 dtϱ with d 0. B5. The power transformation parameter λ belongs to a compact set. If 1 λ (xt β) = 1 λ (xt β ), then β = β and λ = λ with probability one. In addition, the conditional density of Y given x i is continuous and bounded away from zero and infinity uniformly over i in the neighborhood of Q Y (τ x i )forτ = 1 ε. B6. As n, k = k n, k/n 0, k 1/2 n η log k 0 for some η (0, 1), and k 1/2 Ã(n/k) φ R, where Ã( ) is defined in Lemmas 3 5 in the supplementary material for γ0 > 0, = 0, and < 0, respectively. In addition, k 1/2 (k/n) γ 0 = o(1) for γ 0 > 0, (k/n) 1/2 (log n) 2 = o(1) for γ0 = 0, and n1/2 (n/k) γ 0 +max(ϱ, δ) for γ0 < 0. Condition B2 states the form of equivalency between 1 F Y (t x) and 1 F 0 (t) at the right tails, and their main difference can be expressed as the product of a function of t and a function of x. The function K( ) captures the heterogeneity of the model (2.4). Condition B3 assumes the equivalency of the conditional density of Y given x and that of F 0. Similar conditions to B2 and B3 were also posed on the conditional distribution of the untransformed Y in Wang, Li, and He (2012). Condition B4 assumes that U 0 satisfies the second-order condition (3.1) with ϱ<0, which is standard in extreme value theory literature to obtain the asymptotic normality. For the case of γ0 = 0 and ϱ< 0, it is reasonable to consider the model (3.8) by replacing the constant c in (3.3)byγ(x) > 0. Condition B5 is needed to obtain the root-n consistency of λ, and B6 specifies the assumptions on the rate of k. The parameter γ0 is the EVI of F 0, which is related to the extreme value indices of λ (Y ) x and Y x. Under the powertransformed linear QR Model (2.4), Proposition 2.1 implies that γ0 (x) is common across x. From Lemmas 3 5 in the supplementary material, we have γ (x) = γ0.forλ 0, this assumption implies that γ (x) = γ0 /λ is also invariant to x. However, for λ = 0, γ (x) may depend on x even when γ (x) is a constant; see Section 6 for more related discussions. Theorems 3.1 and 3.2 present the asymptotic normalities of the proposed EVI and extreme conditional quantile estimators. Theorem 3.1. Suppose that Model (2.4) and conditions B1 B6 hold. In addition, assume that 0 γ0 < 1/(2λ) or 1/2 <γ0 < 0. Then k 1/2 { γ (x) γ (x)} = φ 1 ϱ + γ (x)xt H 1 W n (1){K(x)} γ 0 + op (1), where H = E[{K(X)} γ 0 XX T ], W n (1) = lim τ 1 W n (τ) converges to a normal distribution with mean zero and variance E(XX T ), W n (τ) ={n(1 τ)} 1/2 n i=1 x i[τ I{y i x T i θ(τ)}], and ϱ is defined in Lemmas 3 5 in the supplementary material for γ0 > 0, = 0, and < 0, respectively. Theorem 3.2. Under the conditions of Theorem 3.1, if np n = o(k) and log(np n ) = o(k 1/2 ), where p n = 1 τ n, then k 1/2 log{k/(np n )} { Q Y (τ n x) Q Y (τ n x) 1 = φ 1 ϱ + γ (x){k(x)} γ 0 x T H 1 W n (1) + o p (1). 3.3 Selection of the Tuning Parameter k The tuning parameter k balances between bias and variance. A smaller value of k leads to larger variance, while a larger k leads to more bias in the estimation of γ (x). Therefore, how to practically select an appropriate k becomes essential. In univariate extreme value theory, the optimal k, denoted by k 0, is often chosen to minimize the mean squared error of the proposed estimator; see de Haan and Ferreira (2006, p. 77) for more details. In our regression setup, we may choose the optimal k 0 by minimizing the integrated mean squared error of the Hill s estimator γ (x), that is, k 0 = argmin k 1 n } { n k 1 var [ γ (x i )x T i H 1 W n (1){K(x i )} γ ] 0 xi i=1 + Ã2 (n/k) (1 ϱ) 2 }. However, the optimal k 0 depends on the unknown index parameter ϱ and regular varying function Ã( ) involved in the second-order condition, which are difficult to estimate in practice. To bypass this difficulty, we propose a flexible and simple approach to select k. Recall that λγ (x) = γ (x) and λ (Y x) D(G γ (x)). Therefore, using the transformed intermediate quantiles {x T θ(τ n j ; λ),j = [n η ],...,k}, we can estimate γ (x) R by, for example, the moment estimator (de Haan and Ferreira 2006) with γ (x) = M (1) 0,n M (i) 0,n = 1 k [n η ] k j=[n η ] ( 1 (M(1) 0,n )2 M (2) 0,n ) 1 { } log xt θ(τ i n j ; λ) x T θ(τ, i = 1, 2. n k ; λ) If k is selected appropriately, the difference between λ γ (x) and γ (x) should be small. This motivates us to select k by k = argmin D n (k), whered n (k) = k 1 n { λ γ (x i ) γ (x i )} 2. The idea of selecting k by minimizing some discrepancy measure was also employed in Wang and Tsai (2009) in a different context. i=1

7 Wang and Li: Estimation of Extreme Conditional Quantiles Through Power Transformation Testing for the Constancy of Extreme Value Index The proposed method allows γ (x), the EVI of Y x, tovary with x. However, the covariate-dependent EVI estimator γ (x) could be unstable due to lack of information if x lies in a datasparse region or the data have small sample size. In some applications, it might be reasonable to assume that γ (x) is common across x or in some region of x. If the constancy assumption is verified, similar as in Wang, Li, and He (2012), we can then take advantage of this commonality and use the pooled EVI estimator γ p = n 1 n j=1 γ (x j )in(2.7), where γ (x j ) is the proposed EVI estimator based on the covariate value x = x j.usingthe pooled estimator often leads to more stable and efficient conditional quantile estimates when the EVI is indeed constant or varies little across covariates; see Section 4 for some numerical evidence. To help identify the commonality of γ (x), we develop a hypothesis testing procedure. We consider the following hypotheses: H 0 : γ (x) = γ for x X versus H a : γ (x) varies across x X, where γ>0issome unknown quantity. Note that the alternative hypothesis is equivalent to H a :var{γ(x)} 0. Here, we focus on testing the constancy of γ (x) across the entire region of x, but the proposed method can be easily modified to test the local constancy of γ (x)forx B, a subset of X. We define the test statistic as T n = 1 n { γ (x j ) γ p } 2, n j=1 where γ p is the pooled EVI estimator. A larger value of T n will suggest violation of H 0. The following theorem presents the large sample properties of T n under both H 0 and H a. Theorem 3.3. Assume all the conditions of Theorem 3.1 hold. (i) Under H 0 : γ (x) = γ for all x X,wehave kt n = γ 2 W T n (1)H 1 VH 1 W T n (1) + o p(1), where V = cov[{k(x)} γ 0 X], H and Wn (1) are as defined in Theorem 3.1. (ii) Under H a :var{γ (X)} 0, we have kt n in probability. Corollary 3.1. Assume K(X) = 1 a.s. or γ0 = 0, and E(X) = (1, 0,...,0) T. Then, under H 0 and all the conditions d of Theorem 3.1, kt n γ 2 χ 2 (p 1). The implementation of the proposed test involves the calculation of the critical values, the upper quantiles of T n under H 0. Theorem 3.3(i) suggests that the null distribution of T n depends on the unknown function K(X), which captures the model heterogeneity. The special case with K(X) = 1 corresponds to the homogenous case, for instance, the location-shift model as discussed in Chernozhukov (2005) and Wang, Li, and He (2012). The more general case with P {K(X) 1} > 0 corresponds to the heterogenous case. Corollary 3.1 shows that the limiting distribution of T n takes a simple form for special cases with K(X) = 1orγ0 = 0. Therefore, for the homogenous case or when γ0 = 0, we can centralize the explanatory variable X to ensure E(X) = (1, 0,...,0) T, and use the critical value C α,1 = k 1 γ p 2χ α 2(p 1) for a level-α test, where χ α 2 (p 1) is the (1 α)th quantile of χ 2 (p 1). For more general cases, we need employ the result in Theorem 3.3(i) by estimating the unknown function K(X). In Theorems 3.1 and 6.1(iv) of Chernozhukov (2005), it was shown that for γ0 0, K(X) = (ξ T X) 1/γ 0 with ξ T E(X) = 1, and for any l>1, s(x,τ) = xt θ(τ) x T θ{1 l(1 τ)} p x T θ(τ) x T θ{1 ξ T x l(1 τ)} for all x X and γ0 0, where x = n 1 n i=1 x i. Therefore, for γ0 0, we propose to estimate ξ by ξ = ( ξ 1,..., ξ p ) T with ξ j = s(x j,τ), where τ = 1 k/n, and x j is the jth unit basis vector of R p, j = 1,...,p. Throughout our implementations we let l = 2. We can then estimate H by Ĥ = n 1 n i=1 ( ξ T x i ) 1 x i x T i and V by the sample covariance of {x i ( ξ T x i ) 1,i = 1,...,n}, denoted by V.The critical value can be finally obtained through simulation. For each repetition, we calculate t n = k 1 γ p 2ωT Ĥ 1 V Ĥ 1 ω, where ω N(0, ) with = n 1 n i=1 x ix T i. For a level-α test, we calculate the critical value, C α,2,asthe(1 α)th quantile of 5000 replications of t n. 4. SIMULATION STUDY 4.1 Estimation of Extreme Conditional Quantiles We consider six different cases to assess the finite sample performance of the proposed estimator of extreme conditional quantiles. For Cases 1 4, the data are generated from λ (y i ) = x T i β + xt i σ ɛ i,i = 1,...,n, (4.1) where x i = (1,x i1,x i2 ) T, and x ij Uniform( 1, 1) for j = 1, 2. The other components are specified as follows. Case 1: λ = 1, β = (2, 2, 2) T, σ = (2, 1.6, 0) T, and ɛ i Pareto(0.5). Thus, γ (x) = γ (x) = 1. Case 2: the same as Case 1 except that λ = 1/2, so γ (x) = 1 and γ (x) = 0.5. Case 3: λ = 1, β = ( 2, 1, 1) T, σ = (3, 1, 1) T, and ɛ i Uniform(0, 1). Thus, γ (x) = 1 and γ (x) = 1. Case 4: λ = 0, β = (2, 1, 1) T, σ = (0.5, 0.25, 0) T, ɛ i are iid random variables with quantile function Q(τ) = τ 1 log(1 τ) for any 0 <τ<1. This is a special case of Example 2 with d = 1, ϱ = 1, γ (x) = x T σ, and γ (x) = 0. In Cases 5 and 6, we first generate a univariate x i from Uniform( 1, 1), and then generate y i conditional on x i from the Pareto distribution with EVI γ (x i ) = exp( 1 + x i ) in Case 5, and from the Fréchet distribution with distribution function F Y (y x i ) = exp{ y 1/γ (xi) } and γ (x) = 1/2[1/10 + sin{π(x + 1)/2}]{11/10 1/2exp( 16x 2 )} in Case 6. The design of Case 6 is the same as in the simulation study of Daouia et al. (2011) except that the latter focused on the estimation of low quantiles. The EVI γ (x) is a constant in Cases 1 3, while it varies with the covariates in Cases 4 6. In Case 1, the conditional quantile of the untransformed Y is linear in x, so the model assumption required by Wang, Li, and He (2012) is satisfied. In Cases 2 4, the QR Model (2.4) holds after power transformation

8 1068 Journal of the American Statistical Association, September 2013 Table 1. The integrated bias (IBias) and root integrated squared error (RIMSE) of different estimators of Q Y (τ n x) in Cases 1 3 with τ n = 0.99 and IBias RIMSE n = 500 n = 2000 n = 500 n = 2000 Method τ n = Case 1, λ = 1, p = 2, constant EVI QR KER Stage Stage StageP Case 2, λ = 1/2, p = 2, constant EVI QR e e e KER 1.1e e e e e e + 4 2Stage 3.5e e e e e e e + 7 3Stage StageP Case 3, λ = 1, p = 2, constant EVI QR e e e e + 5 KER e e e e + 3 2Stage e e e + 3 3Stage StageP {xt θ(τ n ; λ)},the kernel estimator (KER) of Daouia et al. (2011), the estimator of Wang, Li, and He (2012), and the proposed three-stage estimator based on the covariate-dependent EVI estimator γ (x) (3Stage), and the three-stage estimator based on the pooled EVI estimator γ p (3StageP). The method of Wang, Li, and He (2012)involves two steps, estimating the intermediate conditional quantiles by directly regressing untransformed Y on x, and extrapolating to the extreme tails. Therefore, we refer to the approach developed by Wang, Li, and He (2012) as the two-stage method (2Stage). For the proposed 3Stage and 3StageP methods, we let η = 0.1, ε = 0.15 for n = 500, and 0.1 for n = 2000, τ 1 < <τ m be equally spaced between [1 k/n, 1 n 0.9 ], and we choose k from [20, 70] for n = 500 and from [30, 100] for n = 2000 using the proposed selection method as described in Section 3.3. The bandwidth parameter involved in the KER method is chosen by using the cross-validation method proposed in Daouia et al. (2011). We consider two measurements for comparing different estimators of Q Y (τ n x) at τ n = 0.99 and The first measurement is the integrated bias (IBias) defined as the with positive, negative, and zero λ, respectively. In Cases 5 and 6, the power-transformation Model (2.4) assumed by our proposed method is misspecified since Q Y (τ x) = 1 + (1 τ) γ (x) in Case 5 and Q Y (τ x) ={ log(τ)} γ (x) in Case 6. We consider two sample sizes n = 500 and For each scenario, the simulation is repeated 500 times. We include the following estimators for comparison: the direct QR estimator (via transformation) 1 λ average of n 1 n i=1 { Q Y (τ n x i ) Q Y (τ n x i )} across simulated data. The second measurement is the root integrated mean squared error (RIMSE) defined as the square root of the average of n 1 n i=1 { Q Y (τ n x i ) Q Y (τ n x i )} 2 across simulation. Tables 1 and 2 summarize IBias and RIMSE of different estimators in Cases 1 6 for n = 500 and n = Generally speaking, the direct quantile regression method QR leads to inaccurate estimates at high quantiles especially for n = 500. The proposed 3Stage estimator is more efficient than QR in all the scenarios considered. The 2Stage estimator performs slightly better than the 3Stage estimator in Case 1 where the conditional quantile of Y is linear in x, but the former performs much worse when the linearity assumption is violated for instance in Cases 2 4. By using the pooled EVI estimator, 3StageP clearly improves the efficiency over 3Stage in Cases 1 3 where the EVI is common across covariates. However, when the common EVI assumption is violated in Cases 4 6, 3Stage estimator shows higher efficiency than 3StageP at n = The kernel-based method KER in general performs better than QR, but it has higher RIMSE than the 3Stage estimator in Cases 1 5. Note that Cases 1 4 involve two continuous predictors, and this makes the kernel estimation more challenging in the KER method. In Case 6, the quantile of Y is a sine function of x, and there exists no power transformation that can lead to linear or close to linear quantile functions of x throughout the entire region of x. Consequently, the proposed 3Stage method suffers from the model misspecification and has lower efficiency than the KER method at τ = 0.99, and two methods have similar RIMSE at τ = To better understand the performance of KER and 3Stage, in Figure 1 we plot the true and estimated conditional quantiles of Y against x at τ n = and n = 2000 for two typical examples from Cases 5 and 6, respectively. In Case 5, the conditional quantile function is monotonically increasing in x. For such cases, even though the power-transformed Model (2.4) is violated, the proposed 3Stage estimator still

9 Wang and Li: Estimation of Extreme Conditional Quantiles Through Power Transformation 1069 Table 2. The integrated bias (IBias) and root integrated squared error (RIMSE) of different estimators of Q Y (τ n x) in Cases 4 6 with τ n = 0.99 and IBias RIMSE n = 500 n = 2000 n = 500 n = 2000 Method τ n = Case 4, λ = 0, p = 2, nonconstant EVI QR 3.4e e e e KER e e e e + 3 2Stage 8.6e e e e e e + 6 3Stage e StageP Case 5, p = 1, Pareto distribution with nonconstant EVI QR 8.5e e e e e e + 6 KER Stage Stage StageP Case 6, p = 1, Fréchet distribution with nonconstant EVI QR KER Stage Stage StageP Quantile Case 5 (Example 1) x Case 6 (Example 1) Quantile Case 5 (Example 2) x Case 6 (Example 2) Quantile Quantile x x Figure 1. The true (solid) and estimated conditional quantiles from KER (dotted) and 3Stage (dashed) at τ n = and n = 2000, for two examples from Cases 5 and 6, respectively. The online version of this figure is in color.

10 1070 Journal of the American Statistical Association, September 2013 performs competitively well. In Case 6 where the conditional quantile function is not monotonic of x in the entire region, KER captures the local feature of the quantile function better than 3Stage. However, occasionally, due to the unstable EVI estimation, the KER method gives huge extrapolated values of the high quantiles at some region of x, for instance, at the right tail of x in Example 2 of Case 5 and around x = 0.6 in Example 2 of Case 6. The extreme estimates of KER lead to the large integrated mean squared error of KER in Cases 1 5, and mask the advantage of KER over 3Stage in Case 6. In Tables 1 and 2, some of the RIMSEs of QR, 2Stage, and KER methods are very large. These large RIMSEs are caused by the extremely poor estimates of few simulation datasets, but even if these poor cases are removed, these methods are still worse overall than 3Stage (except KER in Case 6). To obtain a more complete picture, we summarize the quantiles of the integrated squared error of the conditional quantile estimates from different methods in Tables S1 S3 of the supplementary material. In conclusion, the proposed 3Stage method performs competitively well with the 2Stage method when the conditional quantiles of Y are linear in the covariates on the original scale. When the conditional quantiles of Y are nonlinear, we would recommend using the 3Stage method, or the KER method. If the conditional quantiles tend to be roughly monotonic in covariates in different regions, an alternative approach is to apply the 3Stage method in each region separately, and this requires further investigations. 4.2 Testing for the Constancy of Extreme Value Index We now assess the performance of the proposed method for testing the constancy of γ (x). The null hypothesis of constant EVI is satisfied in Cases 1 3 while violated in Cases 4 6. Therefore, we study Cases 1 3 to assess the Type I error of the test procedure and Cases 4 6 for power analysis. We employ two types of critical values, C α,1 for the homogenous cases or cases with γ0 = 0, and C α,2 for the general cases, and we refer to the corresponding tests Homo and Heter, respectively. The proportions of rejections for the two tests (with nominal level α = 0.05) are summarized in Table 3 for n = 500 and n = Table 3 shows that the Type I errors from the Heter test are inflated in Cases 1 2 and deflated in Case 3 for the smaller Table 3. Proportions of rejections for testing the constancy of γ (x). The tests Homo and Heter are based on asymptotic critical values for the homogenous and heterogenous cases, respectively n = 500 n = 2000 Case Homo Heter Homo Heter Case 1 (H 0 ) Case 2 (H 0 ) Case 3 (H 0 ) Case 4 (H 1 ) Case 5 (H 1 ) Case 6 (H 1 ) NOTE: The nominal significance level is The null hypothesis of constant EVI is true in Cases 1 3 and false in Cases 4 6. sample size n = 500, but they get close to the nominal level when n increases to The Homo test is based on the χ 2 reference distribution in Corollary 3.1, and it gives slightly inflated Type I errors in Cases 1 2 and a reasonable Type I error in Case 3 at n = Note that in all three cases, the conditions in Corollary 3.1 are violated. Under the alternative models, the power of both Homo and Heter tests increases as n increases, verifying the theory in Theorem 3.3(ii), but the Heter test appears to be slightly more conservative. 5. ANALYSIS OF HEALTH CARE COSTS Health care costs typically present highly skewed and heavytailed distributions due to the presence of a small proportion of high cost patients. Conventional statistical analysis of health care costs has focused on the mean or total cost (Dominici et al. 2005; Welsh and Zhou 2006). However, the study of the upper quantiles of the health care cost distribution also has practical importance. For example, insurance companies are often interested in predicting the number of customers claiming for a reimbursement above a high threshold. In this section, we analyze a subset of the health care costs data in the Department of Veterans Affairs (Yu et al. 2003) to study the impact of chronic conditions on the high quantiles of medical cost Y. Chronic diseases are the leading causes of death and disability in the United States and therefore are major drivers of heath care costs. The dataset consists of 3658 non-hispanic males aged from 40 to 85 years old. We consider the total cost of each subject in the fiscal year 1999 as the response, and the following covariates: age and indicators for diabetes, hypertension, and asthma. We first estimate the power transformation parameter λ at a set of quantiles ranging from 0.9 to The estimated λ against quantiles (Figure 2) suggest that a common log transformation seems appropriate to ensure the linearity of quantile functions at the upper tails. Note that log transformation was also a common practice in mean analysis of health care costs to deal with skewness and heteroscedasticity (Duan et al. 1983; Welsh and Zhou 2006; Wang and Zhou 2010). To further check the adequacy of Model (2.4) with log transformation, we perform a lack-of-fit test using the method developed by He and Zhu (2003). The p-values of the test at the 10 quantile levels τ = 0.9, 0.91,...,0.99 are 0.375, 0.55, 0.076, 0.08, 0.23, 0.25, 0.70, 0.30, 0.80, and 0.55, respectively, which suggest a reasonable fit of Model (2.4) at upper quantiles. Using the procedure described in Section 3.3, the number of upper order statistics involved in the Hill s estimator, k, is selected to be 86; see Figure 3(a) for the plot of the discrepancy measurement D(k) againstk. Therefore, in the following we carry out the analysis for both 2Stage and 3Stage methods with k = 86. Applying the test procedure presented in Section 3.4, we obtain a p-value of (0.905 based on the simpler χ 2 limiting distribution). This suggests that for this cost data, the EVI tends to be constant across covariates. For illustration, we plot γ (x) against age for the non-hispanic males without any of the three chronic conditions in Figure 3(b). The shaded region corresponds to the pointwise 95% confidence band obtained by bootstrap, where the bootstrap samples are obtained by resampling the subjects with replacement for 400 times. The plot

11 Wang and Li: Estimation of Extreme Conditional Quantiles Through Power Transformation Quantile level τ Figure 2. Estimated power transformation parameter λ against quantile levels. The shaded region corresponds to the pointwise 95% bootstrap confidence band for λ based on 400 bootstrap samples. shows that the estimated EVI has little variation across age. Similar patterns are also observed for other subject profiles. In the sequel, we use the pooled index estimator γ p in estimating the extreme conditional quantiles. For comparison, we include results from the following methods: (i) our proposed 3Stage estimator based on the pooled EVI; (ii) the 2Stage estimator proposed in Wang, Li, and He (2012) based on the pooled EVI; (iii) standard quantile regression method (QR) that fits the linear QR model on the log scale and then transforms the estimated high quantiles back to the original scale; (iv) GPD, where a generalized Pareto distribution with constant shape parameter and scale parameter σ (x) = exp(β 0 + x T β 1 ) is fitted to exceedances over the high threshold, chosen as the 95th percentile of costs. Figure 4 shows the estimated quantile functions against age from four methods at τ = for four profiles: (1) subjects without any of the three chronic conditions; (2) subjects with diabetes; (3) subjects with hypertension; and (4) subjects with asthma. All methods show that given the subjects chronic conditions, the high percentile of medical costs tends to be decreasing with age. The estimated quantile functions from 2Stage and 3Stage methods demonstrate similar patterns, but the former shows a weaker decreasing trend over age for all four profiles. Results from the 3Stage estimator suggest that given the same age, veterans with diabetes are most likely to incur high medical costs among four profiles, followed by those with asthma and hypertension, and those without any chronic conditions are least likely to incur high costs. However, the QR/GPD methods show that those with asthma/hypertension have less chance to incur high costs than those without any chronic conditions, which is counter the common expectation. The same phenomenon is also observed at other quantile levels τ> To assess the performance of different methods for predicting the high conditional quantiles, we carry out a cross-validation study. The data are randomly split into a training set with 731 observations (20% of the original sample) and a testing set with the remaining 2927 observations. We apply each method to analyze the training set and predict the high conditional quantiles of medical cost for subjects in the testing set at high quantiles. Let Q(τ x i ) and Q(τ x i ) denote the estimated and true conditional quantiles of costs for subject i in the test set. Conditional on x i, I{Y <Q(τ x i )} has mean τ and variance τ(1 τ). Therefore, we consider the following prediction error measurement PE ={Mτ(1 τ)} 1/2 M i=1 [I{y i < Q(τ x i )} τ], where {(y i, x i ),i = 1,...,M = 2927} are in the testing set. The cross-validation is repeated 500 times. For each repetition, we apply the proposed Heter test procedure for assessing the constancy of the EVI across covariates based on the training data. Among 500 repetitions, nine have p-values below For both 3Stage and 2Stage methods, we use the covariatedependent EVI estimator for prediction for cases with p-values smaller than 0.05, and the pooled EVI estimator otherwise.