Estimation of High Conditional Quantiles for Heavy- Tailed Distributions

Transcription

1 This article was downloaded by: [North Carolina State University] On: 29 January 2015, At: 04:34 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 High Conditional Quantiles for Heavy- Tailed Distributions Huixia Judy Wang a, Deyuan Li b & Xuming He c a Department of Statistics, North Carolina State University, Raleigh, NC, b Department of Statistics, Fudan University, Shanghai, , China c Department of Statistics, University of Michigan Accepted author version posted online: 12 Sep 2012.Published online: 21 Dec To cite this article: Huixia Judy Wang, Deyuan Li & Xuming He (2012) Estimation of High Conditional Quantiles for Heavy- Tailed Distributions, Journal of the American Statistical Association, 107:500, , DOI: / To lin to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis maes 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 mae 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 Huixia Judy WANG, Deyuan LI, and Xuming HE Estimation of High Conditional Quantiles for Heavy-Tailed Distributions Estimation of conditional quantiles at very high or low tails is of interest in numerous applications. Quantile regression provides a convenient and natural way of quantifying the impact of covariates at different quantiles of a response distribution. However, high tails are often associated with data sparsity, so quantile regression estimation can suffer from high variability at tails especially for heavy-tailed distributions. In this article, we develop new estimation methods for high conditional quantiles by first estimating the intermediate conditional quantiles in a conventional quantile regression framewor and then extrapolating these estimates to the high tails based on reasonable assumptions on tail behaviors. We establish the asymptotic properties of the proposed estimators and demonstrate through simulation studies that the proposed methods enjoy higher accuracy than the conventional quantile regression estimates. In a real application involving statistical downscaling of daily precipitation in the Chicago area, the proposed methods provide more stable results quantifying the chance of heavy precipitation in the area. Supplementary materials for this article are available online. KEY WORDS: Downscaling; Extrapolation; Extreme value; High quantile; Quantile regression. 1. INTRODUCTION An important problem in many fields of modern science is the modeling and prediction of events that are rare but have significant consequences. Examples include heavy rainfall, big financial loss, high medical costs, low birth weights, just to name a few. For such events, we are particularly interested in modeling and estimating the tail quantiles of the underlying distribution (e.g., for rainfall) rather than the averages. Extreme value theory (EVT) provides an elegant mathematical tool for analyzing rare events. The extreme value literature on the tail quantiles focuses primarily on independent and identically distributed random variables, see Weissman (1978); Boos (1984); Deers and de Haan (1989); de Haan and Ferreira (2006); Caeiro and Gomes (2009); Li, Peng, and Yang (2010); You et al. (2010), and references therein. In many applications, however, the tail quantiles of the variable of interest Y depend on some covariate X, and thus it is important to incorporate the covariate information in the analysis. For instance, climatologists are interested in examining how the high quantiles of tropical cyclone intensity change over time (Elsner, Kossin, and Jagger 2008) and how they depend on certain climate variables (Jagger and Elsner 2008). Meteorologists wish to predict localized high precipitations based on global climate model projections (Friederichs and Hense 2007; Friederichs 2010). In the analysis of infant birth weights, researchers are particularly interested in studying the impacts of maternal behaviors on the low quantiles of the birth weight distributions, as low birth weight is nown to be lined to subsequent health problems (Abrevaya 2001). Ris management in finance often sees to forecast the low conditional quantiles of a portfolio s future return, or the 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). Xuming He is Professor, Department of Statistics, University of Michigan. This project is partially supported by the NSF (National Science Foundation) Awards DMS (Division of Mathematical Sciences) , DMS , NSF CAREER Award DMS , National Institute of Health (USA) Grant R01GM , and NNSFC (National Natural Science Foundation of China) grants and Value-at-Ris, conditional on information from the past or assumptions on future interest rate changes (Engle and Manganelli 2004; Schaumburg 2010). Without loss of generality, in this article, we focus on the estimation of conditional high quantiles, because a low quantile of Y can be viewed as a high quantile of Y. To our nowledge, relatively little has been done for the estimation of high conditional quantiles. Beirlant, de Wet, and Goegebeur (2004) suggested a two-step procedure based on the univariate EVT and local quantile regression. Gardes and Girard (2010) and Gardes, Girard, and Leina (2010) proposed a local estimation method, where the estimation of an extreme quantile of Y given X = x is based on observations in a small neighborhood of x. The finite sample behavior of these methods depends critically on the richness of data in local neighborhoods. Other researchers extended the generalized extreme value distribution or generalized Pareto distribution to regression by modeling the parameters of the response distributions as parametric or semiparametric functions of the covariates (Davison and Smith 1990; Beirlant and Goegebeur 2003, 2004; Chavez-Demoulin and Davison 2005; Wang and Tsai 2009). Some of these methods can be used to estimate conditional extreme quantiles, but they have to involve choices of covariate-dependent tuning parameters (e.g., threshold values). Our proposal for conditional high quantile estimation integrates quantile regression and the EVT. Quantile regression has been used for estimating tail quantiles in many different studies, such as precipitation downscaling (Bremnes 2004a; Friederichs and Hense 2007), wind power forecasting (Bremnes 2004b), and Value-at-Ris estimation (Taylor 2008). However, due to data sparsity in the tail areas, estimates from quantile regression are often unstable at tails, especially for heavy-tailed distributions. To estimate the conditional quantiles in the very far tails where few observations are available, an additional model for the tail is needed. In this article, we develop two methods of estimating the high conditional quantiles of heavy-tailed 2012 American Statistical Association Journal of the American Statistical Association December 2012, Vol. 107, No. 500, Theory and Methods DOI: /

3 1454 Journal of the American Statistical Association, December 2012 distributions based on different assumptions on the tail behavior. The new methods operate in the following way. We first use quantile regression to estimate the intermediate conditional quantiles at quantile levels τ 0n, where τ 0n is close to one but not in the extremes (mathematically in the sense that τ 0n 1 and n(1 τ 0n ), where n is the sample size). Then we extrapolate these intermediate conditional quantile estimates to the very high quantile level τ n, which may approach one at an arbitrarily fast rate. Our proposed methods distinguish themselves in three ways. First, typical EVT methods required parametric models to lin the tail index parameters to covariates (Davison and Smith 1990; Beirlant and Goegebeur 2003; Chavez-Demoulin and Davison 2005; Wang and Tsai 2009). In contrast, our approach can lead to covariate-dependent tail index estimations without such a parametric model assumption. Instead, linear conditional quantile functions are assumed, which has been shown to be reasonable in the rich literature on quantile regression. Results from our approach can provide a guideline for choosing appropriate lin functions for the tail index regression (TIR) such as in Wang and Tsai (2009). Second, the EVT methods typically model the exceedances of the response over a constant high threshold, while the thresholds in our methods are covariate adaptive. Third, because our proposed methods use parametric quantile regression models to borrow information across covariates, they achieve better convergence rates and higher efficiency than the nonparametric methods of Beirlant, de Wet, and Goegebeur (2004), Gardes and Girard (2010), Gardes, Girard, and Leina (2010), and Gardes, Guillou, and Schorgen (2012), as confirmed by the numerical results in Section 3. The rest of the article is organized as follows. In Section 2,we present two proposed estimation methods and their asymptotic properties. The first method is based on the assumption of common slopes in the upper quantiles, so that the order statistics of the estimated residuals at a given tail quantile level can be used for extrapolation beyond the data range. The second method relaxes this common-slope assumption by estimating the conditional quantiles at a set of intermediate quantile levels and then extrapolating them to the high end. In Section 3, we conduct a simulation study to assess the finite sample performance of the proposed methods. In Section 4, the value of the proposed methods is demonstrated by an example of downscaling daily precipitation to two Chicago stations. Some concluding remars are made in Section 5. All the technical details are given in the online supplementary file. 2. ESTIMATION OF HIGH CONDITIONAL QUANTILES Suppose we observe a random sample {(y i, x i ),i = 1,...,n} of the random vector (Y, X), where y i is the univariate response variable and x i is the p-dimensional design vector. Let Q Y {τ x} =inf{y : F Y (y x) τ} denote the τth conditional quantile of Y given x, where F Y ( x) is the conditional distribution of Y given x. In practice, direct quantile estimates at the tails are often unstable due to data sparseness, and this motivated us to consider new methods of estimation. Let x be a prespecified covariate vector. Our main objective in this article is to estimate the high conditional quantiles Q Y (τ n x), where τ n 1asn.Hereτ n may approach one at any rate, covering special cases such as the intermediate quantiles with n(1 τ n ) and the extreme quantiles with n(1 τ n ) C, where C is some constant. Throughout the article, we assume that F Y ( x) is in the maximum domain of attraction of an extreme value distribution G γ ( ), denoted by F Y ( x) D(G γ ). That is, for a given random sample Z 1,...,Z n from F Y ( x), there exist constants a n > 0 and b n R such that ( ) max1 i n Z i b n P z G γ (z) = exp{ (1 + γz) 1/γ } a n (2.1) as n,for1+ γz 0, where γ is the extreme value index. The maximum domain of attraction assumption is not very restrictive in the extreme value literature, because it covers a wide range of heavy-tailed distributions. Based on the EVT, we develop two methods for estimating the high conditional quantiles associated with heavy-tail distributions with γ> Estimation Based on Common Slopes Proposed Method. Let 0 <τ c < 1 be a fixed constant possibly close to 1. We first assume the following quantile regression model: Q Y (τ x) = α(τ) + x T β c, for all τ [τ c, 1], (2.2) where α(τ) and β c are unnown quantile coefficients. Model (2.2) assumes that the covariate x has a linear and constant effect on the upper quantiles of Y for τ [τ c, 1]. However, the conditional quantile function Q Y (τ x) is unspecified for τ (0,τ c ). The common slope β c can be estimated by the quantile slope estimate at a single quantile level τ [τ c, 1]. To achieve higher efficiency, we consider estimating β c by the composite estimator β c, defined through ( α1,..., α l, β T ) T c = arg min α1,...,α l,β l j=1 i=1 n ( ρ τj yi α j x T i β), (2.3) where τ c = τ 1 <τ 2 < <τ l = τ U is a sequence of quantile levels with τ c <τ U < 1 and l 1, and ρ τ (u) = u{τ I(u < 0)} is the quantile loss function (Koener 2005). The minimization problem in (2.3) can be solved by existing linear programming algorithms such as the function rq.fit.fnb in R pacage quantreg. The composite estimator β c is obtained by pooling information from multiple quantile levels. This idea appeared in Koener (1984), Koener (2004), and Zou and Yuan (2008). In our implementation, we use equally spaced quantiles τ j = τ c + (j 1)(τ U τ c )/(l 1) for j = 1,...,l. Our empirical studies suggest that in cases when the slopes are nearly constant in the upper quantiles, the composite estimator is often more efficient than the estimator obtained at a single quantile level that exceeds τ c. Suppose that F Y ( x) is continuous and strictly monotone, we define u i ={τ : Q Y (τ x i ) = y i }, that is, the latent quantile level at which the quantile function of Y given X = x i crosses y i.itis then easy to see that u i Uniform (0, 1). Let ê i = y i x T i β c,

4 Wang, Li, and He: Estimation of High Conditional Quantiles 1455 i = 1,...,n. Then ê i can be expressed as { α(ui ) + x T i ê i = (β c β c ), if τ c u i 1, Q Y (u i x i ) x T β (2.4) i c, if 0 u i <τ c. Let ê (1) ê (n) be the order statistics of {ê 1,...,ê n }. Then the upper order statistics of ê i can be shown to be asymptotically equivalent to those of Q Y (u i x = 0). Therefore, we propose to estimate the extreme value index γ by γ = 1 j=1 log ê(n j+1) ê (n ), (2.5) where is an integer such that = n and /n 0as n. The estimator γ is the well-nown Hill estimator (Hill 1975) based on the pseudo order statistics of {ê 1,...,ê n }. Define U Y (t x = 0) = inf{y : F Y (y x = 0) 1 1/t} = F 1 Y (1 1/t x = 0) for t 1, the (1 1/t)th quantile of F Y ( x = 0). For a heavy-tailed distribution F Y ( x = 0), we have U Y (tz x = 0) U Y (t x = 0) zγ, as t. Obviously, α(τ) = U Y {1/(1 τ) x = 0}. Motivated by this, we can estimate α(τ n )by ( 1 ) γ τ0n α(τ n ) = α(τ 0n), (2.6) 1 τ n for any τ n arbitrary close to one, where τ 0n = 1 /n and α(τ 0n ) = ê (n ) is an estimate of α(τ 0n ), the τ 0n th quantile of F Y ( x = 0). Therefore, the τ n th conditional quantile of Y given x can be estimated by Q Y (τ n x) = α(τ n ) + x T β c. (2.7) Asymptotic Properties. To obtain the asymptotic normality of γ, we assume that U Y ( x = 0) satisfies the second order condition indexed by (γ,ϱ,a), that is, there exist γ, ϱ 0, and A(t) RV (ϱ) such that for all z>0, { } A(t) 1 UY (tz x = 0) U Y (t x = 0) zγ z γ (z ϱ 1)/ϱ, as t. (2.8) Here A(t) RV (ϱ) means that A(t) is a regularly varying function with index ϱ, that is, lim t A(tx)/A(t) = x ϱ for all x R +. For more details on the second-order condition, we refer to de Haan and Ferreira (2006). Most commonly used families of continuous distributions satisfy the condition (2.8). For example, the normal distribution satisfies (2.8) with γ = ϱ = 0 and the t distribution with degrees of freedom v satisfies (2.8) with γ = 1/v and ϱ = 2/v. Counter examples, which can be found in Hüsler and Li (2006), include Poisson distribution, and the distribution with U(t) = F 1 (1 1/t) = t γ {γ 1 + λt 1 exp(sin t)} for t 1 and 0 <λ 1/e. For notational simplicity, let F i (y) = F Y (y x i ) be the conditional distribution function of Y given x i, f i be the corresponding density function, and D = diag ( f i {Q Y (τ j x i )},i = 1,...,n,j = 1,...,l ). We assume the following conditions. (A1) The distribution functions {F i : i = 1, 2,...,n} are absolutely continuous, and their densities f i are continuous, and bounded away from zero and infinity uniformly over i in an interval that contains Q Y (τ x i )for all τ [τ c,τ U ], where τ c <τ U < 1. (A2) (a) As n, n 1 X T D X 1 in probability, where is some positive-definite matrix, X = (I l 1n, 1 l Xn ), X n = (x T 1,...,xT n )T, 1 n is a n 1 vector of ones, and I l is the l l identity matrix; (b) max 1 i n x i =O p (n 1/2 δ ) with some δ>0. Theorem 1. Suppose that Model (2.2) and conditions A1 and A2 hold. Let be a sequence such that = n and n 1 0asn. In addition, suppose that the second-order condition (2.8) holds with γ>0 and ϱ<0, A(n/) λ R and γ +1 n (γ +δ) 0. Then ( γ γ ) d N ( λ 1 ϱ,γ2 ). Let p n = 1 τ n. Then (2.6) is equivalent to { 1 (1 /n) } γ ( α(τ n ) = α(1 /n) = 1 τ n np n ) γ ê (n ). (2.9) Theorem 2 and Proposition 1 present the asymptotic normality of α(τ n ) and Q Y (τ n x), respectively. In this article, we use a n b n (b n 0) to mean a n /b n 1asn. Theorem 2. Under the conditions of Theorem 1 and assuming that np n = o() and log(np n ) = o( ), we have } ( ) { α(τn ) log{/(np n )} α(τ n ) 1 d λ N 1 ϱ,γ2. Proposition 1. Under the same assumptions as in Theorem 2, for any x with x =O p (n 1/2 δ ), if [log{/(np n )}] 1 pn γ n δ 0, we have log{/(np n )}α(τ n ) { Q Y (τ n x) Q Y (τ n x)} d N ( λ 1 ϱ,γ2 Moreover, if n 1/2 δ pn γ 0, then α(τ n ) Q Y (τ n x) asτ n 1 and hence { } ( ) Q Y (τ n x) log{/(np n )} Q Y (τ n x) 1 d λ N 1 ϱ,γ Estimation Without Common-Slope Assumptions Proposed Method. In Section 2.1, we assume that the quantile slope coefficients are common at the upper quantiles. To relax this common-slope assumption, we focus on the following linear quantile regression model: ). Q Y (τ x) = α(τ) + x T β(τ), for all τ [τ c, 1], (2.10) where the quantile slope coefficients β(τ) may vary across τ [τ c, 1].

5 1456 Journal of the American Statistical Association, December 2012 Define a sequence of quantile levels τ c <τ n <τ n +1 < <τ m (0, 1), where m = n [n η ] with [a] denoting the integer part of a, η>0 is some small constant, and τ j = j/(n + 1). Here we assume = n and /n 0asn, and η>0 such that n η <. For each j = n,..., m, we define ( α(τj ), β(τ j ) T ) T = arg minα,β n ( ρ τj yi α x T i β). (2.11) For a given x, define q j = α(τ j ) + x T β(τ j ), j = n,..., m. The quantities q j can be roughly regarded as the upper order statistics of a sample from F Y ( x). Recall in (2.1) (see also assumption B2 in Section 2.2.2) that we assume F Y ( x) tobein the maximum domain of attraction of an extreme value distribution G γ with γ>0. Therefore, one possible estimator for γ is γ = 1 [n η ] i=1 j=[n η ] Consequently, Q Y (τ n x) can be estimated by ( 1 ) γ τn Q Y (τ n x) = q n. 1 τ n log q n j q n. (2.12) Remar 1. The estimate γ is based on the [n η ] upper quantiles q n,...,q m = q n [n η ]. The trimming of the extreme upper quantile is used to obtain the Bahadur representation of the estimated quantile coefficients ( α(τ j ), β(τ j ) T ) T, which is needed for establishing the asymptotic normality of γ. The asymptotic results hold for any choice of m such that m/n < 1 ɛ n, where ɛ n is a sequence of numbers with ɛ n 0, nɛ n <, and nɛ n as n. In our numerical studies, we choose ɛ n = n 1+η with η = 0.1. Our experience shows that small values of η (0, 0.2) wor well. In the extreme value literature, the selection of is an important and challenging problem. The value can be viewed as the effective sample size for tail extrapolation. A smaller leads to estimators with larger variance, while larger results in more bias. In practice, a commonly used heuristic approach for choosing is to plot the estimate of γ versus and then choose a suitable corresponding to the first stable part of the plot, see section 3 in de Haan and Ferreira (2006) and references therein. We employ this approach for choosing in our data analysis, see Figures 1 and 2 in Section Asymptotic Properties. Let Z = (1, X T ) T, z i = (1, x i ) T, θ(τ) = (α(τ), β(τ) T ) T, and μ z = E(Z). We now mae the following assumptions: (B1) The variable z i has a compact support Z, and E(ZZ T ) is positive definite. (B2) There exists an auxiliary line z z T θ(r) with 0 < r<1 and θ(r) a bounded vector such that for Y = Y z T θ(r) and some heavy-tailed distribution function F 0 ( ) with extreme value index γ>0, 1 F Y (t z) K(z){1 F 0 (t)} 1 ={1 F 0(t)} δ K(z){1 + o(1)}, (2.13) uniformly in z Z as t, where K( ) and K are positive, continuous, and bounded functions on Z and δ>0isaconstant. (B3) (a) τ F 1 Y (τ z) τ F 1 0 {τ/k(z)} uniformly in z Z as τ 1; (b) τ F 1 0 (1 τ) is regularly varying at zero with index γ 1. (B4) U 0 (t) = F 1 0 (1 1/t) satisfies the second-order condition (2.8) with γ>0, ϱ<0, and A(t) = γdt ϱ with d 0. Condition B2 requires some form of equivalency between 1 F Y (t z) and 1 F 0 (t) at the right tails. Under conditions B1 and B2, Theorem 3.1 of Chernozhuov (2005) showed that K( ) taes the following form: K(z) = (z T w) 1/γ for some w R p+1 such that μ T z w = 1 and zt w > 0 for all z Z. Condition B3(b) is a von Mises type condition. The von Mises condition is a very basic condition for the distribution belonging to a maximum domain of attraction, and it is satisfied by many commonly used distributions such as Normal, t, Gamma, Pareto, Beta, see de Haan and Ferreira (2006) for more discussion. Example (location-scale shift regression). The regularity conditions cover most conventional regression settings, including the following location-scale shift model: Y = α + x T β + (1 + x T σ )ɛ, (2.14) where 1 + x T σ > 0forx in its domain, and ɛ F 0 ( ), a heavytailed distribution that satisfies B4 and has a continuous density function. For identifiability, we assume that F 0 ( ) has median zero. By writing z = (1, x T ) T, z T σ = 1 + x T σ, θ = (α, β T ) T, and Y = Y z T θ, we have F Y (y z) = F 0 {y/(z T σ )}. By (S.11) in the online supplementary file and the Taylor expansion, 1 F Y (y z) = 1 F 0{y(z T σ ) 1 } 1 F 0 (y) 1 F 0 (y) = {y(zt σ ) 1 /c} 1/γ [1 + dϱ 1 {y(z T σ ) 1 /c} ϱ/γ {1 + o(1)}] (y/c) 1/γ [1 + dϱ 1 (y/c) ϱ/γ {1 + o(1)}] = (z T σ ) 1/γ [1 + d ϱ {(zt σ ) ϱ/γ 1} ( y c ) ϱ/γ {1 + o(1)} ], where c is a positive constant defined in (S.10) of the online supplementary file. Let K(z) = (z T σ ) 1/γ and K(z) = d ϱ {(zt σ ) ϱ/γ 1}. Then 1 F Y (y z) ( y ) ϱ/γ K(z){1 F 0 (y)} 1 = K(z) {1 + o(1)}. c Since 1 F 0 (y) (y/c) 1/γ,(2.13) in condition B2 holds with δ = ϱ. In addition, it follows from F Y (y z) = F 0 {y/(z T σ )} that U Y (t z) = (z T σ )U 0 (t) and thus condition B3(a) holds. In general, F 0 in condition B2 may differ from F Y ( x = 0), the conditional distribution of Y given x = 0. However, for the location-scale shift model (2.14), the two are equivalent except for a constant shift. More specifically, F Y (y x = 0) = F 0 (y α) for any y. We now provide the asymptotic properties of the proposed estimators without the common-slope assumption. Theorem 3. Suppose that Model (2.10) and conditions B1 B4 hold, and suppose that as n, = n

6 Wang, Li, and He: Estimation of High Conditional Quantiles 1457, /n 0, 1/2 n η log 0, and (n/) ϱ C 0, where ϱ = max( γ, δ, ϱ) and C is a constant. Then λ ( γ γ ) = 1 ϱ + γ zt H 1 W n (1){K(z)} γ + o p (1), where H = E[{K(Z)} γ ZZ T ], W n (1) = lim τ 1 W n (τ) converges to a normal distribution with mean zero and variance E(ZZ T ), W n (τ) ={n(1 τ)} 1/2 n i=1 Z i[τ I{Y i Z T i θ(τ)}], and λ = lim n Ã(n/) with Ã( ) defined in (S.14). Theorem 4. Under the same conditions of Theorem 3, if np n = o() and log(np n ) = o( ), where p n = 1 τ n, then { } Q Y (τ n z) log{/(np n )} Q Y (τ n z) 1 = λ 1 ϱ + γ zt H 1 W n (1){K(z)} γ + o p (1). Remar 2. If we impose a stronger assumption (n/) ϱ 0, then λ = 0 and the asymptotic bias of γ reduces to zero. The statements of Theorems 3 and 4 then become ( γ γ ) = γ z T H 1 W n (1){K(z)} γ + o p (1) and log{/(np n )} respectively. { } Q Y (τ n z) Q Y (τ n z) 1 = γ z T H 1 W n (1){K(z)} γ + o p (1), The difference of the τ n th quantile at two points has a linear form Q Y (τ n z 1 ) Q Y (τ n z 2 ) = (z 1 z 2 ) T θ(τ n ) = (x 1 x 2 ) T β(τ n ). It measures the impact of X on the τ n th conditional quantile of Y. Its estimates does not have the linear form. However, we have the following result. Theorem 5. Assume that all the conditions in Theorem 4 hold. Then γ pn log{/(np n )} {[ Q Y (τ n z 1 ) Q Y (τ n z 2 )] [(x 1 x 2 ) T β(τ n )]} = c γ { z T 1 H 1 {K(z 1 )} γ z T 2 H 1 {K(z 2 )} γ } W n (1) + o p (1), where c = c{k(z)} γ with c defined in (S.10) of the online supplementary file Pooling of Index Estimates. In condition B2, we assume that the conditional distributions F Y ( x) are tail equivalent across covariate values x. That is, F Y ( x) D(G γ ) with the same extreme value index for different x. This condition was imposed mainly for technical reasons to obtain the Bahadur representation of the estimated quantile coefficients at tails, which are needed to establish the normality of the extreme value index estimator. Under the assumption B2, we can estimate the common γ consistently by using the conditional quantile estimates at any given x, for instance, at x = n 1 n i=1 x i, or by the pooled γ estimate γ pool = n 1 n j=1 γ (x j ), where γ (x j ) is the proposed index estimator based on the covariate value x = x j. Under the same conditions assumed in Theorem 3, by applying the law of large numbers and the similar arguments as used in proving Theorem 3, we have ( γpool γ ) = λ 1 ϱ + γn 1 n z T j H 1 W n (1){K(z j )} γ + o p (1) j=1 = λ 1 ϱ + γe[{k(z)} γ Z T ]H 1 W n (1) + o p (1). (2.15) On the other hand, when the common index assumption is violated, γ (x) is expected to vary with x. Therefore, examining γ (x) across different x values provides a diagnostic tool for checing the common index assumption, see Figures 1 and 2 in Section 4. In TIR, for instance Wang and Tsai (2009), the tail index is often assumed to be linear in x after some parametric lin transformation. Our estimator γ (x) can be viewed as a nonparametric estimation of γ (x) and thus can provide some guidance for the choice of lin function required in TIR. For a special case of the location-scale shift model (2.14) with σ = 0, referred to as location shift model, the quantile slope β(τ) is a constant for all τ. In this case, K(z) = 1 and K(z) = 0 for any given z = (1, x T ) T Z. Assuming that conditions in Theorem 3 hold and that (n/) ϱ C with C some constant, we have ( γ (x) γ ) d N ( λ1 1 ϱ,γ2 z T H 1 z ), Ã(n/). where ϱ = max( γ,ϱ), H = E(ZZ T ), and λ 1 = lim n Without loss of generality, assume that E(Z) = (1, 0,...,0) T. Then by (2.15), ( γpool γ ) d N ( λ1 1 ϱ,γ2 For the common-slope-based estimator γ,leta Y x=0( )bethe second-order function corresponding to U Y ( x = 0), see (2.8). Then under the condition that A Y x=0 (n/) λ 2 and other conditions in Theorem 1, ( d λ 2 ( γ γ ) N,γ ), 2 1 ϱ Y x=0 where ϱ Y x=0 is the regular varying index of A Y x=0( ). By F Y (y x = 0) = F 0 (y α), it is easy to chec that ϱ = ϱ Y x=0 and Ã(t) = A Y x=0(t). Thus λ 1 = λ 2, and γ pool and γ have the same asymptotic distribution in the location shift regression. 3. SIMULATION STUDY We conduct a simulation study to investigate the performance of the proposed methods and demonstrate that the proposed methods improve on the usual quantile regression estimates at high tails. The data are generated from the following model: ). y i = x i1 + x i2 + (1 + rx i1 )e i,i = 1,...,n, (3.1) where x ij Uniform( 1, 1), j = 1, 2, e i are independent and identically distributed random variables, and r is a constant controlling the degree of heteroscedasticity. Therefore, the τ th conditional quantile of Y is Q Y (τ x i ) = α(τ) + x T i β(τ),

7 1458 Journal of the American Statistical Association, December 2012 where x i = (x i1,x i2 ) T, α(τ) = Q e (τ), β(τ) = (1 + rq e (τ), 1) T, and Q e (τ) is the τth quantile of e i. We consider three sample sizes n = 200, 500, 1000 and two r values r = 0 and 0.9. The slope coefficients β(τ) are constant in homoscedastic models with r = 0 and they vary across τ in heteroscedastic models with r 0. We use three different cases for generating e i. In Cases 1 and 2, e i are from the Pareto distribution with extreme value index parameters γ = 0.2 and 0.5, respectively. In Case 3, e i are from the t(1) distribution so that the index parameter γ = 1. For each simulated dataset, we apply the proposed methods to estimate Q Y (τ n x) atτ n = and The size of the Monte Carlo datasets is 500 for each scenario. We compare four estimation methods: the usual quantile regression method (RQ), the local estimation method (LOC) of Gardes, Girard, and Leina (2010), the proposed method assuming a common slope on the upper quantiles (CS), and the proposed method without assuming common slopes (NCS). At a given x = (x 1,x 2 ) T,the LOC approach estimates Q Y (τ n x) by extrapolation from a local quantile estimate using the local Hill estimator of γ, which is obtained by using only those cases whose x ij are within distance h of x j for j = 1, 2, respectively, where h is a bandwidth parameter. The number of upper order statistics used in estimating γ ], where n x is the number of observations in the neighborhood of x. We give the LOC method some advantage by choosing h {0.05, 0.1, 0.15,...,0.8} that gives the smallest mean squared error (MSE) of Q Y {0.991 x = (0, 0) T } in each scenario. For NCS, we let η = 0.1 so that m = n [n 0.1 ]. For CS, we set τ c = 0.9 and obtain the composite slope estimator β c by minimizing the combined quantile objective functions at τ 1,...,τ 10 that are equally spaced between [τ c,τ U ]. For comparison, we include two variations of CS, referred to as CS1 and CS2, corresponding to τ U = 0.95 and τ U = 0.99, respectively. For both CS and NCS methods, we let = [cn 1/3 ] with c = 4.5. Our numerical investigation suggests that the index estimation becomes stable around this choice of and the proposed methods clearly outperform RQ for any c [3, 10]. We first tae a quic loo at the asymptotic variance comparison in this simulation setting with r = 0. By the asymptotic theory, the variance ratio of the τ n th conditional quantile estimate at x for LOC versus CS is approximated by var(loc)/var(cs) [{log( x /(n x p n ))} 2 ]/[ x {log(/(np n ))} 2 ], where p n = 1 τ n is set as [4.5n 1/3 x and x = [cn 1/3 x ] with c = 4.5. Therefore, as n, the ratio var(loc)/var(cs) > 1 as long as n x <n, which is always the case for the local method. More specifically, if n x = o(n), then the ratio goes to infinity. Similarly, we have var(ncs)/var(cs) =1 + x1 2/3 + x2 2 /3 1, which shows the efficiency loss of not maing the common-slope assumption in this case. On the other hand, we note that the asymptotic variance comparisons are not fully informative for the estimation of high quantiles, because in the data sparse areas, both bias and variance are important. We focus on the comparisons of the MSE in finite sample simulations in this article. Tables 1 3 summarize the MSE of different estimators of Q Y (τ n x) in different scenarios at x = (0, 0) T or x = (0.5, 0.5) T. The values in the parentheses are the standard errors of the MSE. The MSE results suggest that the proposed methods are quite insensitive to the choice of τ U. The two variations CS1 and CS2 perform very similarly in all scenarios. In most cases, the pro- Table 1. Mean squared errors 100 (standard errors) of different estimators of Q Y (τ n x) in Case 1 with Pareto(0.2) errors, where τ n = and x = (0, 0) T x = (0.5, 0.5) T r Method n = RQ 25.2 (3.3) 47.2 (7.9) 36.7(3.6) 62.9(7.5) LOC 66.1 (3.5) 96.4 (5.8) 31.2(2.4) 42.5(3.6) CS1 8.6 (0.5) 15.5 (1.0) 9.1(0.6) 15.9(1.0) CS2 8.7 (0.6) 15.6 (1.0) 9.3(0.6) 16.1(1.1) NCS 10.9 (1.1) 20.2 (2.1) 19.1(1.6) 33.4(3.0) 0.9 RQ 32.4 (4.1) 55.7 (6.3) 65.7(8.4) (11.9) LOC (12.8) (22.5) 126.3(8.9) (14.2) CS (0.5) 20.6 (0.9) 46.6(1.4) 88.4(2.4) CS (0.5) 21.6 (0.9) 46.1(1.4) 89.8(2.4) NCS 11.8 (1.0) 22.0 (2.1) 29.0(2.1) 51.4(3.8) n = RQ 7.7 (0.7) 18.9 (1.9) 16.1(1.4) 38.5(4.2) LOC 20.5 (1.4) 29.8 (2.3) 15.1(1.3) 20.1(1.8) CS1 4.0 (0.3) 7.8(0.6) 4.2(0.3) 8.0(0.5) CS2 4.0 (0.3) 7.8(0.6) 4.3(0.3) 8.1(0.6) NCS 5.1 (0.4) 10.3 (0.9) 10.2(0.8) 19.2(1.5) 0.9 RQ 9.3 (1.2) 19.7 (2.0) 25.5(3.0) 52.1(4.5) LOC 87.4 (6.7) (11.9) 59.4(5.4) 87.3(8.5) CS1 4.4 (0.3) 8.5(0.6) 23.8(0.8) 42.6(1.5) CS2 4.2 (0.3) 8.1(0.5) 24.2(0.8) 45.1(1.5) NCS 5.8 (0.5) 11.6 (1.1) 15.5(1.1) 29.7(2.3) n = RQ 3.2 (0.2) 8.4(0.9) 6.8(0.5) 18.1(1.8) LOC 9.2 (0.6) 11.6 (0.9) 9.1(0.6) 10.6(0.8) CS1 2.0 (0.1) 4.2(0.3) 2.2(0.1) 4.4(0.3) CS2 2.0 (0.1) 4.1(0.3) 2.2(0.1) 4.4(0.3) NCS 2.3 (0.1) 4.8(0.3) 4.8(0.3) 9.4(0.6) 0.9 RQ 3.6 (0.2) 10.8 (1.2) 9.5(0.6) 26.4(2.6) LOC 33.9 (2.3) 50.2 (3.7) 43.4(4.8) 66.7(7.9) CS1 2.7 (0.2) 6.5(0.4) 16.8(0.6) 28.0(1.0) CS2 2.6 (0.1) 5.9(0.4) 16.5(0.6) 28.6(1.0) NCS 2.8 (0.2) 5.9(0.3) 7.2(0.4) 14.6(0.8) NOTE: RQ is the conventional quantile regression method. LOC is the local estimation method. CS1 and CS2 are two variations of the proposed common-slope-based estimation method. NCS is the proposed method assuming noncommon slopes. posed CS and NCS methods provide more efficient estimation than RQ. Even though the CS method assumes common-slope coefficients in the right tail, it performs competitively well in the heteroscedastic model where this common-slope assumption is violated. In Case 1 with lighter tails, CS is more efficient than NCS in homoscedastic models with r = 0, while NCS performs slightly better in heteroscedastic models with r = 0.9 especially for the estimation at x = (0.5, 0.5) T. For distributions with heavier tails (Cases 2 and 3), the RQ estimates are clearly unstable even at n = The NCS estimates have larger MSE than CS for the sample size n = 200 due to unstable estimation of the intermediate conditional quantiles, but the differences between NCS and CS estimates start to shrin as the sample size increases. The local estimation method is very sensitive to the choice of bandwidth. Even with the bandwidth chosen in favor of LOC, the LOC estimates are overall less efficient than the CS

8 Wang, Li, and He: Estimation of High Conditional Quantiles 1459 Table 2. Mean squared errors (standard errors) of different estimators of Q Y (τ n x) in Case 2 with Pareto(0.5) errors, where τ n = and x = (0, 0) T x = (0.5, 0.5) T r Method n = RQ 76.8 (13.5) (201.0) 77.0(12.9) 274.5(70.4) LOC 17.9 (2.3) 47.2(6.6) 17.0(2.5) 41.1(6.7) CS1 8.8 (0.7) 22.3(1.8) 8.8(0.7) 22.2(1.8) CS2 8.8 (0.7) 22.1(1.7) 8.8(0.7) 22.0(1.7) NCS 17.9 (3.4) 54.1(13.0) 25.8(5.2) 73.1(19.4) 0.9 RQ 99.6 (22.2) (91.3) 146.5(38.0) 367.2(81.6) LOC 22.7 (2.9) 56.2(7.9) 35.2(5.3) 87.0(14.9) CS1 9.6 (0.5) 25.2(1.3) 33.1(0.9) 83.0(2.1) CS2 9.7 (0.4) 25.9(1.1) 33.3(0.9) 85.3(2.0) NCS 17.8 (2.8) 53.9(10.6) 34.4(4.7) 92.6(15.9) n = RQ 12.4 (1.7) 55.3(9.5) 21.2(3.4) 88.8(18.4) LOC 7.8 (0.7) 21.1(2.0) 9.4(0.7) 23.5(1.9) CS1 4.5 (0.4) 13.1(1.4) 4.5(0.4) 13.1(1.3) CS2 4.5 (0.4) 12.9(1.4) 4.5(0.4) 12.9(1.4) NCS 7.1 (0.7) 22.0(2.6) 12.1(1.3) 34.8(4.3) 0.9 RQ 15.7 (3.9) 48.2(7.7) 37.7(9.1) 101.8(14.7) LOC 11.7 (1.0) 28.2(2.7) 18.5(1.3) 46.7(3.5) CS1 4.3 (0.3) 12.1(1.1) 16.4(0.6) 38.6(1.5) CS2 4.1 (0.3) 11.2(0.8) 16.9(0.6) 41.2(1.4) NCS 7.4 (0.9) 23.1(3.8) 17.6(1.9) 50.9(7.6) n = RQ 4.1 (0.4) 19.9(3.4) 7.8(0.7) 34.3(4.9) LOC 3.8 (0.3) 10.6(0.9) 5.2(0.5) 13.8(1.4) CS1 2.2 (0.1) 6.7(0.5) 2.3(0.2) 6.8(0.5) CS2 2.2 (0.1) 6.6(0.5) 2.2(0.1) 6.7(0.5) NCS 2.7 (0.2) 8.6(0.6) 5.1(0.4) 14.6(1.2) 0.9 RQ 4.2 (0.4) 27.3(5.8) 10.4(0.8) 55.8(8.5) LOC 7.1 (0.6) 17.1(1.8) 10.0(0.8) 26.6(2.3) CS1 2.5 (0.2) 8.2(0.6) 12.0(0.4) 26.3(1.0) CS2 2.4 (0.1) 7.7(0.5) 11.9(0.4) 26.7(1.0) NCS 3.2 (0.2) 9.8(0.7) 7.5(0.4) 21.7(1.2) NOTE: RQ is the conventional quantile regression method. LOC is the local estimation method. CS1 and CS2 are two variations of the proposed common-slope-based estimation method. NCS is the proposed method assuming noncommon slopes. estimates due to the smaller effective sample size involved in the local estimation of the tail index. For the NCS method, the extreme value index γ is estimated by using the estimated quantiles q j = α(τ j ) + x T β(τ j ), j = n,...,m. In finite samples, q j may not be monotonically increasing due to quantile crossing. To account for this possible nonmonotonicity, one may first rearrange q j following the idea of Chernozhuov, Fernández-Val, and Galichon (2010) and then estimate γ by using the order statistics of the rearranged quantile estimates q j that are monotonically increasing. Numerical study shows that the two variations of NCS based on unsorted and sorted q j have almost identical performance. Our empirical studies suggest that the proposed method based on the assumption of a common slope at the upper tail is a good choice for small-to-moderate sample problems, even if the common-slope assumption is violated to some extent. The method without the common-slope assumption should be used in larger sample problems, especially when the data exhibit clear heteroscedasticity in the tail. Our estimates are in general better, and sometimes much better, than LOC, which is consistent with what we learn from the asymptotic theory. 4. APPLICATION TO DOWNSCALING OF PRECIPITATION In this section, we apply the proposed extrapolation methods to statistical downscaling of daily precipitations in Chicago urban areas. Statistical downscaling aims to derive a statistical relationship between a global climate model output and the local observations. Our particular interest here is prediction of high conditional quantiles of local precipitation based on the coarser-resolution predictor variables generated from a global climate model. For illustration, we demonstrate the analysis for two stations in Aurora and Midway airport. The response variable Y is the observed daily precipitation (inch) at the station from

9 1460 Journal of the American Statistical Association, December 2012 Table 3. Mean squared errors 10 2 (standard errors) of different estimators of Q Y (τ n x)incase3 with t(1) errors, where τ n = and x = (0, 0) T x = (0.5, 0.5) T r Method n = RQ 332.7(95.5) ( ) (211.7) ( ) LOC 20.5(5.0) (40.2) 30.9(14.4) (135.3) CS1 4.1(0.5) 20.4 (3.0) 4.1(0.5) 20.4 (3.0) CS2 3.6(0.5) 17.1 (2.5) 3.6(0.5) 17.1 (2.6) NCS 27.0(5.5) (50.9) 30.0(8.4) (71.5) 0.9 RQ (278.5) ( ) (648.5) ( ) LOC 36.0(7.8) (70.8) 103.9(48.8) (524.7) CS1 3.5(0.4) 15.4 (2.1) 8.3(0.4) 34.7 (1.8) CS2 3.3(0.3) 14.2 (1.2) 8.3(0.3) 34.7 (1.1) NCS 32.4(7.9) (81.1) 36.5(10.0) (83.4) n = RQ 21.7(5.1) (633.4) 33.5(10.7) (1433.9) LOC 5.9(0.9) 35.3 (5.9) 7.1(1.3) 37.6 (8.0) CS1 2.7(0.3) 15.6 (1.9) 2.7(0.3) 15.5 (1.9) CS2 2.6(0.3) 14.4 (1.8) 2.6(0.3) 14.4 (1.8) NCS 4.8(0.5) 30.1 (3.9) 5.9(0.8) 33.8 (5.0) 0.9 RQ 22.9(9.3) (400.8) 45.7(20.7) (902.9) LOC 5.1(0.7) 29.4 (4.7) 13.9(2.5) 78.6 (16.9) CS1 2.2(0.2) 12.4 (1.5) 4.5(0.2) 19.4 (1.1) CS2 2.0(0.2) 10.7 (1.3) 4.7(0.2) 19.9 (1.0) NCS 4.7(0.6) 29.8 (4.2) 8.2(1.1) 45.2 (7.0) n = RQ 3.6(0.5) 33.6 (6.2) 4.7(0.6) 49.0 (11.8) LOC 2.3(0.2) 13.1 (1.5) 3.1(0.4) 16.0 (2.4) CS1 1.1(0.1) 6.5 (0.7) 1.1(0.1) 6.5(0.7) CS2 1.1(0.1) 6.3 (0.6) 1.1(0.1) 6.3(0.6) NCS 1.6(0.2) 9.7 (1.2) 2.3(0.2) 13.0 (1.6) 0.9 RQ 2.9(0.4) 34.6 (5.6) 5.5(0.6) 52.9 (7.8) LOC 2.1(0.2) 11.8 (1.4) 5.3(0.6) 28.9 (4.2) CS1 1.0(0.1) 6.1 (0.7) 3.1(0.1) 11.8 (0.5) CS2 1.0(0.1) 5.9 (0.7) 3.1(0.1) 11.9 (0.5) NCS 1.5(0.2) 9.1 (1.2) 3.0(0.3) 16.4 (1.8) NOTE: RQ is the conventional quantile regression method. LOC is the local estimation method. CS1 and CS2 are two variations of the proposed common-slope-based estimation method. NCS is the proposed method assuming noncommon slopes. It is well nown that precipitation and temperature have a nonlinear association in most locations. If downscaling is needed for the joint distribution of precipitation and temperature, we refer to He, Yang, and Zhang (2012) for relevant downscaling methods. However, for downscaling of precipitation, the additional benefit of including temperature as a linear predictor is quite insignificant. In this article, we focus on precipitation only, and we tae the predictor X as the simulated daily precipitation from the ERA-40 reanalysis model introduced in Uppala et al. (2005). Since we are interested in predicting high quantiles of precipitation based on X, we consider only the wet days. However, Y is not available and cannot be used to define wet days in the future. A common approach in climate studies is to assume that the percentage of wet days in the future will be the same as that in the past. Therefore, we define wet days as those with X exceeding its 70th sample percentile, resulting in 4816 observations at the Aurora station and 4909 observations at the Midway station. This definition of wet days would just match the common notion of Y>0for the past, but allow us to predict precipitation on wet days in the future. We consider the quantile regression model Q Y (τ X = x) = α(τ) + xβ(τ), (4.1) and we focus on the high quantiles τ = 0.99, 0.995, 0.999, and , where in this example τ = is for events that occur once in about 100 years. 4.1 Aurora Station Figure 1(a) gives the scatterplot, and the fitted RQ lines at four high quantiles. The linear quantile regression appears to provide a decent fit at τ = 0.99 and However, at the very high quantile τ = and , the regression fit is highly influenced by the single extreme event corresponding to x = We first implement the CS method that assumes a common slope at upper quantiles. The common slope is estimated by

10 Wang, Li, and He: Estimation of High Conditional Quantiles 1461 Daily precipitation (Y) (a) Re analysis variable (X) τ=0.999 τ= τ=0.995 τ=0.99 β^(τ) (b) τ (c) (d) γ~ γ^ (x) (e) NCS TIR x γ^ (x=0.01) se of γ^tir (x) TIR (f) x Figure 1. Aurora station. (a) Daily precipitation Y versus the simulated precipitation X and the fitted quantile regression lines at τ = 0.99, 0.995, 0.999, and (b) The RQ estimates of β(τ) atτ [0.95, 0.99] and the composite-slope estimate β c (dashed horizontal line). (c) The index estimates γ for the CS method versus. (d) The index estimates γ (x) from the NCS method versus at x = (e) The index estimates γ (x) from NCS and TIR (the tail index regression method of Wang and Tsai (2009))versusx. (f) The standard error of γ (x) from TIR. The online version of this figure is in color. the composite estimator β c as in (2.3) with τ 1,...,τ 10 being 10 quantile levels equally spaced between 0.95 and 0.99, but the result is stable over variations of the grid choice in τ. Figure 1(b) plots the RQ slope estimates β(τ j ) obtained at each quantile level τ j. The shaded areas correspond to the 95% pointwise confidence band of β(τ) obtained by the inversion of ran score test (Koener 2005). The horizontal line corresponds to β c.the plot shows that β(τ) has little variation across τ [0.95, 0.99]. Furthermore, we used the analysis of variance (ANOVA) test of Koener and Bassett (1982) for testing the null hypothesis of common slopes at τ j,j = 1,...,10. The test yields a p- value of This suggests that the assumption of common slope at upper quantiles is reasonable for this dataset. Following the procedure described in Section 2.1, we obtain the extreme value index estimator γ based on the upper order statistics of the residuals ê i = y i (x i x) β c, where x = n 1 n i=1 x i. For the NCS method, we let τ n <τ n +1 < <τ m, where τ j = j/(n + 1). We choose m such that τ m = 0.995, at which the conventional quantile regression seems to produce reasonable estimates as shown in Figure 1(a). Let x be a prespecified predictor value of interest. We define the covariate-adaptive index estimator as γ (x) = 1 ν j=ν+1 log { qn j+1 q n where ν = n m, q j = α(τ j ) + x β(τ j ) is the estimated conditional quantile at the τ j th quantile obtained by the conventional linear quantile regression. For comparison, we also include the results of the TIR method of Wang and Tsai (2009), which assumes that log{γ (x)} =θ 1 + θ 2 x with θ 1,θ 2 as unnown parameters. Figure 1(c) plots the estimated γ for the CS method against, and Figure 1(d) plots the index estimation γ (x) forthencs method versus at x = For both CS and NCS, the estimated γ s begin to be relatively stable around = 100. In the sequel, we focus on the analysis based on = 100. Figure 1(e) shows that the estimated γ (x) from NCS decreases with x,while that from TIR increases with x. In addition, the TIR estimates of γ (x) are much larger than the NCS estimates. Note from Figure 1(f) that the TIR estimates of γ (x) are associated with large standard errors especially for x>0.02. This discrepancy },

11 1462 Journal of the American Statistical Association, December 2012 is partly because the TIR method uses a constant threshold that does not account for the positive association between y and x. Next we estimate the conditional quantiles Q Y (τ x) byextrapolation for τ = 0.99, 0.995, 0.999, and Let τ 0 = 1 /n = For the CS method, Q Y (τ x) is estimated by Q Y (τ x) = α(τ 0 ){(1 τ 0 )/(1 τ)} γ + (x x) β c, where α(τ 0 ) is the τ 0 th sample quantile of the residuals {ê i }. For the NCS and TIR methods, Q Y (τ x) is estimated by Q Y (τ x) = Q Y (τ 0 x){(1 τ 0 )/(1 τ)} γ (x), where Q Y (τ 0 x)istheτ 0 th conditional quantile of Y estimated by linear quantile regression. Table 4 summarizes the estimation of Q Y (τ x) fromthe CS, NCS, TIR, and the conventional linear quantile regression (RQ) at x = 0.01, 0.02, 0.03, and For NCS method, the covariate-adaptive index estimate γ (x) is unstable for larger values of x due to lac of data points in those areas. Figure 1(e) suggests that the NCS estimates of γ (x) have little variation across x [0.02, 0.04]. Therefore, for extrapolation at x = 0.03 and 0.04, we used γ pool = 0.071, defined as the average of γ (x j )for 100 x j values equally spaced within [0.02, 0.04]. With the large estimated values of γ (x), the TIR method yields much larger quantiles than CS and NCS. The CS, NCS, and RQ give similar estimates at τ = 0.99 and However, at higher quantiles Daily precipitation (Y) γ~ (a) Re analysis variable (X) (c) τ=0.99 τ=0.995 τ=0.999 τ= τ = and , the slope estimates from conventional RQ are inflated due to the high influence of the extreme event around x = Consequently, RQ gives much larger estimates at τ = and especially for larger x values. At the Aurora station, among 4816 wet days in 45 years, only 1 day had daily precipitation reaching the level inches. However, at x = 0.04, the 99.9th (99.99th) percentile of daily precipitation is estimated to be (25.41) inches by RQ and (89.86) inches by TIR, which seem unrealistically high for Aurora. 4.2 Midway Airport Station Similar to the presentation in Section 4.1,weuseFigure 2 for the Midway airport station. The estimated quantile regression lines at τ = 0.99, 0.995, 0.999, and cross each other, possibly due to the sparseness of data or the lacness of fit of linear regression models at high quantiles. Figure 2(b) shows that β(τ) tends to be increasing in τ. Figure 2(c) and 2(d) plots the CS extreme value index estimation γ and the covariate-adaptive estimation from the NCS method γ (x) atx = 0.01, 0.02, respectively. For both CS and β^(τ) γ^ (x=0.01) (b) τ (d) γ^ (x) (e) NCS TIR x se of γ^tir (x) (f) TIR x Figure 2. Midway airport station. (a) Daily precipitation Y versus the simulated precipitation X and the fitted quantile regression lines at τ = 0.99, 0.995, 0.999, and (b) The RQ estimates of β(τ) atτ [0.95, 0.99] and the composite-slope estimate β c (dashed horizontal line). (c) The index estimates γ for the CS method versus. (d) The index estimates γ (x) from the NCS method versus at x = (e) The index estimates γ (x) from NCS and TIR (the tail index regression method of Wang and Tsai (2009)) versusx. (f) The standard error of γ (x) from TIR. The online version of this figure is in color.