Finite-sample distribution of regression quantiles

Statistics and Probability Letters 80 (2010) 1940–1946

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Finite-sample distribution of regression quantiles✩ Jana Jurečková ∗ Department of Probability and Statistics, MFF UK, Charles University, Sokolovská 83, CZ-186 75 Prague 8, Czech Republic

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Article history: Received 8 June 2010 Received in revised form 25 August 2010 Accepted 26 August 2010 Available online 8 September 2010 MSC: 62E15 62J05 62G30 62H10

The finite-sample distributions of the regression quantile and of the extreme regression quantile are derived for a broad class of distributions of the model errors, even for the noni.i.d case. The distributions are analogous to the corresponding distributions in the location model; this again confirms that the regression quantile is a straightforward extension of the sample quantile. As an application, the tail behavior of the regression quantile is studied. © 2010 Elsevier B.V. All rights reserved.

Keywords: Order statistic Regression quantile Extreme regression quantile Regression rank scores Score function

1. Introduction Consider the linear regression model Yi = x⊤ i β + ei ,

i = 1, . . . , n

(1)

with an unknown parameter β, where xi ∈ R , xi1 = 1, i = 1, . . . , n and where e1 , . . . , en are i.i.d. errors with distribution ⊤ function F and density f . We assume that matrix X with rows x⊤ 1 , . . . , xn is of rank p and that f (z ) is absolutely continuous and positive for z ∈ (a, b) where a = inf{z : F (x) > 0} and b = sup{z : F (z ) < 1}. ˆ n (α) minimizing the criterion The regression α -quantile (0 < α < 1) of Koenker and Bassett (1978) is any vector β p

n −

ρα (Yi − x⊤ i b),

b ∈ Rp

(2)

i=1

ˆ n (α) in the linear programming sense is the vector where ρα (z ) = |z |{α I [z > 0] + (1 − α)I [z < 0]}. Dual to β aˆ n (α) = (ˆan1 (α), . . . , aˆ nn α)⊤ of the regression rank scores, defined as a solution of the linear programming problem n −

aˆ ni (α)Yi = max

i=1

✩ Research was supported by the Grants GAČR201/09/0133 and IAA101120801, and by Research Projects MSM 0021620839 and LC 06024.

∗

Tel.: +420 221913285; fax: +420 222323316. E-mail address: [email protected]. URL: http://www.karlin.mff.cuni.cz/∼jurecko.

0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.08.023

J. Jurečková / Statistics and Probability Letters 80 (2010) 1940–1946

under

n −

1941

aˆ ni (α) = n(1 − α)

(3)

i=1 n −

xij aˆ ni (α) = (1 − α)

n −

xij ,

j = 2, . . . , p

i=1

i=1

0 ≤ aˆ ni (α) ≤ 1,

i = 1, . . . , n.

Because of the restrictions in (3), the aˆ ni are unchanged if Yi is replaced with ei = Yi − x⊤ i β, i = 1, . . . , n. As they are

ˆ n (α) have the following relation: mutually dual, aˆ n (α) and β aˆ ni (α) =



ˆ · · · Yi > x⊤ i β(α) ⊤ˆ · · · Yi < xi β(α),

1 0

i = 1 , . . . , n,

(4)

ˆ while 0 < aˆ ni (α) < 1 for Yi = x⊤ i βn (α), i = 1, . . . , n; the latter correspond to the optimal basis of p vectors among x1 , . . . , xn . Moreover, the identity n −

ˆ x⊤ i β(α) = −

i =1

n − i=1

Yi

daˆ i (α)

(5)

dα

ˆ (see e.g., Hallin and Jurečková (1999)). The derivative of aˆ i (α) is nonzero only if holds at the point α of continuity of β(·) ˆ . Yi = x⊤ β(α) i In the location model, where X = 1n , the regression α -quantile reduces to the ⌈nα⌉-order statistic Yn:⌈nα⌉ , and the vector of α -regression rank scores reduces to the Hájek’s rank scores which can be expressed as follows: 0 · · · Ri < nα Ri − nα · · · (Ri − 1)/n < α ≤ Ri /n 1 · · · α ≤ (Ri − 1)/n i = 1, . . . , n

 a∗ni (Ri , α) =

(6)

where Ri is the rank of Yi among Y1 , . . . , Yn . Rank scores a∗n (Ri , α) were first used by Hájek (1965) as a starting point for the daˆ (α)

construction of the nonlinear rank tests. We see that the identity (5) is also true in the location model, where di α ̸= 0 only for i such that nα < Ri < nα + 1, i = 1, . . . , n. In the location model we have i.i.d. observations Y1 , . . . , Yn with the ranks R1 , . . . , Rn , and the vector of ranks is independent of the vector of order statistics Y(·) = (Yn:1 , . . . , Yn:n ), Yn:1 ≤ · · · ≤ Yn:n . The kth order statistic Yn:k has the density n! (F (y))k−1 (1 − F (y))n−k f (y), (k − 1)!(n − k)! while P {R1 = r1 , . . . , Rn = rn } = n1! for any permutation r1 , . . . , rn of numbers 1, 2, . . . , n. Generally, if the vector Y1 , . . . , Yn has density p(y1 , . . . , yn ), the ranks and order statistics can be mutually dependent; their joint distribution can be written in the following way: P (Y(·) ∈ B, R1 = r1 , . . . , Rn = rn ) =

∫

∫ ···

p yn:r1 , . . . , yn:rn dyn:1 . . . dyn:n





B

for any Borel subset of the ordered quadrant of Rn and for any permutation r1 , . . . , rn . The asymptotic distribution of regression quantiles and of regression rank scores under n → ∞ was studied by more authors, under various conditions on f and on the regressors (let us mention Koenker and Bassett (1978), Ruppert and Carroll (1980), Gutenbrunner and Jurečková (1992), Gutenbrunner et al. (1993), among others). The asymptotic properties of autoregression quantiles and the rank scores were studied by Koul and Saleh (1995). However, the proofs of the asymptotic distributions available in the literature impose various restrictions on the tails of the distribution of errors and on the regression matrix, even though the numerical evidence demonstrates that e.g. the regression rank scores’ tests work under quite general conditions. For instance, the asymptotic distribution of the regression rank scores’ criterion was proved only when either the tails of the basic distribution are lighter than those of the t-distribution with 4 degrees of freedom or when the score generating function ϕ : (0, 1) → R1 is zero outside (ε, 1 −ε) ⊂ (0, 1) (cf. e.g. Gutenbrunner and Jurečková (1992) and Gutenbrunner et al. (1993)). Moreover, the asymptotic distribution, which is typically normal, does not provide the full ˆ , and it can stretch its true behavior under heavy-tailed f . information on the behavior of β(α) We expect that the finite-sample distribution of the α -regression quantile reminds the distribution of the sample quantile ˆ in the location model. Koenker and Bassett (1978) provided one possible form of the finite-sample distribution of β(α) (see also Koenker (2005)). However, these authors themselves do not find their form very tractable and adaptable to practical statistical inference. A possible form of the finite-sample distribution, similar to the distribution of the sample quantile, is given in the present paper. Using the regression rank scores, dual to regression quantiles, we first derive the joint distribution ˆ of (ˆan (α), β(α)) in the linear regression model (1). Unlike in Koenker and Bassett (1978), our main tool is using the score

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J. Jurečková / Statistics and Probability Letters 80 (2010) 1940–1946

ˆ function of (ˆan (α), β(α)) and the fact that the score function of a statistic S = S (Y) is equal to the conditional expectation of the score function of Y under given S, provided it exists (see, e.g., Kagan et al. (1972) and Jurečková and Milhaud (2003)). We also benefit from the fact that the regression quantiles and rank scores are mutually dual optimal solutions of a linear programming problem. Because 0 ≤ aˆ i (α) ≤ 1, i = 1, . . . , n, the vector aˆ (α) can be interpreted as a test function, and we can use the generalized Neyman–Pearson lemma. 2. Distribution of α-regression quantile under i.i.d. errors Consider the linear regression model (1) with deterministic regression matrix X with the first column equal to 1n . Consider first the situation that the errors e1 , . . . , en are i.i.d. with distribution function F and density f . We assume ⊤ that matrix X with rows x⊤ 1 , . . . , xn has rank p and that f (z ) is absolutely continuous and positive for z ∈ (a, b) where

ˆ a = inf{z : F (x) > 0} and b = sup{z : F (z ) < 1}. The first step is to derive the score function of (ˆa(α), β(α)) . Generally, the score function of the vector Y1 , . . . , Yn with density p(y1 , . . . , yn , θ) = p(y, θ), θ ∈ 2 ⊂ Rp , is the vector function 

∂ ln p(y, θ) , j = 1, . . . , p ∂θj

⊤

.

The score function is in 1:1 correspondence with the probability distribution, and it is one of its main characteristics. The use of the score functions was recommended by Hampel (1973), Field and Hampel (1982), Field and Ronchetti (1990), among others. A natural argument for the use of the score function is that it is linear for the normal distribution, which is considered as a unit in the family of probability distributions. Fix α and consider the set

 An (α) =

a : 0 ≤ ai ≤ 1, i = 1, . . . , n,

n −

xi ai I [ai > 0] = (1 − α)

n −

i=1

xi and 0 < aij < 1, j = 1, . . . , p

i=1

 for 1 ≤ i1 , . . . , ip ≤ n, such that xi1 , . . . , xip is a basis of R

p

.

(7)

ˆ The score function of the vector (ˆa(α), β(α)) follows. ˆ Lemma 2.1. Under the conditions stated above, the score function of (ˆa(α), β(α)) has the form r(a, b, β) = −

n −



xi f (x⊤ i [b − β])

i =1



I [ai = 0] F (x⊤ i [b − β])

−

I [ai = 1] 1 − F (x⊤ i [b − β])



+ I [0 < ai < 1]

f ′ (x⊤ i [b − β])



f (x⊤ i )[b − β]

a ∈ An (α), b ∈ Rp .

, (8)

Proof. The score function of model (1) is the vector function

   

∂ ln

p ∏

⊤

f (yi − x⊤ i β)

∑ n

  i =1  , j = 1 , . . . , p  =

i=1

∂βj

∂ ln f (yi − x⊤ i β) ∂βj

⊤ n −  f ′ (yi − x⊤ i β) , j = 1, . . . , p = − x . i ⊤  f (yi − xi β) i=1

(9)

ˆ ˆ Regarding inequalities (4), the conditional distribution of Yi , given [ˆai (α) = 0 ⇔ Yi < x⊤ i β(α)] and [β(α) = b] has the density f (y − x⊤ i β) F (x⊤ i [b − β])

I [ˆani (α) = 0].

ˆ ˆ Similarly, the conditional density of Yi given [ˆai (α) = 0 ⇔ Yi > x⊤ i β(α)] and [β(α) = b] is f (y − x⊤ i β) 1 − F (x⊤ i [b − β])

I [ˆani (α) = 1].

This further implies for i = 1, . . . , n

 ′    f (Yi − x⊤ i β)  ˆ Eβ − βn (α) = b, aˆ ni (α) = 0 f (Yi − x⊤ i β)  ∫ x⊤ b  ′ i 1 f (y − x⊤ −f (x⊤ i β) i [b − β]) ⊤ = − f ( y − x β) dy = i ⊤ ⊤ ⊤ F (xi [b − β]) −∞ f (y − xi β) F (xi [b − β])

(10)

J. Jurečková / Statistics and Probability Letters 80 (2010) 1940–1946

1943

and similarly,

 −

Eβ

=

f ′ (Yi − x⊤ i β) f (Yi − x⊤ i β)

   βˆ n (α) = b, aˆ ni (α) = 1  ∞

∫

1

 −

1 − F (xi [b − β]) ⊤

x⊤ b i

f ′ (y − x⊤ i β)



f (y − xi β) ⊤

f (y − x⊤ i β)dy =

f (x⊤ i [b − β]) 1 − F (x⊤ i [b − β])

.

(11)

Moreover,

 −

Eβ

f ′ (Yi − x⊤ i β)

  ′ ⊤  βˆ n (α) = b, 0 < aˆ ni (α) < 1 = − f (xi [b − β]) .  ⊤ f (Yi − xi β) f (x⊤ i [b − β])

ˆ n (α)) follows from (10)–(12). The score function of (ˆan (α), β

(12)



Integrating (8) over β, we obtain the joint distribution of the α -regression rank scores aˆ n (α) and of the α -regression ˆ n (α): quantiles β Lemma 2.2. Under the above conditions, the joint distribution of the α -regression rank score and α -regression quantile has the form Pβ



∫ ∏ n    ⊤ I [ai =0]  I [ai =1] ˆβn (α) ∈ B, aˆ n (α) = a = C F (xi [b − β]) 1 − F (x⊤ i )[b − β] B i=1

I [0
a ∈ An (α), B ∈ B p , C > 0 is a normalizing constant . Proof. Denote H =

 ∑

n i =1

xi1 ai ,

∑n

i=1

xi2 ai , . . . ,

∑n

i =1

xip ai

⊤

(13)



: 0 ≤ ai ≤ 1, i = 1, . . . , n . Then H is closed and convex.

Let a ∈ An (α), 0 < aij < 1 for j = 1, . . . , p, and let xi1 , . . . , xip be linearly independent rows of matrix X. Fix b ∈ Rp , b ̸= 0, and choose y1 , . . . , yn so that ⊤

yij = x⊤ ij b,

⊤

j = 1, . . . , p

yi > xi b if ai = 1, ⊤

yi < x⊤ i b if ai = 0, i = 1, . . . , n. Then a maximizes ∈ An (α), and b is the α -regression quantile pertaining to (y1 , . . . , yn ). On the other i=1 yi ai under a ∑ n hand, let a solve (3). Because (1 − α) i=1 xi is an inner point of H , there exists the vector b of Lagrange multipliers such that

∑n

yi > x⊤ i b if ai = 1, yi < x⊤ i b if ai = 0, i = 1, . . . , n, the remaining ai corresponding to yi = x⊤ i b and the values of b are determined so that Neyman–Pearson lemma (Lemma 3 in Lehmann (1986, Chapter 3))). 

∑n

i =1

xi ai = (1 − α)

∑n

i =1

xi (cf. the

Lemmas 2.1 and 2.2 finally give the density of the α -regression quantile. Theorem 2.3. Consider the linear regression model (1) with deterministic regression matrix X of rank p, with the first column equal to 1n . Assume that the errors e1 , . . . , en are i.i.d. with absolutely continuous distribution function F and with density f , absolutely continuous and positive for z ∈ (a, b) where a = inf{z : F (x) > 0} and b = sup{z : F (z ) < 1}. Then the ˆ α -regression quantile β(α), 0 < α < 1 has the density g (b; α) =

n  − ∏ 

F (x⊤ i [b − β])

I [ai =0] 

1 − F (x⊤ i )[b − β]

I [ai =1] 

I [0
f (x⊤ i [b − β])

,

b ∈ Rp .

(14)

a∈An (α) i=1

3. Remarks and application to the tail behavior of regression quantiles In the location model with X = 1n , density (14) reduces to the density of the order statistic Yn:k with k = ⌈nα⌉. Indeed, then Xn ≡ 1n and the regression rank scores reduce to the Hájek scores (6). Then aˆ ni (α) ∈ (0, 1) iff the rank of Yi equals Ri = ⌈nα⌉, which happens just for one among Y1 , . . . , Yn with probability 1, and aˆ nk (α) = 0 or 1 according to whether Rk is greater or less than ⌈nα⌉, respectively.

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J. Jurečková / Statistics and Probability Letters 80 (2010) 1940–1946

Even in the location model the moments of order statistics must be calculated numerically. In spite of that, the BLUE estimators of Lloyd (1952) (see also Sarhan and Greenberg (1962), David (1981)) are considered useful in various contexts. For instance, the Shapiro and Wilk (1965) test is a surprisingly good intuitive test of normality based on the ratio of the MLE and BLUE estimators of σ under normal f . Sen et al. (2003) extended the test to the model with nuisance linear regression, where it also works well. ˆ 1 ), β(α ˆ 2 ). Their expectations With the above methods we can derive the joint distribution of two regression quantiles, β(α and covariances are sufficient just under β = 0, because of the affine equivariance of regression quantiles and of the invariance of regression rank scores. Even if the finite-sample distribution is more complex than the asymptotic one, and the finite-sample moments are calculated via numerical integration, we cannot limit ourselves to the asymptotics, which can stretch the truth and is sometimes informative only for a very large number of observations. As an illustration, we apply the methods of Section 2 to the study of the finite-sample tail behavior of regression quantiles. Typically, the quantiles and other estimators inherit their tail behavior from the parent distribution, though their asymptotic (normal) distribution is light tailed. Following He et al. (1990), consider the following finite-sample measure of the right-tail performance of a regression estimator Tn in model (1) (assume that 0 < F < 1, f > 0, for brevity)

 − ln Pβ

max x⊤ i (Tn − β) > γ







1≤i≤n

, γ > 0. (15) − ln(1 − F (γ )) The probability in (15) tends to 0 as γ → ∞, but the rate of this convergence depends on the tail of F . Then the bounds of  (15) under finite n and γ → ∞ are of interest, because they show how faster is the convergence Pβ max1≤i≤n x⊤ i (Tn − β) > γ → 0 than 1 − F (γ ) → 0 as γ → ∞. Put β = 0, without loss of generality, and assume for simplicity that nα = k > 0, an integer, and k + p < n. Regarding B(Tn , γ ) =

(3) and (4), we can write

 P0

ˆ max{x⊤ i β(α)} ≤ γ



i

≤ P0 (at least k + 1 among Y ’s are ≤ γ )

hence

 P0

ˆ max x⊤ i β(α) > γ



i

≥ P0 (Yn:k+1 > γ ) = ≥

n! k!(n − k − 1)!

∫

n! k!(n − k − 1)!

(F (γ ))k

1−F (γ )

∫ 0

1 F (γ )

uk (1 − u)n−k−1 du

v n−k−1 dv =

n! k!(n − k)!

(F (γ ))k (1 − F (γ ))n−k .

It implies that

  ˆ − ln P0 max x⊤ β(α) > γ i i

limγ →∞

− ln(1 − F (γ ))

≤ n − k.

Similarly,

 P0

max xi β(α) ≤ γ ⊤ˆ



i

  ≥ P0 Yn:k+p ≤ γ ,

hence

 P0

ˆ max x⊤ i β(α) > γ i



n!

∫

1

uk+p−1 (1 − u)n−k−p du (k + p − 1)!(n − k − p)! F (γ ) n! × (1 − F (γ ))n−k−p (1 − (F (γ ))k+p ) (k + p − 1)!(n − k − p)! n!(k + p − 1) (1 − F (γ ))n−k−p+1 ≤ (k + p)!(n − k − p)!

≤

and this implies that

  ˆ − ln P0 max x⊤ β(α) > γ i limγ →∞

i

− ln(1 − F (γ ))

≥ n − k − p + 1.

Summarizing, we conclude that

ˆ ˆ n − k − p + 1 ≤ limγ →∞ B(β(α), γ ) ≤ limγ →∞ B(β(α), γ) ≤ n − k which is consistent with the tail behavior of the order statistic Yn:k in the location model (where p = 1).

(16)

J. Jurečková / Statistics and Probability Letters 80 (2010) 1940–1946

1945

4. Distribution of the extreme regression quantile After a slight modification, the above method applies to the extreme regression quantiles, corresponding to α = 0, 1. ˆ 1) is a solution of the minimization problem: The maximal regression quantile β( min

b∈Rp

n −

+ (Yi − x⊤ i b)

(17)

i =1

where z + = max(z , 0) denotes the positive part of z. The minimization (17) can be alternatively described as any solution to the linear program: min

b∈Rp

n −

⊤ {x⊤ i b} s.t. Yi ≤ xi b,

i = 1, . . . , n.

(18)

i =1

This statistic was studied by Smith (1994) who derived its asymptotic distribution under heavy-tailed distribution F and under some conditions on the xi . Portnoy and Jurečková (1999), Knight (2001), Chernozhukov (2005), Jurečková (2007) (among others), derived various forms of the asymptotic distributions of the extreme regression quantiles under various conditions. As in the extreme value theory, the asymptotic distribution of the extreme regression quantile depends on the domain of attraction of the model errors, and the results in the literature are proved under various additional conditions on f and on X. ⊤ˆ ˆ If (xi1 , . . . , xip )⊤ = x˜ is the optimal basis of problem (18), then Yij = x⊤ ij β(1) for j = 1, . . . , p and Yi < xi β(1) for the remaining components. Denote the set Cn = {c = (c1 , . . . , cn )⊤ : ci1 , . . . , cip = 1 for 1 ≤ i1 < · · · < ip ≤ n such that (xi1 , . . . , xip ) is a basis of Rp , and ci = 0 otherwise}.

(19)

Proceeding analogously as in Lemmas 2.1 and 2.2 and Theorem 2.3, we obtain the density of the extreme regression quantile:

ˆ 1) has density Theorem 4.1. Under the conditions of Theorem 2.3, the extreme regression quantile β( g (b; 1) =

n  −∏ 

F (x⊤ i [b − β])

I [ci =0] 

I [ci =1] 

f (x⊤ i [b − β])

,

b ∈ Rp .

(20)

c∈Cn i=1

ˆ 1) = b of (18) determines the optimal basis (xi , . . . , xip ) and Yi = x⊤ b, j = 1, . . . , p. For Proof. The optimal solution β( ij j 1 the remaining i we have the conditional probability





ˆ 1) = b = P Yi < y|β(

F (y − x⊤ i β) F (x⊤ i (b − β))

I [Yi < x⊤ i b].

Following the steps in the proofs of Lemmas 2.1 and 2.2 and Theorem 2.3, we arrive at (20). In the location model with X = 1n , density (20) reduces to n(F (x))n−1 f (x), the density of the maximal order statistic.  5. Distribution of α-regression quantile under non-i.i.d. errors The distribution of the regression quantile can be obtained even when the errors ei are non-identically distributed, but still independent. Consider model (1) with the change that the error ei has a distribution function Fi with density fi , hence Yi has distribution function Fi (y − x⊤ i β), i = 1, . . . , n. The distribution functions F1 , . . . , Fn are generally different, but all are assumed to be absolutely continuous and strictly increasing on R1 with absolutely continuous densities. A special case is the heteroscedastic model Yi = x⊤ i β + σ i ei with e1 , . . . , en i.i.d. with distribution function F , i.e. Fi (z ) = F

  z

σi

, i = 1, . . . , n.

The score function of the random vector (Y1 , . . . , Yn ) in the non-i.i.d. model is

−

n − i =1

xi

fi′ (yi − x⊤ i β) fi (yi − x⊤ i β)

.

The definitions (2) and (3) of the regression quantile and of the regression rank scores are unchanged, and so are their algebraic properties. So, following the steps of proofs of Lemmas 2.1 and 2.2, we shall arrive at the following form of distribution of the regression quantile in this situation: Theorem 5.1. Consider the linear regression model (1) with deterministic regression matrix X of rank p, with the first column equal to 1n . Assume that the errors e1 , . . . , en are independent with absolutely continuous and strictly increasing distribution functions Fi , 0 < Fi (x) < 1, x ∈ R, which have absolutely continuous densities fi > 0, i = 1, . . . , n. Then the α -regression

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J. Jurečková / Statistics and Probability Letters 80 (2010) 1940–1946

∗

ˆ (α), 0 < α < 1 has density quantile β g ∗ (b; α) =

n  − ∏ 

Fi (x⊤ i [b − β])

I [ai =0] 

1 − Fi (x⊤ i )[b − β]

I [ai =1] 

I [0
fi (x⊤ i [b − β])

,

b ∈ Rp .

(21)

a∈An (α) i=1

We obtain as a corollary the distribution of α -regression quantile in the heteroscedastic model. 6. Concluding remarks The finite-sample distributions of the α -regression quantile and of the extreme regression quantile are analogous to the corresponding distributions in the location model; this again confirms that the regression quantile is an extension of the sample quantile. Moreover, these distributions are true for a broad class of underlying distributions of errors, even in the non-i.i.d. case, while the asymptotic distributions of regression quantiles were derived only under various restrictive conditions. References Chernozhukov, V., 2005. Extremal quantile regression. Ann. Statist. 33, 806–839. David, H.A., 1981. Order Statistics, 2nd ed.. J. Wiley, New York. Field, C.A., Hampel, F.R., 1982. Small-sample asymptotic distribution of M-estimators of location. Biometrika 69, 29–46. Field, C.A., Ronchetti, E., 1990. Small Sample Asymptotics. In: IMS Lecture Notes, Monograph Series, vol. 13. Hayward, CA. Gutenbrunner, C., Jurečková, J., 1992. Regression rank scores and regression quantiles. Ann. Statist. 20, 305–330. Gutenbrunner, C., Jurečková, J., Koenker, R., Portnoy, S., 1993. Tests of linear hypotheses based on regression rank scores. Nonparametr. Stat. 2, 307–331. Hájek, J., 1965. Extension of the Kolmogorov–Smirnov test to the regression alternatives. In: Neyman, J., LeCam, L. (Eds.), Bernoulli-Bayes-Laplace. In: Proc. Intern. Res. Seminar, Springer-Verlag, Berlin, pp. 45–60. Hallin, M., Jurečková, J., 1999. Optimal tests for autoregressive models based on autoregression rank scores. Ann. Statist. 27, 1385–1414. Hampel, F.R., (1973). Some small-sample asymptotics. In: J. Hájek (Ed.), Prof. Prague Symposium on Asymptotic Statistics. Charles University. pp. 109–126 (in Prague). He, X., Jurečková, J., Koenker, R., Portnoy, S., 1990. Tail behavior of regression estimators and their breakdown points. Econometrica 58, 1195–1214. Jurečková, J., 2007. Remark on extreme regression quantile. Sankhya 69 (Part 1), 87–100. Jurečková, J., Milhaud, X., 2003. Derivative in the mean of a density and statistical applications. In: M. Moore S. Froda C. Léger (Eds.), IMS Lecture Notes. Monograph Series, vol. 42. Hayward, CA. pp. 216–230. Kagan, A.M., Linnik, Ju.V., Rao, C.R., 1972. Characteristic Problems of Mathematical Statistics. Nauka, Moscow. Knight, K., 2001. Limiting distributions for linear programming estimators. Extremes 4, 87–103. Koenker, R., 2005. Quantile Regression. Cambridge University Press, Cambridge. Koenker, R., Bassett, G., 1978. Regression quantiles. Econometrica 46, 466–476. Koul, H.L., Saleh, A.K.Md.E., 1995. Autoregression quantiles and related rank scores processes. Ann. Statist. 23, 670–689. Lehmann, E.L., 1986. Testing Statistical Hypotheses, second ed.. Springer-Verlag, New York. Lloyd, E.H., 1952. Least squares estimation of location and scale parameters using order statistics. Biometrika 34, 41–67. Portnoy, S., Jurečková, J., 1999. On extreme regression quantiles. Extremes 2, 227–243. Ruppert, D., Carroll, R.J., 1980. Trimmed least squares estimation in the linear model. J. Amer. Statist. Assoc. 75, 828–838. Sarhan, A.E., Greenberg, B.G. (Eds.), 1962. Contributions to Order Statistics. J. Wiley, New York. Sen, P.K., Jurečková, J., Picek, J., 2003. Goodness-of-fit test of Shapiro–Wilk type with nuisance regression and scale. Aust. J. Stat. 32 (1–2), 163–177. Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance for normality (complete samples). Biometrika 52, 591–611. Smith, R., 1994. Nonregular regression. Biometrika 81, 173–183.