franparametric Siaiisiics, Vol. 2. pp. 307-33!                © 1993 Gordon and Breach Science Publishers
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TESTS OF LINEAR HYPOTHESES BASED ON REGRESSION RANK SCORES
C GUTENBRUNNER1, J. JUREČKOVÁ2, R. KOENKER3 and
S. PORTNOY3
'Philipps Universität, Marburg, Germany, Charles University, Prague, Czechoslovakia und ^University of Illinois at Urbana-Champaign, USA
(Received September 25, 1992; in final form December 16, 1992) Dedicated to the memory of Jaroslav Hájek
We propose a general class of asymptotically distribution-free tests of a linear hypothesis in ihc linear regression model. The tests are based on regression rank scores, recently introduced by Gutenbrunncr and Jurcčková (1992) as dual variables to the regression quantiles of Koenker and Bassctt (1978). Their properties arc analogous to those of the corresponding rank tests in location model. Unlike Ihc other regression tests based on aligned rank statistics, however, our tests do not require preliminary estimation of nuisance parameters, indeed they arc invariant with respect to a regression shift of (he nuisance parameters.
KEYWORDS: Ranks, regression quantiles, regression rank scores.
1. INTRODUCTION
Several authors including Koul (1970), Puri and Sen (1985) and Adichie (1978) have developed asymptotically distribution-free tests of linear hypotheses for the linear regression model based upon aligned rank statistics. Excellent reviews of these results including extensions to multivariate models may be found in Puri and Sen (1985) and the survey paper of Adichie (1984). The hypothesis under consideration typically involves nuisance parameters which require preliminary estimation; the aligned (or signed) rank statistics are then based on residuals from the preliminary estimate. Alternative approaches to inference based on rank estimation have been considered by McKean and Hettmansperger (1978), Aubuchon and Hettmansperger (1988) and Draper (1988) among others.
A completely new approach to the construction of rank statistics for the linear model has recently been introduced by Gutenbrunner and JureČková (1992). Their approach is based on the dual solutions to the regression quantile statistics of Koenker and Bassett (1978). These regression rank scores represent a natural extension of the "location rank scores" introduced by Hájek and Šidák (1967, Section V.3.5), which play a fundamental role in the classical theory of rank
AMS 1980 subject classifications: 62GI0, 62J1Ü.
The work was partially supported by NSF grants 88-02555 and 89-22472 to S. Portnoy and R. Koenker
and by support from the Australian National University to J. Jurčíková and R. Koenker.
*>.
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statistics. In this paper we consider tests of a general linear hypothesis for the linear regression model based upon regression rank scores. These tests have the advantages of more familiar rank tests; they are robust to outliers in the response variable and they are asymptotically distribution free in the sense that no nuisance parameter depending on the error distribution need be estimated in order to compute the test statistic. Furthermore, they are considerably simpler than many of the proposed aligned rank tests which require preliminary estimation of the linear model by computationally demanding rank estimation methods. The robustness of the proposed tests and the sensitivity of the aligned rank procedures to response outliers is illustrated in the sensitivity analysis of the example discussed in Section 2. In the classical linear model,
Y = X/3 + E,                                           (1.1)
the vector $(a) ■*($!(<*)> - • • • ßpi**))' e Rp of «th regression quantiles is any solution of the problem
min f, pa(Y,-x;t),       t€Rp                              (1.2)
/-i
where
PÁ») = I«I i(l - *)/[« <0] + «l[u >0]},       u e R'.              (1.3)
Least absolute error regression corresponds to the median case with a = 2. In the one-sample location model, with X = l„. solutions to (1.2) are the ordinary sample quantiles: when not is an integer we have an interval of solutions between two adjacent order statistics. Computation of the regression quantiles is greatly facilitated by expressing (1.2) as the linear program
alX + (1 - a)l>- : = min                                (1.4)
\ß + u* - iT = Y
ßeRp,       u\ireR1
and 1„ ■ (1,. .. , 1)' e R", with 0< a< 1. Even in this form, the problem of finding all the regression quantile solutions may appear computationally demanding, since there would appear to be a distinct problem to solve for each a e (0, 1). Fortunately, there are only a few distinct solutions. In the location model we know, of course, that there arc at most n distinct quantiles. In regression, Portnoy (1991) has shown that the number of distinct solutions to (1.2) is Op {n log/i). Finding all the regression quantiles is a straightforward exercise in parametric linear programming. From any given solution for fixed a we may compute the interval containing a for which is solution remains optimal, and one simplex pivot brings us to a new solution at either endpoint of the interval. Proceeding in this way we may compute the entire path /?(-) which is a piecewise constant function from (0,1] to Rp. Detailed descriptions of algorithms to compute the regression quantiles may be found in Koenker and d'Orey (1990), and Osborne (1992). Finite-sample as well as asymptotic properties of ß(a) are studied in Koenker and Basset! (1978), Ruppert and Carroll (1980), Jurečková (1984), Gutenbrunner (1986), Koenker and Portnoy (1987), Gutenbrunner and Jurečková (1992), and Portnoy (1991b).
I I SIS   HASH)   ON   KANK   S( OKI S
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The regression rank scores introduced in Gutenbrunner and Jurečková (1992) arise as a n-vector ä„(o) = (änl(a),... , änn{a))' of solutions to the dual form of the linear program required to compute the regression quantiles. The formal dual program to (1.4) can be written in the form
Y'ä(o-): = max
X'á(tt) = (l-a)X'lfl                                    (1.5)
ä(tt)e[0, If,       0<*<1
As shown in Gutenbrunner and Jurečková (1992), many aspects of the duality of order statistics and ranks in the location model generalize naturally to the linear model through (1.4) and (1.5). Moreover, as pointed out there, ä is regression invariant with respect to XI, in the sense that a(a) is unchanged if Y is transformed to Y + XI y for any y e R'.
To motivate our approach, consider {Án(a), 0< a< 1} in the location model with X ■ 1„. In this case, anl{a) specializes to
ŮJa)mal{Ritet)~
1             if <*<(/*,- \)ln
R.-an   if (R,-l)fn< or sRJn            (1.6)
0             if RJn < a
where R, is the rank of Y, among Vj.....Y„. The function a*{j, a), j = 1,... ,n
0<oe<\, coincides exactly with that introduced in Hájek and Šidák (1%7, Section V.3.5). Under the general model (1.1), both the finite-sample and asymptotic properties of the regression rank scores and of the process {á„(o), 0< a< 1} are described in the next section. The regression rank score process may be efficiently computed by standard parametric linear programming techniques, essentially as a byproduct of the regression quantile computation requiring no additional computational effort and only some additional storage. See Koenkcr and d'Orey (1990) for algorithmic details. The formal duality between ß(a) and a(a) implies that for i = 1,.... n
ifl5>Ě*«A<*> /-I
if y, <£**&*)
(1.7)
/-í
while the components of a„(a) corresponding to {*' \ Y, = x',ß(a)} are determined by the equality constraints of (1.5). Thus, as in the location model, the regression rankscore for observation í is one while y, is above the ath quantile regression plane, and zero when v, falls below this plane, and taking an intermediate value while y, falls on the ath plane. Integrating the regression rankscore function for each observation over [0, 1] yields a vector of (Wilcoxon) ranks: observations falling "below" most the the others receiving small ranks, while those falling "above" the others, and thus having rankscore one over a wide interval, receive large ranks. This observation is completely transparent in the location model where "above" and "below" have an obvious interpretation. In regression, the
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interpretation of these terms relies on the optimization problem defining the regression quantiies. The resulting rank scores illustrated, for example, in Figure 2.1, are, we believe, a useful graphical diagnostic in linear regression in addition to their role in formal hypothesis testing.
The next section of the paper surveys our results, establishes some notation, and provides an illustrative example. Section 3 develops some theory of the regression rank score process. Section 4 treats the theory of simple linear rank statistics based on this process, and Section 5 contains a formal treatment of the proposed tests.
2. NOTATION AND PRELIMINARY CONSIDERATIONS
We will partition the classical linear regression model
Y = X/? + E                                           (2.1)
as
Y = X,/J,+X2ft + E                                     (2.2)
where ß, and ß> are p- and ^-dimensional parameters, X = X„ is a known, nx(p + q) design matrix with rows x^ = x', - (x'lt, xá) e Rpr", i=l,...,n. We will assume throughout that x,, =■ 1 for i = 1,.. . , n. Y is a vector of observations and E is an n x I vector of i.i.d. errors with common distribution function F. As in the familiar two-sample rank test, our test statistics is shift-invariant and hence independent of location. Thus like other rank tests, hypotheses on the intercept cannot be tested. This is immediately apparent from the regression invariance of the test statistic noted above. The precise form of F need not be known but we shall generally assume that F has an absolutely continuous density /on (A, B) where -»s/1 = sup{*: F(x) = 0} and +<»s B = inf{.r: F(x) = I}. Moreover, we shall impose some conditions on the tails of/assuming, among other conditions, that / monotonically decreases to 0 when x—*A + , or x—*B—. Define D„ =
B"lXiXj,
H^X^XlX.r'X;    and   Q„ = /r'(X2-X2)'(X2-X2)            (2.3)
with X2 = H,X3 being the projection of X2 on the space spanned by the columns of X,. We shall also assume
limD„ = D,       limQn=Q                                (2.4)
n—•»                                   n—»or
where I) and Q are positive definite {p x p) and (q x q) matrices, respectively. We are interested in testing the hypotheseis
H0:ß2 = 0,       ß, unspecified                               (2.5)
versus the Pitman (local) alternatives
Hm-fa^n'^fio                                (2.6)
with ßo being a fixed vector in R*.
TESTS BASED ON RANK SCORES
Ml
As in (he classical theory of rank tests, we shall consider a score-function <p:(0, 1)-»R which is nondecreasing and square-integrable on (0,1). We may then construct scores based on the regression rankscore process following Hájek and Šidák, (1967) as,
Bni = - ľ <p{i) äájfí,       i = l,...,«.                        (2.7)
Jo
Defining
S„ = «-w(XIl2-ÄB2)'6fl                                   (2.8)
where b„ = (Bn1,... , S„n)', we propose the following statistic for testing H„ against //„:
Tn = $&;%fA\<p)                                     (2.9)
where
X<f)= f(9>(0-*)2ď.       9= í <p{t)át                   (2.10)
and with Q„ defined as in (2.3). An important feature of the test statistic T„ is that it requires no estimation of nuisance parameters, since the functional A(q>) depends only on the score function and not on (the unknown) F. This is familiar from the theory of rank tests, but stands in sharp constrast with other methods of testing in the linear model where typically some estimation of a scale parameter of F is required to compute the test statistic. See for example the discussion in Aubuchon and Hettmansperger (1989) and Draper (1988).
We shall show in Section 5, that the asymptotic distribution of 7^, under Ha is central x! w',n 9 degrees of freedom while under H„ it is noncentral %z with q degrees of freedom and noncentrality parameter
where
*r= [A<P> F)M2(V)]/%Q/3o                              (2.11)
y(«P.F)=-f<P(0d/(F-l(0).                            (2-12)
Jo
Like -4, y is also familiar from the classical theory of rank tests. The test statistic T„ is first-order asymptotically distribution free in the sense that the first-order term in its asymptotic representation is exactly distribution free, as follows from (4.2). Moreover, it follows from (2.11) that the Pitman efficiency of the test based on T„ with respect to the classical F test of H0 coincides with that of the two-sample rank test of shift in location with respect to the Mest. For / unimodat, we obtain an asymptotically optimal test if we take
f'(F~l(t)\ *« = *»--^jFIflj,'      0<l<1-                 <2',3>
Thus for Wilcoxon scores (see below) the asymptotic relative efficiency of the test based on T„ relative to the classical F test is 3/*t = 0.955 at the normal distribution and is bounded below by 0.864 for all F. When F is heavy tailed this asymptotic efficiency is generally greater than one, and can in fact be unbounded.
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For normal (van der Waerden) scores (<p{u) = 4>~'(h)) the situation is even more striking. Here the test based on 7;, has asymptotic efficiency greater than one, relative to the classical F test, for all symmetric F, attaining one at the normal distribution. See e.g. Lehmann (1959, p. 239), and Lehmann (1983, pp 383-87). Let us now examine more closely the scores (2.7), which can be written as
4, = -/<P('K,(/)d/       ŕ-1,....*                        (2.14)
where Ihe functions a'llt(t)^dani{t)ldt are piecewise constant on [0.1]. The piecewise linearity of the regression rank scores follows immediately from the linear programming formulation (1.5) of the dual, greatly simplifying the computation in (2.14). In the location model, using (2.13) this reduces to the well-known Hájek and Šidák (1967) scores
q>(t)di,       i = l, ...,n
There are three typical choices of <p:
(i) Wilcoxon scores: tp(t) = t-{, 0<r<l. The scores are &ni = -l(t-\)dai(t) = j äj(t) d/ - % while A\<p) = •£, and y(<p. F) = J f2(x) dx. Wilcoxon scores are optimal when / is logistic. (ii) Normal (van der Waerden) scores: <p(t) = <£"'(/), 0 < f < 1, * being the d.f. of standard normal distribution. Here A\<p) = l and y(q>>F) = //(ír"'(<p(Ar))) dr. These scores are asymptotically optimal when / is normal, (iii) Median (sign) scores: <p(t) = { sign(f - \), 0<t<\, then (2.7) leads to the form S„,: = á„,(i) ~ h which is J if the ith /, residual is positive and —J if it is negative, and between —\ and í otherwise.
Remark. Using the standard reduction to canonical form e.g. Scheffe (1959, Section 2.6) or Amemiya (1985, Section 1.4.2), we may consider a more general form of the linear hypothesis
R'ß - r e R"                                         (2.15)
where B is a (p + q) x q matrix of rank q <p. Let V be a (p + q) x p matrix such that A = [V;R]' is nonsingular and R'V = 0. Set y = \ß and Z = XA"\ Partitioning y = [y!, y2]' where y, = \'ß and yz~R'ß, under the hypothesis (2.15) we have
Y-XR(R'R)"lr = XV(V,V)'Vi + E.
Thus, in view of the equivariance of regression (juantiles, see Koenker and Bassett (1978), Theorem 3.2, we may define V = Y-XR(R'R)~'r, X,= XV(V'V)"1, XaÄXR(R'R)"', and proceed as previously discussed with (Ý, X,,X2) playing the roles of (Y, X|,X2). By this device the tests described above and detailed in Section 5 may be extended to a wide range of applications including, for example, the hypotheses of parallelism and coincidence of regression lines discussed by Adichie (1984) and others.
To illustrate the tests proposed above we consider briefly an example taken from Adichie (1984, Example 3) dealing with the combustion of tobacco. The log
TESTS BASED ON RANK SCORES
313
of the leaf burn (in seconds) of 30 batches of tobacco is thought to depend upon the percent composition of nitrogen, chlorine, and potassium. Adichie suggests testing the potassium effect and describes an aligned rank version of the test. We are unable to reproduce some details of his calculations, however, using his approach we get least squares estimates of the nitrogen and chlorine effects of -0.529 and -0.290 with an intercept of 2.653. With these preliminary estimates we obtain aligned (Wilcoxon) ranks
7 17 2 18 6 1 11 3 30 13 25 16 4 29 26 27 21 23 19 12 28    10   8   15   24   20   22     5    14     9
which yield a test statistic of 13.59 highly significant relative to the 1% x\ critical value of 6.63.
The full set of regression rank scores á,{i) for the restricted model excluding potassium for this data are illustrated in Figure 2.1. There are 34 distinct regression quantile solutions and therefore each a„,(r) is a piecewise linear function with at most 34 distinct segments. Recall that áni(i) = 1 while the observed y, is above the rth regression quantile plane, 0 while below, and takes some intermediate value when y- falls on the rth plane. The plots are ordered according to their Wilcoxon rank score, which may be computed as £,= -Sl(t-h)ääi(t) = 5oäi{t)dt-l While the Wilcoxon rank scores provide an unambiguous ranking of the observations, since the regression rank score functions typically cross in regression applications, in contrast to the location model, this ranking depends upon the score function employed. The regression rank score plots give some further visual evidence concerning the ranking of the sample observations. Note that if änt{t) ^ anf(t) for all /, then Bni =: 6ni for any montone score function <p. Numerical calculations give Wilcoxon ranks
-0.27 0.06 -0.41 0.09 -0.32 -0.48 -0.17 -0.38 0.48 -0.06 0.23 0.04 -0.37 0.42 0.28 0.37 0.19 0.41 0.15 -0.26 0.38   -0.16   -0.23   -0.01       0.33      0.12      0.15   -0.42   -0.10   -0.06
and yield a test statistic of 13.17. In view of Theorem 5.1 the approximate ^-value is 0.0003. The two vectors of Wilcoxon ranks correspond closely. Observation 6 is smallest in both rankings and observations 14 and 9 are largest in both. The simple correlation between the two rankings is 0.978. Note that as a practical matter when q> = J*J <p(i) d/ = 0, we may omit the X2 term in the computation of S„ in (2.8) since bn is orthogonal to X,. This is in contrast with the aligned rank situation where the use of X2 - H2 'S essential. Corresponding calculations for the normal scores using
where <p denotes the standard normal density, and I, is the ith regression quantile
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C. GUTENBRUNNER et at
OüsNo 6 tank--0.48
Obs No 28 rar*- -0.42
Obs No :: .■..--■; ■-:
00        03      c.       06        «4      i«                   00      03        04      O«      44      10                 00      03      O«        «6        08       1.0
Obs No 8 rank- -038
Obs No 13 rank- -0-37
Obs No $ rank- -0.32
oo       oj       o«       06       o*       i«                   oo      03       oj       od       oa       i«
O?      0.«        0 6        OB      <«
Obs No 1 tank- -027
Obs No 20 rank--026
Obs No 23 rank- -0.23
02      0.'      06
>6      0«      II
ObsNo 7 rank--0.17
COS NO 22rank=-0.16
Obs No 29 rank--01
-
10        03      0"
0.4        42        O*      06      08       10
0.0      02        0"        OB        OS       ľ
ObS No 30 rank- -0.06
ObsNo 10 rank--0.06
Obs No 24 rank- -0.01
0?      4*       06      OS
0.4      42        °A      06        49
04      02        0«      04        M       U
Figure 2.1. Regression rank scores for tobacco data.
TESrS HASED ON RANK SCORES
315
Obs No 12 tank-0<j4
Obs No 2 far*- 0.06
ObsNo A nmk- 0.09
«O      02      O*      °t,      0*      14
«O        02        O'        0"        O*        «O
go      03        O«      0«        O»       10
ObsNO 26 rank-0.12
ObsNo 19 radk-0.15
ObsNo 27iank=0.15
oi)     o 2     0.4     oe    oi     u
oč    oj     o.     «e    «a     io
o.O      O»      o'      06      on       i
ObsNo 17 rank-0.19
Oba No 11 rank-033
Ote No 15ranh-056
02        O*        04        OB      i»
00        «?        04        06        O»        lO
o*      o«      oe       ío
ObsNg 25 rank-033
OtjsNo 16 rank* 0.3?
ObS No 21 rank- 0.38
OJ)       0?       04       00        08       10
oo      «2        •>
ObsNo 16 rank-0.41
ObsNo H rank-0-42
oo    ol     o*     o«     o«      io                o«     02     0.4      oo     oa     n
Figuře 2.1. cont'd.
Obs N<5 9 rank. 048
00      02        O«        OB        OB       1.0
TESTS BASED ON RANK SCORES»
.ip
original dala. The same perturbation of yi changes the Wilcoxon regression rankscore test statistic from 13.17 to 14.70 with a correlation between the two rank vectors of 0.87. A more robust initial estimator would improve the performance of the aligned rank test somewhat. The regression rank score version of the test is seen to be relatively insensitive to such perturbations. One should be aware that comparable perturbations in the X7 design observations may wreck havoc even with the rank score form of the test. Recent work of Antoch and Jurečková (1985) and deJongh, deWet, and Welsh (1988) contain suggestions on robustifying regression quantiles and therefore the corresponding regression rank scores to the effect of influential design points.
Compulation of the tests was carried oul in S+ using ihe algorithm described in Koenker and d'Orey (1987, 1990) to compute regression quantiles.
3. PROPERTIES OF REGRESSION RANK SCORES
Consider ihe linear regression model (2.1) with design X„ of dimension n x p. Let ß(&)eW be the a-regression quantile and ä(a)e/?" be the vector of ath regression rank scores defined in (2.7). We see from ihe form of the linear constraints in (1.5) that the regression rank scores are regression invariant, i.e.,
än(ff, Y + Xb) ^ ä„(a. Y),       b e R".                         (3.1)
Moreover, in view of the invariance, we may assume
JU-0,       / = 2,...,p                        (3.2)
/-i
without loss of generality.
Our primary interest in this section will be the properlies of ihe regression rank scores process
(M* 0*1*1}.                                        (3-3)
Gutenbrunner and Jurečková (1992) studied the process
W* = [wJO = V^ t *AW: 0 a r < 1 j                 (3.4)
and showed that
Vft(t)^VÍ(t) + op{l)                                    (3.5)
where
I*) = n~u2 Í áJlE, > F-'OM                            (3.6)
ř-l
as n—»oc uniformly on any fixed interval [e, 1 — e\, where 0<£<2 for any appropriately standardized triangular array {áni: i = 1,... , n] of vectors from R*. They also showed that the process (3.3) (and hence (3.4)) has continuous trajectories and, under the standardization E^i dnr = 0, (3.5) is tied-down to 0 at r = 0, and / = 1. The same authors also established the weak convergence of (3.4) to ihe Brownian bridge over [e, 1 - s]. Note however that Theorem V.3.5 in Hájek and Šidák (1967) establishes the weak convergence of (3.4) to the
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Brownian bridge over the entire interval [0,1] in Ihe special case of the location submodel. Here we extend the results of Gutenbrunner and Jurečková (1992) in the tails of [0,1], in order to find the asymptotic behavior of the rank scores and the test statistics (2.7) and (2.8), for which the score functions are not constant in the tails.
H may be noted that this extension is rather delicate. If the rank scores involved integration from e to 1 — t (i.e., if (p were constant near 0 and 1). then the earlier Gutenbrunncr-Jurečková (1992) representation theorem could be used to obtain the asymptotic distribution theory here under somewhat weaker hypotheses (see the remark following Theorem 5.1). It is the desirability of treating such tests as the Wilcoxon and Normal Scores Tests that requires the extensions here. Nonetheless, the fact shown here that the rank score process can be represented uniformly on an interval (a*, 1 — a*) with a* decreasing as a negative power of n (precisely, o-'= ŕí"1'<l",4ŕ) for some b>Q) is rather remarkable and of independent theoretical interest.
To this end, we will assume that the errors Bif..., £„ in (2.1) are independent and identically distributed according to the distribution function F(x) which has an absolutely continuous density /. We will assume that / is positive for A<x<B and decreases monotonically as x-*A+ and x-*B-where
-co<J4 = sup{x:F(^) = 0}    and    +™>B = '\ni{x\ F(x) = l}.
For 0<a< 1, let y>a denote the score function corresponding to (1.2):
V*(x) = a - I[x <0],       x € R1.                             (3.7)
We shall impose the following conditions on F:
(F.l) \F-l(a)\*c(a(l-a))~a for 0<aSflo,   l-a0Sa<l,  where 0<a<
\-e, e>0 and c>0. (F.2) 1/'/'(F'l{a))sc(a(\ - a))~l~a for 0<asa0 and l-a0sa<l, c>0. (F.3) f(x) >0 is absolutely continuous, bounded and monotonically decreasing as
x—*A+ and x—* B-. The derivative /' is bounded a.e. (F-4)
fix)
fix)
£cbr|       for \x\* K ^0,       c>0.
Remark. These conditions are satisfied, for example, by the normal, logistic, double exponential and t distributions with 5, or more, degrees of freedom. Condition (F.l) implies J |i|4T,1dF(/)< +« for some 0>Q. Hence using (F.4) also, F has finite Fisher Information, a fact to be applied in Theorem 5.1. The following design assumptions will also be employed.
(X.l)*i«l,i-l,...tii
(X.2) UmM_DM = D where D„ = n  'X;,X„ and D is a positive definite pXp matrix.
(X.3) B-IS?-i|M4,= 0(i)a*"-*w-
(X.4) maxl*JKn\\xl\\ = 0(n(2l'>-a>-Ay<,+4b)) for some 6>0 and Ô>0 such that 0 < b - a < e SI (hence 0 < b < J - e/2).
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C. GUTENBRIJNNER et at.
The following theorem which follows from Theorem 3.2 is an extension of Theorem V,3.5 in Hájek and Šidák (1967) to the regression rank scores. Some applications of this result to Kolmogorov-Smirnov type tests appears in Jurecková (1991).
Theorem 3.3. Under the conditions of Theorem 3.2, as n-*x
sup    irw 2 cU<M«) -M*))   ^°
Moreover, the process
A-V^IUit^OSffSl
i=i
(3.47)
(3-48)
converges to the Brownian bridge in the Prokhorov topology on C[0, \\. Proof. By Theorem 3.2,
sup       n^SMô-W-M«))^0-
(3.49)
Further, using the fact that £ľ~i (1 - ä^(a)) = not, due to the linear constraints in (1.5),
sup   \n'll2^dnA„(a)
OSo^o,* 1           .     i
sup
« ,'22'Ul-<aa))
i-i
á n"2 max K,| «; =■- ory«*«»-)-««**)-»*»**)) = 0{n'™)   (3.50)
and we obtain an analogous conclusion for supi.,*.^^ I«"1" E"-t 4«á»í(#)l- On the other hand,
sup
«-■ a" >",;
,-',
!!-■.-    .v-
«-inE*rÄ«) "■   W    iTwS<WW<*_1(*H~*)
i=i
s max \dj. 0„(*S(1 - a*))™ = op{\)       (3.51.)
! -■ r ■. íl
and analogously
sup
n "* X ^Bfäii«)

= 0.(1).
Thus (3.47) follows, and consequently (3.4B).   D
4. ASYMPTOTIC PROPERTIES OF SIMPLE LINEAR REGRESSION RANK SCORES STATISTICS
Maintaining the notation of Section 3, let q>{l):0<t<l be a nondecreasing square-integrable score-generating function and let b„„ i = 1,.. -, n be the scores denned by (2.7). Let {d„} be a sequence of vectors satisfying (D.1)-(D.3) of Lemma 3.2.
TESTS BASED ON RANK SCORES
327
Following Hájek and Šidák (1967), we shall call ihe statistics
«.-«-"S^Ai                          (4-1)
1=1
simple linear regression rank-score statistics, or just simple linear rank statistics. Our primary objective in this section is to investigate the conditions on q> under which we may integrate (3.47) and obtain an asymptotic representation for S„ of the form
Sn = n',a Í d,MHEd) + 0,(1).                       (4.2)
i-i
We shall prove (4.2) for <p satisfying a condition of the Chernoff-Savage (1958) type; thus our results will cover Wilcoxon, van der Waerden (Normal), and median scores, among others.
Theorem 4.1. Let <p(t):0<t<\, be a nondecreasing square integrable function such that <p'(f) exists for 0 *c t < aQ, 1 — aQ < / < 1 and satisfies
l<p'(OI*c</(l-0)-'-fl*                                   (4.3)
for some Ô"<Ô where Ô is given in condition (X.4), and for te(0, <v0)U (1-tfo, 1)- Then, under (R1)-(F.4), (X.1)-(X.4) and (D.l)-(D.3) of Lemma 3.2, the statistics S„ admits the representation (4.2) and hence is asymptotically normally distributed with zero expectation and with variance
A2( J( <p\t) ót-py       <P = { <p{t) dr.                       (4.4)
Proof- Let us consider 5„ defined in (4.1) with the scores (2.7). Integrating by parts (notice that a„t(t) - ä<(t) = 0 for t = 0, 1), we obtain
-n*idtikf <P« <H<M0 - 5,(0) - h""2 t änl \\äm(t) - Ář(0) Wň.    (4.5)
which we must show is ofr(l). We shall split the domain of integration into the intervals (0, a*]t {a*, a0), [a0,1 - a0], (1 - «0> 1 - *")> U - «1,1) and denote the respective integrals by /,,... , 75. Regarding Theorem 3.2, we immediately get that /v^*0 by the dominated convergence theorem. Similarly, for some
«•>t
ran                        «
I4is     IVC01 »""S«*-«-««)
J«.;                        --i
*c  wi^or^'wi-o)"2- »^wi-or^x-UMO-w» *
J«;                                                                                i-1
JOD «i-or-%-w*-*,(i)-«v(i). t?
328                                             C. GUTENBRUNNER et al.
Finally,
|/,|<«-Iŕ2 max 14*1 fV(')l Ž lU0-M0ld^/.. + /i2
' - 'i             JQ                       ŕ=l
= «-|ainax|rfJ      l?>'<Ol2(l-<U0)dŕ                   (4.6)
a = n-ia max K,.| ply'íOI Í (l-4(0)dŕ.                   (4.7)
and
/,* =
Then
/„s«10 max |dfl,| rV^dí-Ofr^^^W^-G-*-)««**))
= O(n-2tó'ó'i'0','Ab)). Finally,
fe = "~"3 S 4* f"<í>'(0/[< > FfäÜ *
ŕ-1        ^o
= n""2 i <[?(*:) - v(F(£,))lflf(£») < *:]
i-l
Now wt may assume that q>(a*)<0 for n^nn, since otherwise if q> were bounded from below then la^* 0- Hence
Var(/12) s«"1 Í ^£([2(p(ŕ-(£())]2/[F(E,) < «„*]) * f" <p2(«) du ■ O(l)->0
/-I                                                                       -"0
due to the square-integrability of ip. Treating the integrals /4, /5 analogously, we arrive at (4.5) and this proves the representation (4.2)    D
5. TESTS OF LINEAR SUBHYPOTHESES BASED ON REGRESSION RANK SCORES
Returning to the model (2.2), assume that the design matrix X = (X,: X3) satisfies the conditions (X.1)-(X.4), (2.3) and (2.4). We want to test the hypothesis ri0:ß2 = 9 OS, unspecified) against the alternative Hn:ß2„ = n~l'2ß0 {ßoeR" fixed).
Let %„(<*) = (á„i(íV)(.... ä„„(a))   denote  the  regression  rank  scores  corresponding to the submodel
V = X,& + E under /&.                                   (5.1)
Let ?(f):(0, 1)~>R' be a nondecreasing and square integrable score-generating function. Define the scores Sni, i = I,... , n by the relation (2.7), and consider the test statistic
T^S'&'S JA\q>)                              (5.2)
where
S„ = «-l/2(Xn2-Xn,)'bn                                   (5.3)
TESTS BASED ON RANK SCORES
329
and where Q„ and A2(<p) are defined in (2.4) and (2.10), respectively. The lest is based on the asymptotic distribution of T„ under //„. given in the following theorem. Thus, we shall reject H0 provided Tn s^(oí), i.e. provided Tn exceeds the (o critical value of the %2 distribution with q d.f. The same theorem gives the asymptotic distribution of T„ under Hn and thus shows that the Pitman efficiency of the test coincides with that of the classical rank test.
Theorem 5.1. Assume that X, satisfies (X.1)-(X.4) and (X,iX3) satisfies (2.3) and (2.4). Further assume that F satisfies (F.1)-(F.4). Let T„ defined in (5.3) and (5.4) be generated by the score function q> satisfying (4.3), and nondecreasing and square-integrable on (0,1).
(i) Then, under M,, the statistic T„ is asymptotically central y} w',n q degrees of
freedom, (ii) Under H„, Tn is asymptotically nonccntral x with q degrees of freedom and
with noncentrality parameter.
T)2 = ß'oQß«YH<P.F)tA2(<P)                               (5.4)
with
),F) = -fV(0d/(F-'(0)-
y(<p,F)=-    <p(i)df(F-'(t)).                              (5.5)
Remarks. (I) If <p is of bounded variation and is constant near 0 and 1. the representation given in Theorem 2(ii) of Gutenbrunner and Jurcčková (1992) could be used to provide the conclusion of Theorem 5.1 under somewhat weaker hypothesis; namely, (X.l), (X.2), max, ||x,|| = o(nir2), F has finite Fisher Information, and0</<^on {x:0<F(x)<l).
(ii) The analogy between the location and regression models concerning the noncentrality parameter y(<p, F) may be extended in the following way: instead of defining local alternatives via (2.6). the definition of Behnen (1972) can be generalized to the regression model. That is, with F,(0~ F(l ~ x\ßi) and Gi = L(Yj), consider
H0:Gi = F,   vs.   Wn:~ = l+xL&A,(/v) where
fa-n-Mßo.       h„-j+h€L2(0,l).   and   max fell ||/iJ|i = o(«L2).
In this setting, even without the assumption of finite Fisher Information, (4.2) implies that the conclusion of Theorem 5.1 holds with y(rp, F) in (5.4) replaced by the F-independeni constant
Y*(<p A)=        f(<p(u)-<p)(h(u)-h)du
(J(«P(W)-^)2dU/(y.(H)-Ä)^d^),'3•
i.e., the correlation of the functions q> and h. Such local alternatives provide insight into the structure of the regions of constant efficiency for regression rank
330
C. GUTENBRUNNER et at.
Proof, (i) II follows from Theorem 4.1 that, under Hq, S„ has the same asymptotic distribution as
šn=n-lfl(xn2-xn2yba
where b„ = (BnU. .., £„„)' and S,u = <p{f{E,)), i = 1,..., n. The asymptotic distribution of S„ follows from the central limit theorem and coincides with (/-dimensional normal distribution with expectation 0 and the covariance matrix
(ii) The sequence of local alternatives Hn is contiguous with respect to the sequence of null distributions with the densities {H"=i/(e\)}. Hence, (4.1) holds also under //„ and the asymptotic distributions of S„ under H„ coincide. The proposition then follows from the fact that the asymptotic distribution of Š„ under Hn is normal Nq{y(<p, F)Qß0. QA2(<p)).    O
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