Ui\iV£RSlTY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
 BOOKSTACKS 
 
o 
 
 Z (0 
 
 - — c 
 
 >•! 
 21 
 
 u 
 
 CD (/) 
 
 5 5 
 
 < o 
 
 2 c 
 *5 
 
 M 
 
 ijil 
 
 <T\ 
 
 
 00 
 
 
 <x> 
 
 
 rH 
 
 >* o 
 
 1 
 «5l« 
 
 Eh S 
 
 r- 
 
 JHtf 
 
 V© 
 
 « D u: w 
 
 O r-l 
 
 PQUS^o, 
 
 0\ • 
 
 w < o < 
 
 en o 
 
 oq wm 3: a. 
 
 rH 2 
 
 C4 r-i o o\ 
 
 00 f- 
 
 CM CM C\J iH 
 
 
 Oh 
 
 oo 
 
 A VO 
 E r-l 
 > I 
 
 v 
 
 m 
 
 o oo 
 
 n ro 
 
 • ft 
 o o 
 
 A ^ 
 E-« *3 
 2 H 
 H 
 
 ai4-i 
 ft, o 
 
 s 
 
 M • 
 v D 
 
 OS < 
 
 § < 
 
 H OS 
 
530 
 
 385 
 
 d. 1676 COPY 2 
 
 STX 
 
 BEBR 
 
 FACULTY WORKING 
 PAPER NO. 90-1676 
 
 Tests of Linear Hypotheses Based 
 on Regression Rank Scores 
 
 
 C. Gutenbrunner 
 J. Jurcekovd 
 R. Koenker 
 S. Portnoy 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois Urbana-Ci lampaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 90-1676 
 
 College of Commerce and Business Administration 
 
 University of Illinois at CIrbana-Champaign 
 
 August 1990 
 
 Tests of Linear Hypotheses 
 Based on Regression Rank Scores 
 
 C. Gutenbrunner 
 
 J. Jureckova 
 
 R. Koenker 
 
 S. Portnoy 
 
 Dedicated to the memory of Jaroslav Hajek 
 
 Philipps Gniversitat, Marburg, Germany 
 
 Charles University, Prague, Czechoslovakia 
 
 University of Illinois at (Jrbana-Champaign, USA 
 
 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. Jureckova and R. Koenker. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/testsoflinearhyp1676gute 
 
ABSTRACT 
 
 We propose a general class of asymptotically distribution-free tests of a linear hypothesis in the linear 
 regression model. The tests are based on regression rank scores, recently introduced by Gutenbrunner 
 and Jureckova (1990) as dual variables to the regression quantiles of Koenker and Bassett (1978). 
 Their properties are analogous to those of the corresponding rank tests in location model. Gnlike the 
 other regression tests based on aligned rank statistics, however, our tests do not require preliminary 
 estimation of nuisance parameters. 
 
 AMS 1980 subject classifications: 62G10, 62J05 
 
 Key words and phrases: Regression quantile, regression rank scores 
 
TESTS OF LINEAR HYPOTHESES BASED ON REGRESSION RANK SCORES 
 
 C. Gutenbrunner, J. JureSkovl, R. Koenker, S. Portnoy 
 
 Philipps University, Marburg, Germany 
 
 Charles University, Prague, Czechoslovakia 
 
 University of Illinois at Urbana-Champaign, U.S.A 
 
 1. Introduction 
 
 Several authors including Koul (1970), Puri and Sen (1985) and Adichie (1978) have 
 proposed and investigated asymptotically distribution-free tests of some types of linear 
 hypothesis for the linear regression model based upon aligned rank statistics. A good review 
 of these results including extensions to multivariate models may be found in Puri and Sen 
 (1985) and Adichie (1984). The hypothesis under consideration typically involves nuisance 
 parameters which should be estimated by a preliminary estimate; the aligned (or signed) rank 
 statistics are then based on residuals from the preliminary estimates. Alternative approaches 
 to inference based on rank estimation have been considered by McKean and Hettman- 
 sperger(1978), Aubuchon and Hettmansperger (1988) and Draper (1988). 
 
 In this paper we explore an alternative approach to the construction of rank tests for the 
 linear regression model based on the regression rank scores introduced in Gutenbrunner and 
 JureSkovd (1990). Regression rank scores represent a natural extension of location rank 
 scores introduced in Hdjek and Siddk (1967, Section V.3.5) which stand in one-to-one rela- 
 tion to the ranks of sample observations. The tests based on regression rank scores offer a 
 natural extension of rank-based methods of testing to the general linear model and avoid many 
 of the difficulties introduced by preliminary estimation of nuisance parameters in prior propo- 
 sals. 
 
 The next section of the paper surveys our results and establishes notation. 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. Section 5 contains a formal treatment of the 
 proposed tests. And Section 6 describes an example. 
 
 2. Notation and preliminary considerations 
 
 Consider the classical linear regression model 
 
 Y = X0 + E (2.1) 
 
 which u c will partition as 
 
 Y = X& + X^ 2 + E (2.2) 
 
 where fi t and B 2 are P~ an d <7 -dimensional parameters, X = X n is a known. nx(p+q) design 
 matrix with rows x ra -' = x, ' = (x lf -\ x 2 , ') € R p+1 , i«l,..., n . We will assume throughout that 
 x tl = 1 for i = 1 ,...,«. Y is a vector of observations and E is an nxl vector of i.i.d. errors 
 with common distribution function F. 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 
 -oo < A = sup{x: F(x) = 0) and +oo > B = inf{x: F(x) = 1}. Moreover, we shall impose 
 some conditions on the tails of / assuming, among other conditions, thai / monotonically 
 decreases to when x — ► /I + , or x — > B-. Denoting D n = n~ 1 X 1 'X 1 and 
 
 Hi = X^X/X^X/ and Q„ = n^(X 2 - X 2 )'(X 2 - X 2 ) (2.3) 
 
 with X 2 = H X X 2 being the projection of X 2 on the space spanned by the columns of X l5 we 
 shall also assume 
 
 lim D„ = D, lim Q„ = Q (2.4) 
 
 n— mx> n— K)o 
 
 where D and Q are positive definite (pxp) and (qxq) matrices, respectively. 
 We are interested in testing the hypothesis 
 
 #o : 02 - 0, Pi unspecified (2.5) 
 
 versus the Pitman (local) alternatives 
 
H n : /? 2n =«- 1/2 /?o (2.6) 
 
 with O being a fixed vector from R q . 
 
 The regression rank scores introduced in Gutenbrunner and Jureikova (1990) arise as a 
 vector of solutions 
 
 Ua) = (a nl (a),..., a TO (a))', < a < 1 (2.7) 
 
 of the dual form of the linear program required to compute the regression quantile statistics of 
 Koenker and Bassett (1978). More precisely, the vector p^a) = (^(a),..., p (a))' e R p of 
 regression quantiles corresponding to the submodel 
 
 Y = X^ + E (2.8) 
 
 is any solution of the minimization 
 
 EpJXi - Xu'tX teR* (2.9) 
 
 t'=l 
 
 where 
 
 P a (x) = \X | {(l-a)/[x<0] + al[x>0]}, x 6 R 1 . (2.10) 
 
 Koenker and Bassett (1978) characterized the regression quantile as the component of the 
 optimal solution (/?, r + , r~) of the linear program 
 
 al m 'r + + (l-a)l„'r~ : = min 
 
 X^ + r + -r-=Y (2.11) 
 
 0€R P , r+, r~eR^ 
 
 and 1„ = (1,..., 1)' g R n , < a < 1. Finite-sample as well as asymptotic properties of 0(a) 
 are studied in Koenker and Bassett (1978), Ruppert and Carroll (1980), Jureckova (1984), 
 Gutenbrunner (1986), Koenker and Portnoy (1987), Gutenbrunner and Jureckova (1990) 
 among others. 
 
The dual program to (2.10) can be written in the form 
 
 Y'a(a) : = max 
 
 X 1 'S(a) = (l-a)X 1 'l n (2.12) 
 
 a(a) e [0, l] n , < a < 1 
 
 As shown in Gutenbrunner and Jurefckovd (1990) many aspects of the duality of order statis- 
 tics and ranks in the location model generalize naturally to the linear model through (2.11) 
 and (2.12). 
 
 To motivate the approach, let us illustrate (a„(a), < a 1} in the location model which in 
 the present notation may be viewed as (2.1 1) with X x = l n . Then S ra (a) specializes to 
 
 a^(a) = a n *(/?,, a) = 
 
 1 if a < (/?,-!)/« 
 
 Ri-an if (*,— l)/w < a < RJn (2.13) 
 
 if RJn < a 
 
 where R { is the rank of Y,- among Y lv .., Y n . The function a„0',a), 7=1,..., n, 0<a< 1, 
 coincides with that introduced in Hajek and Siddk (1967, Section V.3.5). Under the general 
 model (2.1) the finite-sample as well as asymptotic properties of the regression rank scores and 
 of the process {a n (a), < a < 1} are studied in the next section. 
 
 As in the classical theory of rank tests, we shall consider a score-function 
 4> : (0, 1) — ► R 1 which is nondecreasing and square-integrable on (0, 1). We may then con- 
 struct scores in the following way: 
 
 £» = -£ 4>it)d^{t), / = !,..., n. (2.14) 
 
 Defining 
 
 S^n-^iX^-^Yk (2.15) 
 
 where b„ = (b nlr ., o^)', we propose the following statistic for testing H against H n : 
 
r.-s,'Qi%/^) (2.16) 
 
 with 
 
 A\4>) = £ m)-??dt % $ = £ 4>(t)dt (2.17) 
 
 and with Q„ defined as in (2.3). An important feature of the test statistic T n is that it requires 
 no estimation of nuisance parameters, since the functional A (<f>) depends only on the score 
 function and not on F. This is familiar from the theory of rank tests, but stands in sharp con- 
 trast 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 (1988) and Draper (1988). 
 
 As we shall show in Section 5, the asymptotic distribution of T n under H is central x 2 
 with q degrees of freedom while under H n it is noncentral x 2 with q degrees of freedom and 
 noncentrality parameter 
 
 r? 2 - [T 2 ^, F ) / A\<j>)WQPo (2.18) 
 
 where 
 
 li*. F)--£+(t)df(F*(t)) (2.19) 
 
 The quantities i and A are familiar from the theory of rank tests. The test based on T n 
 is asymptotically distribution free in the sense that, under H , neither T n nor its asymptotic 
 distribution depend on F. Moreover, it follows from (2.18) that the Pitman efficiency of the 
 test based on T n with respect to the classical F test of H coincides with that of the two- 
 sample rank test of shift in location with respect to the / -test. For / unimodal, we obtain an 
 asymptotically optimal test if we take 
 
 ^(0 = ^/(/) = - | /f ,( - F ~y^ , 0</<l. (2.20) 
 
 f(F~Kt)) 
 
Thus the asymptotic relative efficiency of the test based on T n , relative to the classical F 
 test is, for Wilcoxon scores (<t>(u) = u - 1/2). is 3/ir = .955 at the normal distribution and is 
 bounded below by .864 for all F '. When F is heavy tailed this asymptotic efficiency is gen- 
 erally greater than one, and can in fact be unbounded. For normal (van der Waerden) scores 
 (<f>(u ) = $ -1 (w )) the situation is even more striking. Here the test based on T n 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 look at the scores (2.14), which can be written as 
 
 £.=-/*(')£* W i = \,.,n (2.21) 
 
 where the functions a ra - '(/) are piecewise constant on [0,1]. In the location model, using (2.13) 
 this reduces to 
 
 There are three typical choices of 4>: 
 
 (i) Wilcoxon scores: <j>(t ) - t - 1/2, < / < 1. The scores are 
 
 L = -I \t - U2)dUt) = J \u)dt - 1/2 
 
 while A\<f>) = 1/12, and i(<f>, F) = Jf 2 (x)dx. Wilcoxon scores are optimal when / is 
 the logistic distribution. 
 
 (ii) Normal (van der Waerden) scores'. <j>{t) = $ -1 (0> < t < 1, $ being the d.f. of stan- 
 dard normal distribution. Here A 2 (<f>) = 1 and i(<f>, F) = jf{F~ l (^{x)))dx. These scores 
 are asymptotically optimal when / is normal. 
 
 (iii) Median (sign) scores: 4>{t) = ^sign^-te), 0</ < 1, then (2.14) leads to the form 
 ^m = Sm'(Vz) which is non-zero if and only if the l x estimate passes through the /th 
 observation. 
 
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'0 = reR« (2.22) 
 
 where Risa(/?+<?)x# matrix of rank q < p. Let Vbea(/?+#)xp matrix such that 
 A = [V ; R]' is nonsingular and R'V = 0. Set 7 = Ap and Z = XA -1 . Partitioning 
 7 ■ [7i' i 7a'l' where i x = V'P and 7 2 = R'/?» under the hypothesis (2.22) we have 
 
 Y - XR(R'R)" 1 r = XV(V'V)- 1 7 i + E. 
 
 Thus, in view of the cquivariance of regression quantiles we may define Y = Y - XR(R'R) _1 r, 
 \ x = XV(V'V)- 1 , X 2 = XR(R'R) -1 , and proceed as previously discussed with (Y, X l5 X 2 ) play- 
 ing the roles of (Y, X x , X 2 ). By this device the tests described above and detailed in Section 5 
 below 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. 
 
 3. Properties of regression rank scores 
 
 Consider the linear regression model (2.1) with design X„ of dimension n x p. Let 
 p{a) gR p be the a-regression quantile in (2.1 1) and a(a) e R n be the vector of ath regression 
 rank scores defined in (2.15). We see from (2.12) that the regression rank scores are regression 
 invariant, i.e., 
 
 *.(<*, Y+Xb) = a„(a, Y), beR". (3.1) 
 
 Moreover, in view of (2.12), we may assume 
 
 £*<, =0, y=2,...,p (3.2) 
 
 without loss of generality. The formal duality between £(a) and a(a) implies that for / = 1, ..., n 
 
a m (a) = 
 
 p 
 
 1 if l>E**0y(«O 
 
 T 
 
 o if r,<x;^^(a) 
 1=1 
 
 while the components of a n (a) corresponding to {;' | K, = x, '/?(«)} are determined by the 
 equality constraints of (2.12). 
 
 Our primary interest in this section will be the properties of the regression rank scores 
 process 
 
 {Kit): 0</ < 1}. (3.4) 
 
 Gutenbrunner and Jure£kova(1990) studied the process 
 
 W* = {W n d (/) = v^ EdJUO : < t < 1} (3.5) 
 
 and showed that W*{t) = U*(t) + o p (1) where 
 
 Ufr) = /I" 1 / 2 Ed ra /[£, > F~\t)] (3.6) 
 
 as n — * oo uniformly on any fixed interval [e,l-e], where < e < 1/2 for any properly stand- 
 ardized triangular array {d ra - : / = 1,.„, n) of vectors from R«- They also showed that the pro- 
 cess (3.4) (and hence (3.5)) has continuous trajectories and, under the standardization 
 
 n 
 
 J]df» = 0, (3.5) is tied-down to at / =0, and / = 1. The same authors also established the 
 
 weak convergence of (3.5) to the Brownian bridge over [e, 1-e], Note however that Theorem 
 V.3.5 in Hajek and Sidak (1967) establishes the weak convergence of (3.5) to the Brownian 
 bridge over the entire interval [0, 1] in the special case of location submodel. Here we extend 
 the results of Gutenbrunner and JureSkova (1990) into the tails of [0,1], in order to find the 
 asymptotic behavior of the rank scores and the test statistics (2.14) and (2.15). 
 
 To this end, we will assume that the errors E lt ..., E n in (2.1) are independent and identi- 
 cally 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 mono- 
 tonically as x — > A +, and x — ► B- where -oo < A = sup {x: F(x) = 0} and +00 > B = inf 
 {x: F(x) = 1). For < a < 1 denote rp a the score function corresponding to (2.10) 
 
 $ a (x ) = a - I[x < 0], x 6 R 1 . (3.7) 
 
 We shall impose the following conditions on F: 
 
 (FA) |F _1 (a)| < c(a(l-a))^* for < a < a , l-a <a < 1, where < a < % - e, e > 
 
 and c > 0. 
 
 (F.2) 1//(F-Ha)) < c(a(l-a))" 1 ^ for < a < a and l-a < a < 1, c > 0. 
 
 (F.3) / (x ) > is absolutely continuous, bounded and monotonically decreasing as x —* A + 
 and x —> B-. The derivative / ' is bounded a.e. 
 
 (F.4) 
 
 /'(*) 
 
 fix) 
 
 <c\x\ 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. 
 
 The following design assumptions will also be employed. 
 
 (X.1) x $1 -l, i=l » 
 
 (X.2) lim D„ = D where D„ = n _1 X n 'X n and D is a positive definite p x p matrix. 
 
 (X.3) n - 1 £ 11^,- II 3 = O ( 1 ) as n -+ 00. 
 
 (X.4) max||x,|| = CK/iW^M^ 1 -* 6 )) f or some b>0 and £>0 such that < b-a <e/2 (hence 
 
 l<t'<7l 
 
 < b < y 4 - e/2). 
 We may now define 
 
 a ; = n -i/(i-Hb) (38) 
 
 and 
 
10 
 
 Mi^L, <,<«<!. (3.9) 
 
 /(F-'(a)) 
 
 and prove the following crucial lemma. 
 
 LEMMA 3.1 Assume that F satisfies (F.l) - (F.4) and that X„ satisfies (X.l) - (X.3). Then, 
 as n — ► oo, 
 
 sup{|r n (t, a)\ : ||(|| < C, a n # < a < 1-a.V (3-10) 
 
 for any fixed C > 0, where 
 
 r n (t, a) = (ad-a^/V-^tPaC^.a - n^a^'i)- p a (E ia )) 
 
 i=i 
 
 + n-^ad-ar^ExrWfJ - %t'D„t (3.11) 
 
 »=i 
 
 and 
 
 E ia = E, ,- F-Ha), 1=1, ...,». (3.12) 
 
 PROOF. 
 
 (i) First fix a e [a tt *, l-a„*] and t such that ||t|| < C. 
 Define 
 
 fl B = minfo-^-J/W 1 -" 6 )), /j- 2 */(i-4M) (3.13) 
 
 We wish to show that for any A > 
 
 P(\r n (t,a)\ >(X+l)B n )<Kn- x (3.14) 
 
 with a fixed K > 0. To do this, we will use the Markov inequality 
 
 P(K(t, a)\ > s n ) < exp(-us n )(M(u) +M(-u)\ u>0 (3.15) 
 
 where M{u) = Eexp(ur n (t, a)). 
 
11 
 
 Denote 
 
 c ra - = e„(t, a) = w 1/2 a a x,'t 
 
 (3.16) 
 
 and 
 
 tf,(t, a) = (a(l-a))- 1 VW£ ia -"- 1/ V,'t)-p a (£ ia )] 
 
 + fi^ad-a))" 1 ^ >«(£* J-Mi/i^Cx,- 't) 2 i =1 «. (3.17) 
 
 By definition of E ia , a a , p a and V'c 
 
 ^•(t,a)+%/t- 1 0c,'t) 2 = (ct(l-a))-*' !i <T« 1 {(E ic re ai )Ile ni <E ia <0] 
 
 + (e„-£,J/[0<£, a <e m ]} 
 
 (3.18) 
 
 and hence, uniformly for a n * < a < 1-a^, ||t|| < C and / = 1,...,«, 
 
 |tf,(t, a) + 1 kn-\x i 'if\ < 2«- 1 / 2 (a(l-a)r 1/2 |x,'t|=(9(«-( 2o -^/( 1 - K6 )). (3.19) 
 
 If uR { is bdd, that is < u < «( 2a+5 )/( 1 - K6 ) may be expanded to obtain. 
 
 log Mgfii) < uERi(t, a) + c« 2 Var(/?,(t, a)) 
 
 (3.20) 
 
 for some constant c > 0. By (3.18) and conditions F.1-F.4, for e„- > and for a n *< a < a , 
 l-a n *> a > l-a , 
 
 ERi(t, a) + %«- 1 (x f 't) a < (a(l-a))- 1 / 2 a-7j m '(e m -z)/ o *(|F- 1 (a)| +W )/(F- 1 (a))^^z 
 
 < c(a(l-a))- 1 / 2 - 2a «^/ 2 |x,'t| 3 +c(a(l-a))- 1 - 2a «- 2 |x,'t| 4 (3.21) 
 
 and we get the same inequality for c m < 0. The same expressions are 
 0(«^/ 2 |x,'t| 3 ) + 0(>7- 2 |x.-'t| 4 ) if a < a < l-a . Hence, 
 
 ££/?,(t,a) + %t'D n t = <9 
 
 ' -2(6 -a) ~\ 
 1+46 
 
 (3.22) 
 
 Similarly, 
 
12 
 
 £Var/?,-(t,a)< £E/tf(t,a) 
 
 «=i 1=1 
 
 a(l-a) 
 
 ^2 
 
 i=i 
 
 (3.23) 
 
 -(26 -o) 
 
 = 0(/7- 1 / 2 -(a(l-a)) 1 / 2 //(/ r_1 a))) = C>(« l+lt ). 
 
 These results hold uniformly in a, t. 
 
 Hence, using (3.15) and (3.13) with u = log n/B n = 0(n 3a ^ 1M % so that 3.20 holds, 
 
 -2(6 -a) 
 
 -(2b- 
 
 P(\r n (t,a)\>(\ + l)B n )<exp{-(\ + l)\ogn+(K\ogn/B n )-n l+ " + (* log 2 « /5 2 ) • n ^ } 
 
 < n 
 
 -x 
 
 (3.24) 
 
 for n >n where K > and « do not depend on a and t. 
 
 (ii) Following the proof of Lemma A.2 in Koenker and Portnoy (1987), choose intervals 
 [a,, a, +1 ] of length \/n h covering [a B *, l-a„*| and balls of radius 1//2 5 covering {t: ||t|| < C}. 
 Let (a x , a 2 ) c [a,, a, +1 ] and t u X^ lie in one of the balls covering {t:||t|| < C). For /' £ {l x , / 2 } we 
 use (3.17) (and the boundedness of/ and / ') to obtain 
 
 \R i {h,a l )-R l {X 2 ,a 2 )\< 
 
 a 1 (l-a 2 ) 
 
 -i^ * 
 
 |(F- 1 (a-) + 6 ra )| 
 
 + /(F"»(a") + € ra )^- 1 (a ) | 
 
 -V~*< 
 
 1 
 
 1 
 
 a 1 (l-a 1 ) a 2 (l-a 2 ) 
 
 •-U~*' 
 
 Ofi(l-a 2 ) 
 
 + /7- 1 / 2 (a 1 (l- Ql ))- 1 /2|| x ,|||| tl -y 
 
13 
 
 + Cn-^WxiW IMl-cO)- 1 /* - (a 2 (l-a 2 ))-^\ = CK*" 2 ); 
 
 |ai-a 2 | 
 
 here a * e (a, , a, +1 ) and we have used the inequalities F 1 (a 1 ) - F l (a 3 )| < 
 
 f(F-\a)) 
 
 s + -}^ 
 
 = 0{n 1446 ) = Oin- 37 ^. Hence, on any ball in the covering, \r n (t lt a x ) - r n (t 2 , a 2 )l < K/n. 
 Since the number of sets needed to cover the set S =[a„, 1-cO x{t : ||t|| < C) is bounded by 
 « 5 (p+ 1 ) wc obtain from (3.14) for A > 50 + 1) 
 
 P(sup |r„(t, a) | > (A + l)5 n + K /n) < n 5 ^^n^ -* ■ 
 
 (a,t)€S 
 
 LEMMA 3.2. Assume the conditions of Lemma 3.1 and let d„ = (d nU - - • ,d m Y be a sequence 
 of vectors satisfying 
 
 X„'d n = 0, -i-£42 - A 2 , < A 2 < oo (D.l) 
 
 n i=i 
 
 n~ l Y,\ d ni\ Z = 0{\) as n-^oo (D.2) 
 
 i=i 
 
 max |rf„- | =0 |„(2(6-a)^)/(i-K6) (D 3) 
 
 Ki<n 
 
 
 then 
 
 p 
 
 sup {(ad-a))- 1 /^- 1 ^^ l^^aCE.^-VV^/t) - a (£ ta ) + n" 1 / 2 ^,- 't]| } - (3.25) 
 
 as n -» oo for any fixed C > and for a* given in (3.8). 
 PROOF. Consider the model 
 
 Y = X'p* + E (3.26) 
 
 where X* = (X„ : dj, p' = (ft, • • • ,ft,,/? p+1 ). Then 
 
X*'X* = 
 
 14 
 
 X n 'X n o 
 
 d„'d„ 
 
 and the conditions of Lemma 3.1 are satisfied even when replacing X by X*. Computing the 
 right derivative of (3.1 1) with respect to teR p+1 , we arrive at (3.25). ■ 
 
 Let fi n (a) be the a-regression quantile corresponding to the model (2.1) with the design 
 matrix of order (n xp ); i.e. , /?»(<*) is a solution of the minimization 
 
 YsPJXi ; - x t - 't) : - min, / € R p . 
 
 (3.27) 
 
 The following theorem establishes the rate of consistency of regression quantiles, and is 
 needed for the representation of the dual process. 
 
 THEOREM 3.1. Under the conditions (F.l) - (F.4) and (X.l) - (X.4), 
 
 n l ft*?@{a) - 0(a)) = n^\a{\-a))-^Ji n Y l rl> a (E ia ) + o,(l) 
 
 »=i 
 
 uniformly in a n * < a < l-a n *. Consequently, 
 
 . sup . tffivf&Jfx) - fta))\\ = 0,(1). 
 
 <a<l-a„ 
 
 *».:«-= ^^"n 
 
 (3.28) 
 
 (3.29) 
 
 PROOF. If /9„(a) minimizes (3.27), then 
 
 T na = « 1 /V- 1 (J9„(a)-^(a)) 
 minimizes the convex function 
 
 (3.30) 
 
 Gna(t) = (a(l-a))- 1 / a a- 1 E[P«(^«-n- a/4 ^x i -'t)-p a (E itt )] 
 
 8=1 
 
 with respect to t e R p . By Lemma 3.1, for any fixed C > 
 
 min G na (t) = min{-t'Z na + %t'D B t} + o p (l) 
 
 n<c \\4<c 
 
 (3.31) 
 
 (3.32) 
 
15 
 
 uniformly in a n * < a < l-a„*, where 
 
 Z tta = B^a(l-a))^J]^a(U (3.33) 
 
 Denoting 
 
 V na = arg min{-t' Z na +%t'D„t}, (334) 
 
 we immediately get 
 
 U^-D^Z^-OpO) (3.35) 
 
 uniformly in a^ < a < 1-a^ and 
 
 min {-t'Z^+fct'D.t} = -%Z na T)?Z na . (3<36) 
 
 From (3.35) and (3.36), we can write 
 
 t'Z na + 1 / 2 t'D n t= %{(t-U na )'D n (t-U„ a )-U na 'D„U na ) (3.37) 
 
 and hence we could rewrite (3.10) in the form 
 
 sup K(t, a) I = sup {C na (t)-%[(t-U na )'D n (t-U na )-U na 'D ft U„J} - 0. (3.38) 
 
 (a,t)€S (a,t)es 
 
 Inserting U na = O p (l), for t, we further obtain 
 
 . sup jUG^flO + %U na 'D„U n J } = o p (l). (3 39) 
 
 a n <a<l-a n 
 
 We would like to show that 
 
 . sup ,{||T na -UJ|} = o p (l). (340) 
 
 ar„<a<l-a n 
 
 Consider the ball B na with center U na and radius 8 > 0. This ball lies in a compact set with 
 probability exceeding (1 - e) for n > n ; actually, for t G B na , 
 
 lltll < l|t-U na || + ||IU| <6 + K, 
 
 for some K t with probability exceeding 1 -e fjorn > n . Hence, by (3.10), 
 
16 
 
 p 
 A na = , sup , sup |r„(t, a)| — 0. (3.41) 
 
 Following Pollard (1988), consider the behavior of G na (t) outside B„ a . Suppose 
 t a = U n01 + k £, k > 6 and ||£|| = 1. Let t* be the boundary point of B„ a that lies on the line from 
 U na to t a , i.e., t a *=U na + *£ Then t a * = ( 1 - (6/k ))U na +(6/k )t a and hence, by (3.38) and 
 (3.39), 
 
 S/kCJX) + (W/*)CJUJ > G na (0 > WXo + G na (U na ) - 2A na 
 
 where A is the minimal eigenvalue of D. Hence, 
 
 inf <7 na (t)> (7 na (U na ) + (k /S)(WX - 2A na ). (3 42) 
 
 Using (3.39) the last term is positive with probability tending to one uniformly in a for any 
 fixed 6 > 0. Hence, given 6 > and e > 0, there exist n and r\ > such that for n > n , 
 
 />( . inf J inf C na (t) - C na (U na )] > V )> 1 - e 
 
 *n ^" ^ * "n 
 
 (3.43) 
 
 and hence (since the event in (3.43) implies that G na must be minimized inside the ball of 
 radius 6) P( t sup t \\T na - U„J < S) — ► 1 for any fixed 6 > 0, as n -* oo. ■ 
 
 a n <a<l-a n 
 
 The following theorem approximates the regression rank score process by an empirical 
 process. 
 
 THEOREM 3.2. Let d n satisfy (D.l) - (D.3), X n satisfy (X.l) - (X.4) and F satisfy (F.l) - 
 (F.4). Then 
 
 . sup < {|/t- 1 / 2 (a(l-a))- 1/2 E^m(S n ,(")-3,(a))|}-0 (3.44) 
 
 as n -+oo, where 
 
 a, (a) = /[£,- >F~\a)], / = l,...,/i. (3.45) 
 
 PROOF. Insert n l l 2 o- l (p n {a) - 0(a)) for t in (3.25) and notice (3.29) and the fact that 
 
17 
 
 .sup .{n-^aO-cOr^Erfrf/m -x,'i§(a)]}-» 0, ( 3 .46) 
 
 from which (3.44) follows. ■ 
 
 The following theorem which follows from Theorem 3.2 is an extension of Theorem V.3.5 in 
 Hajek and Sidak (1967) to the regression rank scores. Some applications of this result to 
 Kolmogorow-Smirnov type tests will appear in Jureckova(1990). 
 
 THEOREM 3.3. Under the conditions of Theorem 3.2, as n — ► oo, 
 
 sup (In-^E^CM") " S»(«))l> * (3.47) 
 
 0<a<l 
 
 Moreover, the process 
 
 {A -i„-i/2 E ^.g ra(a) : o < a < 1} (3.48) 
 
 converges to the Brownian bridge in the Prokhorov topology on C[0, 1]. 
 PROOF. By Theorem 3.2, 
 
 . sup . Iw-^ErfidUW - *«(«))! - 0. (3.49) 
 
 t=i 
 
 Further, 
 
 inpJn-^Erf-M*)!- sup JB^S^O-Ua))!^" 1 ^ maxilla; 
 
 <a<a tt »=1 0<a<a n , = x !<«'<+* 
 
 = o 
 
 w ' 1-H6 1+46 
 
 = 0(«-«) (350) 
 
 and we obtain an analogous conclusion for syp l« _1 ^]C^mS»(a)|. On the other hand, 
 
 l-<*ri<<*<l »=i 
 
 sup , |n^E^«S;(a)l = sup . \n-*/*Zd ml (I[E i < F^(a)] - a)\ 
 
18 
 
 < max !</.. | • O p ( a ;{l-aW* = o p (\) (3.51) 
 
 l<l<n 
 
 and analogously 
 
 sup |»-^E<f^(o)| -0,(1). 
 
 1 -a n <a<l ,' 
 
 Thus (3.47) follows, and consequently (3.48). 
 
 i=l 
 
 4. Asymptotic properties of simple linear regression rank scores statistics 
 
 Maintaining the notation of Section 3, let ^(/ ) : < / 1 be a nondecreasing square- 
 integrable score-generating function and let o m , / = !,...,« be the scores defined by (2.14). Let 
 {d n } be a sequence of vectors satisfying (D.l) - (D.3) . 
 
 Following Hajek and Siddk (1967), we shall call the statistics 
 
 S n = n-^dj^ (4.1) 
 
 the simple linear regression rank-score statistics, or just simple linear rank statistics. Our pri- 
 mary objective in this section is to investigate the conditions on 4> under which we may 
 integrate (3.47) and obtain an asymptotic representation for S n of the form 
 
 S n = n-^d^HFiEi)) + 0,(1). (4.2) 
 
 We shall prove (4.2) for <f> satisfying a condition of the Chernoff-Savage(1958) type; thus our 
 results will cover Wilcoxon, van der Waerden, and median scores, among others. 
 
 THEOREM 4.1. Let 4>(t) : < / < 1, be a nondecreasing square integrable function such that 
 <f>'(t) exists for < t < a , l-a < t < 1 and satisfies 
 
 l^'(/)l <c(/(l-/))" lV (4.3) 
 
19 
 
 for some 6* < 6 where 6 is given in condition (X.4), and for / e (0, a ) u (l-a , 1). Then, 
 under (F.l) - (F.4), (X.l) - (X.4) and (D.l) - (D.3), the statistic S n admits the representation 
 (4.2) and hence is asymptotically normally distributed with zero expectation and with variance 
 
 ^\jJ\t)dt-^\ $~f Ht)dt. 
 
 (4.4) 
 
 PROOF. Let us consider S n defined in (4.1) with the scores (2.14). Integrating by parts 
 (notice that £»•(<) - a,(0 = for / = 0, 1), we obtain 
 
 -"- 1/2 XX- jjW&At)- 3,(0) = n" 1/2 E^ / Wo - S.-(OW(0. (4.5) 
 
 «=i 
 
 i=i 
 
 which we must show is o p (l). We shall split the domain of integration into the intervals 
 (0, a n *], (<*„*, a ), [a , l-a ], (l-a , l-a n "), [1-a^, 1) and denote the respective integrals by 
 
 F 
 
 / x , ... ,/ 5 . Regarding Theorem 3.2, we immediately get that / 3 — ► by the dominated conver- 
 gence theorem. Similarly, 
 
 |/ 2 | </? 1^(01 |«- 1 / 2 E^»(S„,(/)-a,(/))|^ 
 
 .«o. 
 
 <cj .(/(l-or 14 * (/(i-0) 1/2 - l»- 1/ft (/(i-0)-*/ a x;^--(S«(0-ff.-(/))l* 
 
 Finally, 
 
 where 
 
 ,°o 
 
 = c/.(/(l-0) 5 - 1 /2^. 0p (l) = 0p (l). 
 
 *^» ° ,=i 
 
 and 
 
 ,«o 
 
 / u = w- 1 / 4 max \d*\ / |*'(/)| Ed - MOW 
 
 f'=l 
 
 (4.6) 
 
20 
 
 7 12 = n" 1 / 2 max |J ra | C \4>'{t)\ £(1 - Ut))dt. (4.7) 
 
 Then 
 
 / u < n 1 ' 2 max |c/ ra | / V dt = 0(w l+ " 1 " K6 ) = 0(n~ 2 ^ >). 
 
 Finally, 
 
 * 
 /12 = «- 1/2 E^« / °V(O/[/ > F(£,)]^ = n-^Zd^M*^ - ^(F(£,))]/(F(£:,) < o^ 
 
 »=i «=i 
 
 Now we may assume that <£(<*„*) < for n > w , since otherwise if 4> were bounded from below 
 
 p 
 then 1 12 — * 0. Hence 
 
 * 
 ^r(/ 12 ) < n^Erf^C^CFC^))] 2 /^^) < a„*]) < C<t>\u)du> (9(1) - 
 
 • =1 
 
 due to the square-integrability of <f>. Treating the integrals 7 4 , 7 6 analogously, we arrive at (4.5) 
 and this proves the representation (4.2). ■ 
 
 5. Tests of linear subhypotheses based on regression rank scores 
 
 Returning to the model (2.2), assume that the design matrix X = (X x • X 2 ) satisfies the 
 conditions (X.l) - (X.4), (2.3) and (2.4). We want to test the hypothesis H : fi 2 = (Pi 
 unspecified) against the alternative H n : p 2n = « _ly,2 ^o (Po^ R q fixed). 
 
 Let S„(a) = (S nl (a), ..., & m (a)) denote the regression rank scores corresponding to the 
 submodel 
 
 Y = X 1 £ 1 + E under 77 . (5.1) 
 
 Let <£(/):(0, 1) — > R 1 be a nondecreasing and square integrable score-generating function. 
 Define the scores &„, / = !,...,« by the relation (2.14), and consider the test statistic 
 
 T n =S n 'Q?SJAH<(>) (5.2) 
 
21 
 
 where 
 
 S n =n- l /HX n2 -St n2 Yb n (5.3) 
 
 and where Q„ and A 2 (<f>) are defined in (2.4) and (2.17), respectively. The test is based on the 
 asymptotic distribution of T n under H , given in the following theorem. Thus, we shall reject 
 H provided T n > x 2 (<*>), te. provided T n exceeds the w critical value of the x 2 distribution 
 with q d.f. The same theorem gives the asymptotic distribution of T n under H n and thus 
 shows that the Pitman efficiency of the test coincides with that of the classical rank test. 
 
 THEOREM 5.1. Assume that Xj satisfies (X.l) - (X.4) and (X x \ X 2 ) satisfies (2.3) and (2.4). 
 Further assume that F satisfies (F.l) - (F.4). Let T n defined in (5.3) and (5.4) be generated by 
 the score function <f> satisfying (4.3), and nondecreasing and square-integrable on (0, 1). 
 
 (i) Then, under H , the statistic T n is asymptotically central x 2 with q degrees of freedom. 
 
 (ii) Under H n , T n is asymptotically noncentral x 2 with q degrees of freedom and with non- 
 centrality parameter, 
 
 •7 3 -A>'Q/W(*.nA4 2 (*) (5.4) 
 
 with 
 
 Tf(rf,F) --£«/*// (F-*(/)). (5.5) 
 
 PROOF. 
 
 (i) It follows from Theorem 4.1 that, under // , S n has the same asymptotic distribution as 
 
 S n =n-^(X n2 -X n2 yb n 
 
 where b n = (b nl ,...,b m )' and F m = ^(F(£,)), i = l,...,n. The asymptotic distribution of S„ 
 follows from the central limit theorem and coincides with <7 -dimensional normal distri- 
 bution with expectation and the covariance matrix Q • A 2 {4>). 
 
22 
 
 (ii) The sequence of local alternatives H n is contiguous with respect to the sequence of null 
 
 n 
 
 distributions with the densities {n /(?,)}. Hence, (4.1) holds also under H n and the 
 
 «=i 
 
 asymptotic distributions of S n under H n coincide. The proposition then follows from the 
 fact that the asymptotic distribution of S n under H n is normal NA~i{4>, F)Q0 o , QA 2 (<j>)). 
 
 6. An Example 
 
 To illustrate the tests proposed above we consider briefly an example taken from Adi- 
 chie (1984, Example 3). The log of the leaf burn (in seconds) of 30 batches of tobacco is 
 thought to depend upon the percent composition of nitrogen, chlorine, and potassium. Adi- 
 chie 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 -.529 and -.290 with an intercept 
 of 2.653.. With these preliminary estimates we obtain aligned 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% xi critical value of 
 6.63. 
 
 In contrast the Wilcoxon regression rank scores computed as 
 ft = -£(t - l/2)dUO= f Q \(t)dt - 1/2 
 and based on the restricted model excluding potassium, are 
 
 -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. The full set of regression rank scores £,(/) for this data are 
 illustrated in Figure 6.1 with the plots ordered according to their Wilcoxon rank score. Note 
 
23 
 
 that as a practical matter when ? = / 4>{t )dt = 0, we may omit the X 2 term in the computation 
 of S„ in (5.3) since b n is orthogonal to X v This is in contrast with the aligned rank situation 
 where the use of X 2 - X 2 is essential. 
 
 Corresponding calculations for the normal scores using 
 
 1.00 
 
 1.00 
 
 -1.00 
 
 1.00 
 
 -1.00 
 
 -1.00 
 
 -1.00 
 
 1.00 
 
 1.00 
 
 -1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 1.00 
 
 -1.00 
 
 -1.00 
 
 -0.37 
 
 1.00 
 
 1.00 
 
 1.00 
 
 yields 
 
 -0.74 0.15 -1.41 0.23 -0.91 -2.13 -0.45 -1.17 2.08 -0.15 
 0.63 0.10 -1.25 1.44 0.78 1.15 0.50 1.35 0.40 -0.72 
 1.41 -0.40 -0.61 -0.03 0.94 0.30 0.39 -1.45 -0.26 -0.18 
 
 and a test statistic of 12.87. The corresponding normal score aligned rank statistic is 1 1.72. 
 
 Finally, regression rank score version of the sign test yields the scores 
 
 ■1.00 1.00 -1.00 
 
 1.00 0.16 -1.00 
 
 ■1.00 -1.00 -0.79 
 
 and a test statistic of 8.42 while the aligned rank sign scores yield 10.20. Obviously, all ver- 
 sions of the tests lead to a decisive rejection of the null. Note that for the sign scores the test 
 coincides with the l x Lagrange multiplier test discussed in Koenker and Bassett(1982). 
 
 Since an important objective of the proposed rank tests is robustness to outlying obser- 
 vations, it is interesting to observe the effect of perturbing the first v observation of the Adi- 
 chie data set on the aligned and rank scores versions of the test statistic. This sensitivity 
 analysis is illustrated in Figure 6.2. Even a modest perturbation in y l is enough to confound 
 the initial least squares estimate and reverse the conclusion of the aligned rank test. However 
 the regression rank score version of the test is seen to be relatively insensitive to such pertur- 
 bations. One should be aware that comparable perturbations in the design observations may 
 wreck havoc even with the rank score form of the test. Recent work of Antoch and 
 Jureikovd (1985) and deJongh, deWet, and Welsh (1988) contain suggestions on robustifying 
 
24 
 
 regression quantiles to the effect of influential design points. 
 
 Computation of the tests was carried out in 5+ using the algorithm described in 
 Koenker and d'Orey (1988, 1990) to compute regression quantiles. 
 
 REFERENCES 
 
 Adichie, J.N. (1978). "Rank tests for sub-hypotheses in the general linear regression," Am. 
 Statist., 6, 1012-1026. 
 
 Adichie, J.N. (1984). "Rank tests in linear models," in P.R. Krishnaiah and P.K. Sen (eds.), 
 Handbook of Statistics,, vol. 4, Elsevier, New York. 
 
 Amemiya, T. (1985). Advanced econometrics, Harvard University Press, Cambridge. 
 
 Antoch, J. and Jureckova, J. (1985). "Trimmed LSE resistant to leverage points," Comp. Sta- 
 tist. Quarterly, 4, 329-339. 
 
 Aubuchon, J.C. and Hettmansperger, T.P. (1989). "Rank based inference for linear models: 
 Asymmetric errors," Statistics and Probability Letters, 8, 97- 1 07. 
 
 Chernoff, H. and Savage, I.R. (1958). "Asymptotic normality and efficiency of certain non- 
 parametric test statistics," Ann. Math. Statist., 29, 972-994. 
 
 Draper, D. (1988). "Rank-based robust analysis of linear models. I. Exposition and Review," 
 Statistical Science, 3, 239-271. 
 
 Gutenbrunner, C. (1986). Zur Asymptotik von Regression quantileprozessen und daraus 
 abgeleiten Statistiken, Ph.D. Dissertation, Universitat Freiburg. 
 
 Gutenbrunner, C. and Jureckova, J. (1990). "Regression rank-scores and regression quan- 
 tiles," Ann. Statist (to appear). 
 
 Hajek, J. and Sidak, Z. (1967). Sidak, Z. (1967). "Theory of rank tests," Academia, Prague. 
 
 deJongh, P.J.,deWet, T. and Welsh, A.H. (1988). "Mallows type bounded-influence-regression 
 trimmed means," J. Amer. Statist. Assoc, 83, 805-810. 
 
 Jureikova, J. (1971). "Nonparametric estimate of regression coefficients," Ann. Math. Statist., 
 42, 1328-1338. 
 
 JureSkova, J. and Sen, P.K. (1989). "Uniform second order asymptotic linearity of 
 A/-statistics in linear models," Statistics & Decisions, 7, 263-276. 
 
 Koenker, R. and Bassett, G. (1978). "Regression quantiles," Econometrica, 46, 33-50. 
 
 Koenker, R. and Bassett, G. (1982). "Tests of linear hypotheses and /x-estimation," Econome- 
 trica, 50, 1577-83. 
 
25 
 
 Koenker, R. and d'Orey, V. (1987). "Computing regression quantiles," Applied Statistics, 36, 
 383-393. 
 
 Koenker, R. and d'Orey V. (1990). "Remark on algorithm 229," preprint. 
 
 Koul, H.L. (1970). "A class of ADF tests for subhypotheses in the multiple linear regression," 
 Ann. Math. Statist., 41, 1273-1281. 
 
 Lehmann, E.L. (1959). Testing Statistical Hypotheses, Wiley, New York. 
 
 Lehmann, E.L. (1983). Theory of Point Estimation, Wiley, New York. 
 
 McKean, J.W. and Hettmansperger, T.P. (1978). "A robust analysis of the general linear model 
 based on one-step R -estimates," Biometrika, 65, 571-79. 
 
 Pollard, D. (1988). "Asymptotics for least absolute deviation regression estimators," 
 Econometric Tlieory, to appear. 
 
 Puri, ML. and Sen, P.K. (1985). Nonparametric Methods in General Linear Models. J. Wiley, 
 New York. 
 
 Ruppert, D. and Carroll, R.J. (1980). "Trimmed least squares estimation in the linear model," 
 J. Amer. Statist. Assoc, 75, 828-838. 
 
 Scheffd, H. (1959). The Analysis of Variance, Wiley, New York. 
 
Figure 6.1 
 Regression Rank Scores for Tobacco Data 
 
 Obs No 6 rank= -0.48 
 
 Obs No 28 rank= -0.42 
 
 Obs No 3rank= -0.41 
 
 02 0.4 6 8 1.0 
 
 0.0 02 0.4 0.6 8 1.0 
 
 0.2 4 0.6 8 10 
 
 Obs No 8 rank= -0.38 
 
 Obs No 13rank=-0.37 
 
 Obs No 5 rank= -0.32 
 
 02 0.4 0.6 8 1.0 
 
 02 4 0.6 0.8 1.0 
 
 0.0 02 4 6 8 1.0 
 
 Obs No 1 rank= -0.27 
 
 Obs No 20 rank= -0.26 
 
 Obs No 23 rank= -0.23 
 
 0.0 02 0.4 6 8 1.0 
 
 2 0.4 6 8 1.0 
 
 0.0 0.2 4 06 08 1 
 
 Obs No 7rank=-0.17 
 
 Obs No 22rank=-0.16 
 
 Obs No 29 rank= -0.1 
 
 0.0 0.2 0.4 6 8 
 
 05 0.4 0.6 8 1.0 
 
 0.2 4 6 8 10 
 
 Obs No 30 rank= -0.06 
 
 Obs No 10rank=-0.06 
 
 Obs No 24 rank= -0.01 
 
 00 02 4 6 08 1 
 
 2 4 6 8 10 
 
 0.0 02 4 6 8 10 
 
Figure 6.1 
 (continued) 
 
 ObsNo 12rank=0.04 
 
 Obs No 2 rank= 0.06 
 
 Obs No 4 rank= 0.09 
 
 0.2 0.4 6 8 1.0 
 
 02 0.4 6 8 1.0 
 
 02 4 6 
 
 Obs No 26rank=0.12 
 
 Obs No 19 rank= 0.15 
 
 Obs No 27rank=0.15 
 
 0.0 2 0.4 0.6 8 1.0 
 
 02 0.4 6 8 1.0 
 
 0.0 02 4 6 0.8 10 
 
 Obs No 17rank=0.19 
 
 Obs No 1 1 rank= 0.23 
 
 Obs No 15 rank=0.28 
 
 02 4 6 8 10 
 
 0.2 0.4 6 8 1.0 
 
 0.2 4 6 8 1.0 
 
 Obs No 25 rank= 0.33 
 
 Obs No 16rank=0.37 
 
 Obs No 21 rank= 0.38 
 
 0.2 0.4 6 8 10 
 
 2 0.4 6 8 1.0 
 
 00 0.2 04 06 08 10 
 
 Obs No 18rank=0.41 
 
 Obs No 14 rank=0.42 
 
 Obs No 9 rank= 0.48 
 
 02 4 6 8 10 
 
 0.2 4 6 8 10 
 
Figure 6.2 
 
 Sensitivity Curves for Rank Tests 
 
 CM 
 
 -*— ' __ 
 
 0) 
 
 CD O O 
 
 XI -^ i- 
 ■♦- CO 
 
 o CO 
 
 CD to CO 
 
 _^ 
 
 CO 
 > 
 
 CD - 
 
 M- - 
 
 
 h\ 
 
 
 /: \ 
 
 
 
 
 
 
 Aligned 
 Rankscore 
 
 
 
 
 
 ■10 
 
 
 
 10 
 
 Perturbation of y(1) 
 
HECKMAN 
 
 BINDERY INC. 
 
 JUN95 
 
 Bound -To -Pie 
 
 N MANCHESTER 
 INDIANA 46962