Chapter VI
LIMITING DISTRIBUTIONS OF TEST STATISTICS UNDER THE ALTERNATIVES
1. Contiguity
1.1. Asymptotic methods. Contiguity. The asymptotic approach consists in, regarding a given testing problem as a member of a sequence {Hv, Kv}, v ^ 1, of similar testing problems. In this sequence the v-th testing problem concerns Nv observations Xlt ...,Xfiv wilh Nv -* co as v -> co. As a rule, Hv depends on v through A',, only, i.e. H, = H(N^), whereas Kr depends on some parameters dvl, 0 í f í Nv, in addition. For example, we might assume that Hv = H0, H0 being applied to N = A'tf observations, and that JC, consists of a single density qt,
«»° UMxi - dvt) •
Of course, there are infinitely many such sequences, and we try to choose one which resembles the given testing problem as much as possible. First of all it would be desirable to keep the envelope power function ß(z, Hv, Kv) independent of v. Since this is usually difficult or even impossible, we shall be satisfied with the existence of a limit ß(x):
(0                            Kmß(a,HvtKv) = ß(x)t   Oáaál.
As in the previous chapter, we shall also consider indexed sets of testing problems {Hd,Kd,deD}, where the convergence will be equivalent to the convergence to a fixed limit for all sequences selected from the set and satisfying certain requirements. As a rule, Kä will be simple, consisting of a density qd.
The limiting relation (l) entails that ß(v,Hv,KT) will approximately equal ß(a) for v ä v0. The usefulness of the asymptotic results will depend on whether the problems "#» against K" with v ^ v0, may occur in practice or not. The value of v0 is usually guessed on the basis of numerical calculations for selected v's and the assumption that the convergence is more or less monotone.
In this book we shall not investigate the somewhat degenerate cases in which
(2)
/9(«) =1   for aU   <x > 0.
202
VI. 1.2
Wc shall even exclude the cases in which
(3)                                           ß(x)-**0   for    a-»0.
However, it may be shown, that in problems dealt with in the sequel, (3) implies (2).
The requirement thai (3) should not take place finds its theoretical expression in the notion of contiguity, which is due to LeCam (1960). The notion of contiguity is basic for the asymptotic methods of the theory of hypothesis testing.
Consider a sequence {pv, q,} of simple hypotheses p, and simple alternatives qv defined on measure spaces (Xv, sft, /jv), v ž 1, respectively.
Definition. If for any sequence of events {A,}, A„ € stv,
(4)                                      IPJA,) - 0] * IQJIA.) -> 0]
holds, wc say that the densities qv are contiguous to the densities pw, where dPv = = pvdft„dQ, = qvdttv, v ž 1.
If Hv is composite, we say that qy is contiguous to Hv if for each v the convex hull J7, of //, contains a density pv such that (4) holds.
If both Ht and K, are composite, we say that Kt is contiguous to Ht if (4) holds for some pv e Bv and q, e Kt.
Contiguity implies that any sequence of random variables converging to zero in Pv-probability converges to zero in £)T-probability, v -+ co.
1.2. LeCam's first lemma. According to the Ncyman-Pearson lemma, for any event Ar there exists a critical function >PV such that
(1)                                     #, - 0 ,    if   qy< fc.p,,
where 0 á { £ 1, and that
= 1,    if   qt> krpv,
P,«) = kdP,,
QiA^Ú UrdQv.
Thus contiguity will follow if we show that
(2)                                      f*, dPv -> ol => I" j*v dö, -+ ol
for critical functions of the type (l).
VI. I. 2
203
Introduce the likelihood ratio L, = 4,/p,, or more precisely,
(3)                    M*,) = 4,(x,)M*v).   if   ftfc)>0,
= 1,                    if    pr(jcj = 9,(xJ = 0 ,
= co ,                  if   p,(x,) = 0 < 4,(*v) ,
where x¥ denotes the typical point of the space X¥. » ž 1. Let F, be the distribution function of Lv under P¥:
(4)                                                  F^x)-P^Sx), where L¥ = Z^), v^l.
Lemma. Assume that F, given by (4) converges  weakly [at continuity points) to a distribution function F such that
(5)
rW(x)=K
Jo
Then the densities qt are contiguous to the densities pv, v 2; 1-
Proof. Take a sequence of critical functions <PV of the type (l) and suchthat
(6)
Then note that
(?)
ŕ
fdP,->0.
šy|V,dP¥ + í       dߥ = = y|#,dJ\ + 1 - ľ       dô.--yU9dP9+l- í       I*dPr-
=>'f*vdpv+i - r.xdpv.
Now for any e > 0 we can find a continuity point y of F such that, in view of (5),
1 -
x dF < le .
:ui
VI. J. 2
Since F„-> F entails
we shall have for some v0 (8)                                      1-
íXdF-íXdF
x df¥ < \i,   v £ v0 .
Furthermore, (6) ensures the existence of v-j such that
M
0, dPy < j£ ,   v £ v,.
Finally, from (7) through (9) il follows that
*> dÖ. < £   for   v £ max (v0, yj .
J-
Thus J\rv dö, -+ 0, which concludes the proof.
Remark. Note that contiguity does not entail that the Q, are absolutely continuous with respect to the P,. The singular part of Qv. however, must tend to zero,
Ô,(p, = 0) - 0
as a consequence of Pv(pv = 0) = 0 -* 0.
The asymptotic distribution of the likelihoods Lv will regularly be log-normal. We shall say that a random variable Vis log-normal (/*, <r2), iľlog Y is normal (/(, o2). Now let us establish the condition under which we obtain
ZY
xáF = 1
for a log-normal random variable V. We obviously have
El'= Eexp(logy) = (2n)-*ff-1 ľ" exp [x - 4(.v - nf <*~2] dx = e"_io1. This equals 1 for
(10)                                             „=  -±a2.
Thus wc have the following
Corollary- U *-> >s asymptotically log-normal ( — \<j2,o2), then the densities q, are contiguous to the densities pv.
VI. 1.3
205
13.   LeCam's second lemma. Assume that x, = (x1(..., xÄJ and
(i)                                   pJM - ŤÍM*b
t=i
and
(2)                             f&)-ÖMh)-
From (l) and (2) we have
i=l
(3)                         logL^Slog^^.O/AW]
i-i
which makes sense even if log L, = ±co, since on the right side the summands are <co with P,-probabiIity 1 and are > — x with (^-probability I.
Thus we may regard log Lv as an extended random variable allowed to attain — co with positive probability under P„. However, asymptotic normality of logL,, is defined in the same way as for an ordinary random variable, i.e. as convergence of PT(Iog L, < x) to a normal distribution function in every real point x. Thus asymptotic normality entails ^„(IogL, = — co) -* 0.
In what follows we restrict ourselves to cases in which the summands in (3) are uniformly asymptotically negligible, i.e.
(4)                                lim   max P,
»-as  l<i£AV

>  E      =0
Under this condition necessary and sufficient conditions of asymptotic normality arc well-known. These conditions arc considerably simpler if the summands have Unite variance. However, this fails sometimes to be satisfied in (3) within the class of problems considered below. For this reason we instead consider the statistic
.=]
which always consists of summands with finite variances, as may be easily seen, and has additional advantages. The following lemma, due to LeCam, shows that asymptotic normality of log L, may be established by proving asymptotic normality of If,.
Lemma. Assume thai (4) holds and lhal the statistics l^,vž 1, are asymptotically normal ( — $<7Z, o1) under Pv. Then the statistics log L, satisfy
(6)                        limP,(|logLv- »; + ä<T2|> s) = 0,   e>0.
and are asymptotically normal ( — \g2, a2) under Pt.
206                                         ________________________________VI. 1. 3
Proof. If a function h(x) has a second derivative /»"(x), then
(7)                                   h(x) = h(xo) + (x-xo)h'(x0) +
+ i(x - xoy ľ 2({ - X) h«[x0 + X(x - x0)] dX,
as may be easily seen by integration by parts. Thus, putting
(8)                                      Tvi = 2[gvl(Xl)!frl(Xiy]*-2, we obtain
',   r "ibx/ x                  iogfe„//„) = 2iog(i + ir„) =
** *TfH                = t« - m ľ'[2(i - m + i^wH d..
Consequently,
(9)                   log Lv = W, - - £ 7* f [2(1 - X)i(l + UTvif] dX .
This holds even for logLv = -co. Introduce
Tf., = rw, if |r„|$af
= 0 ,      otherwise .
As is well known (Louvg (1955), p. 316), asymptotic normality ( — £<t2, a2) of \VV implies under (4) that for every ô > 0
(io)                               Ži\(Irw[>á)->o,
(n)                                 EetS-,-iff2,
(12)                                      SvwTS-*^.
i-l
Now (10), holding for eoéry ô > 0, entails
(13)                       Z ii f [2(1 - i)/(i + iÁr¥í)2] dA ~ £ (r£)
n*
VI. i. 3
207
where ~ denotes thai the ratio of both sides tends to 1 in /^-probability. Thus, in order to prove (6), it remains to show that
1-1
in Pv-probability. For this purpose it suffices to prove
(15)                                                ŽE(7Í()'-„!
1-1
and
(16)                                     lim lim sup £ var (7j)2 = 0,
6-0      »-»      (-1
since then (14) will follow by the Chebyshev inequality. Further, in view of (12), (15) is equivalent to
(17)                                                Í"(E72)2-0.
We first prove (17). If Ô > 2, then 7j $ Tti, since Tvl £ -2, in view of (8). Consequently,
(18)                           E7J š ET,, = 2E{gvi(Xi)//w(*i)}i - 2 £
ú 2{E[ff,^)//vi(X1)]}i - 2 =
M
= 2 jí       ff„(x)d.xj   -2 SO.
Ui;.i>oi            J
Thus, for ô > 2,
S(ElS)aS  min  EISIeiS,
and (17) follows from (11) and from the fact that
min E7Í-*0,
which is an easy consequence of (4). Now it remains to note that the validity of (17) for any Ô > 2 entails its validity for any Ô > 0, because of (12) and of
E e(ü)! š E E(rt')3,  ô, < ô2 .
1*1                      1=1
As for (16), first note that
E «r [(Iff] S E E(7?,)4 S í* J E(7?r)'.
i-1                               (=1                            (=1
208
VI. I. 4
Thus, on account of (15),
(19)                                    limsup£var(Tj)2á<iV.
»-»   i = i
Consequently, (16) holds.
Finally, asymptotic normality (—Iff2, it2) is an immediate consequence of (6). Q.E.D.
The reader, having finished the above proof, can observe that we did not evade niceties connected with the truncation of summands. However, they are concentrated in the above proof and will not trouble us any more.
1.4. LeCam's third lemma. Limiting distributions of test statistics Sy under the alternative arc important from the point of view of the power properties of the respective tests. Unfortunately, their derivations are considerably more difficult than the proofs of limiting distributions under the hypothesis. Nonetheless, in the contiguity case, the difficulties are essentially diminished by the following lemma due to LcCam.
We say thai the pair (S,, log Lr) is asymptotically jointly normal (/(,, /i2, a\, o\> ffl2) if it converges in distribution to a normal vector (Zlt Z2) such thai EZ,- = /((, varZ, = fff, / ■ 1,2, and cov (Zt,Z2) = ax2. (For convergence in distribution in k ž 2 dimensions see the definitions of Section V.2.1.)
Lemma. Assume that the pair (S,, logLv) is under i% asymptotically jointly normal (/i,, //2, a\, a\, ff12) with jt2 = ~\o\-
Then S, is under Q, asymptotically normal (/i, + ffI2, a]).
Proof. Obviously,
(1)              Q,(Sy g x) - f       dßv -
J {5„S*}
|         L,dP,+ Ô¥(pv = 0,Sv^.v) =
J IS.S*)
- r     ľ e" áh\(u, v) + QXp, = 0, S, á x) ,
J —to J — to
where Fv{u, v) denotes the distribution function of (Sv,logL,). Now /i2 = — \o\ implies contiguity (Corollary 1.2), and hence
(2)                                              Qv(p, = 0,S,Sx)-»0,
VI. 1.4
209
since Pv(pv = 0, 5¥ á x) = 0 -»0. Furthermore, for any c > 0
J"*       ("ť                                    ŕX       fC
(3)                                         e'dF/«,»)-* j           e"d<D(u,i;),
J—toj—c                            J-toJ—c
where 4>(u, ľ) denotes the two-dimensional normal distribution function with parameters (u,, p2l a\, o\, ffu). Actually, Fv -* <b according to our assumption, and the function
h(u, v) = e",    — co < u < x,     -cSdíc
= 0,    otherwise,
is uniformly bounded and continuous except on the set {{u, v) : v = — c or v — c or u=x], which obviously has ^-probability 0. Thus we may apply DI of Section V.2.1. Now (1), (2) and (3) will imply
(4)                                   Ô,(S,ž:c)-r    r  e'd*(«,p)
J -10 J -<•>
if we show that for every e there exist c0 and v0 such that
(5)                                ľ 'V dFv + T     fV dFw < s,    v ä v0 .
J   — CC J — M                 J—«Jco
In other words we must show that the truncated parts of the integral are uniformly small if c0 's sufficiently large. However, (5) is an easy consequence of contiguity. Actually, if (5) were not true for some c > 0, we would have a sequence of pairs (cj, Vj) such that
(6)                                           lim Cj = co ,    lim vj = co ,
j-*n                          J—a>
and
ôv/log LVJ < -Cj or log Lt] > cj) =
ľ pVdFv+r pdFv,ä
J — co J - co                      J — co J c j
r ľvM* í
J —to *) — ")                      J — to J<
On the other hand, since log L, is asymptotically normal under Pv,
Pv,(Iog LtJ < -cj or log LVJ > Cj) -» 0 , because of (6). This contradicts contiguity, and thereby (4) is proved.
14 — Hijelc-Sidik: Theorv of Rank TeM*
VI. 1. 4
209
since Pv(pv = 0, S, g x) = 0 -» 0. Furthermore, for any c > 0
(3)                       T    ľC"dFT(<M>)->r    fe-d^r),
J—coJ—c                                    J—aiJ—e
where 0(«, ľ) denotes the two-dimensional normal distribution function with parameters (/i,, fi2, a\, a\, <71?). Actually, Fv -* O according to our assumption, and the function
h(u, v) = e",    - co < u < x,    - c ž t> 5 c = 0,    otherwise,
is uniformly bounded and continuous except on the set {(u, v):v= — c or v = c or u=x}, which obviously has 0-probability 0. Thus we may apply Dl of Section V.2.1. Now (1), (2) and (3) will imply
(4)                                       ß/S,Sx)->r    f" e'ďb(u,v)
J -<B J —m
if wj show that for every £ there exist c0 and v0 such that
ŕx      f~CQ                     px       pso
(5)                                           e°dF,+             e'dFt<e,   vžv0.
J—«J--«              J — at J eo
In other words we must show that the truncated parts of the integral are uniformly small if c0 is sufficiently large. However, (5) is an easy consequence of contiguity. Actually, if (5) were not true for some c > 0, we would have a sequence of pairs {cj, Vj) such that
(6)                                           lim Cj = co ,   lim v, = co ,
J-<©                  J-«
and
Qtfi°S Lvj< -Cj or log LfJ > Cj) -
r*    c~et               p«    f«
C*dF„ +           e'dF„š
J -«J -<o              J -xjcj
ľ     |"V<U\,+ ľ     fVdF„ä
J — <a J — »                      J — co J ci
■"í
On the other hand, since log L, is asymptotically normal under Pr,
PTJ(Iog Lv, < -cj or log Lr, > cj) -♦ 0, because of (6). This contradicts contiguity, and thereby (4) is proved.
14 — Hájcle-Sidik: Theory of Rank Teiu
210
VI. 2.1
Now, by easy computations, we derive lhal
w
r ľ e-d*=r r h*,. &)-' kí - e!)]-».
J — n J — co                 J ~ m «I — to
.exp^-Ri-e2)]-1^-^)2^2-
- 2tf(u - nt)(í + i^Xa,^)-' + (» + tá)2 *í2]} du du = = ar,(2*ri f  exp[-i(«-^-ffia)2«rr3]dii(
where g = ff^^i^)'1- Combining (4) and (7), we easily conclude the proof.
Remark. The above Lemma holds even if a\ = 0, i.e. if logL, converges to 0 in probability.
2. Simple linear rank statistics
2.1.    Location alternatives for // . Wc shall consider alternatives
(0
«i-n/o(*i-4).
where f0 is a known density with I{fo) < °o, and ä = (dlf.... ds) is an arbitrary vector. Recall that Ihe vector d runs through the set of all real vectors of all finite dimensions, and lhal the asymplotic statements concern sequences {</, = (<f?l».» ..., </,#„)} selected from this set. However, to simplify ihe notation, we shall drop ihe index v. First of all, we shall establish conditions under which sequences of such alternatives are contiguous with rcspeel to corresponding sequences of ihe hypotheses Hq — Hon- F°r ^is purpose we associate with each qd the following density
(2)
p< - n/o(*i -3) i=i

Obviously pd e Hos, where N is the dimension of d, and it depends on d only through 3 = N~l Yßv If we show that under certain conditions the densities qd are contiguous with respect lo the densities pd, then, a forliori, they will be contiguous with respect to the hypotheses Hos. The densities pd have been chosen so as to be least favourable for Hos against q4, in an asymptotic sense.
According to LeCam's first lemma, the densities qd are contiguous to the densities pd. if log Lj, where Ld = qdjpd, is asymptotically normal ( — \al, a2). Moreover, on
216
VI. 2. 4
Application 2. Approximate scores corresponding to some density / such that <p{u,f) is a sum of monotone and square intcgrablc functions. If, in addition, <p(u,f) is skew symmetric, then
(10)
R:
2>
i=i   \m + n + 1
./.
with (S) and (9) still applicable. If <p{u,f) is not symmetric, we have to subtract the expectation under H0, i.e. to put
R(
2>,
i=i   \wi + n + 1
./)-=2>
I
N (-i   \m + n -f 1
./ ■
However, since Jôo>(u,/)di/ = 0,  the  correction is asymptotically negligible, as may be shown.
2.4.   Rank statistics for //._ against regression. Now we shall investigate the limiting distribution under qä of the statistics
(0
5e = Z(c,-č)flff(Ä().
f=i
Theorem. Let qd be given by (2.1.1) and assume that (2.1.4) and (2.1.5) hold. Then, under qd the statistics Se given by (1), where the scores satisfy (2.3.1), are for
N
£ (c; - č)2/ max (c,- - č)2 -* co asymptotically normal (uäc, ff2) with
i- :
(2) and
(3)
I 5 ig A'
N                           pi
toe = [ E (cf - Č) («*l - 3)]      <?(") 9>(«>/o) <*"
'=»                                       Jo
^ = [E(^-a)2]fw«)-^]2d«.
1-1               Jo
77ie assertions remain true if we replace (2.1.1), (2.1.5) and q>(u,f0) by (2.2.1), (2.2.3) and <Pi(u,f), respectively.
Proof. Without loss of generality, we may assume that
(4)
and
(5)
£K-g>2 = 1
i=i
EOv- *0(rf,~ a)-*"*„.
1=1

VI. 2. 4
217
Note that under (4) £(c, - č)2/max (c, - č)2 -» co is equivalent to
(6)                                               max (c, - a)2 -* 0 . Furthermore, if S* is given by
(7)                               Sj-ffo-íJaSílO,
where the scores arc related to <p by (V. 1.4.12), then (Sc - S£)(varSj)~* -* 0 in probability under II0 (see the proof of Theorem V.I.Ó.a). We shall denote this briefly by Se - S*. Furthermore, inspecting the proof of Theorem V.1.5.a, we see that S? - Te, Te given by
(8)                               r.-ffo-4 f(V0.
where U, = F^X,), F/x) - P/AT, ^ x) with Prf given by (2.1.2). Thus Se ~ Tc, and Sc may be replaced by Tc in considerations concerning the limiting distribution. On the other hand, we know from Theorem 2.1 that log L, - (Tj - $b2), T4 given by (2,1.13), or equivalently, by
(9)                                     Td-£(</,-3) vfl/o/o).
(■1
Thus
(10)                                    (S^logA,)-^,?-,-^3),
where it should be noted that Te and T4 differ not only in their regression constants but also in their ^-functions. Consequently, if we show that (Te, Td) is under Pd asymptotically jointly normal with /í, = ft2 = 0, variances a\ = JJ [<p(u) - y]2 d« and Oj = &2> arid covariance a12 = bl2 JÔ <?(") <Ku'/o)du, we can conclude that (Se, log ly) is asymptotically jointly normal with the same parameters except for /x2 = -Jí»2, and the theorem will follow immediately from LeCam's third lemma. Now, since the Ut's are independent and uniformly distributed under Pd, we have ET, = ETj = 0, and in view of (2.1.5) and (5),
var
'=i             Jo
var Td = Y, {di - 3)1 í fl>a(«,/o) du - b1, i-i              Jo
cov (Te, ij - [J] (c, - č) (rf, - a)] ľ 9>(u) 9>(«,/0) du 1-1                          Jo
->b12\ <p(u)q>(u,f0)du.

218
VI. 2. 4
Thus ihe limiting parameiers have the required values. In view of D3 of Section V.2.I. it remains to show, for all real Xt and X2, that XiTe + X2Td i$ either asymptotically normal (0, o2cd) with c2ci = var(A,Tc + X2Td), or var(^Tť + X2Té) -> 0. Write
xtTe + jar, - z M<> - *) -p(^i) + 4(* - 3) <p(tfb/o)] -
and assume that
(II)                                 v*r(XlTe + X2T4)-*P2>0.
Now put
Zu   =Xt{cl-č)[<p(U ,)-$],
Z2i   =Xi(di-3)<p(UiJ0),
Zi    = zu + z2i,
Z{5) = Zit   if   [Z,l>$, = 0,     if   |Z,|áä,
and define Z(i(<5) and Z3ť(í>) similarly. In view of (ll) the Lindeberg condition for XiTe + X2Td may be expressed as follows:
(12)                             IEpiMP-0,   ô>0.
However, we obviously have
so that (12) follow« from
.v ('3)
and

(14)
£E[Z2((Ó)]2^0,   5>0.
i-i
Finally, observe that (13) and (14) are equivalent to the Lindeberg condition for XXTC and X*Td, respectively, in view of (4) and (2.1.5). Moreover, from (6) and (2.1.4) it follows ihat this condition is satisfied in both cases (see the proof of Theorem V.1.2). This concludes the proof for qd given by (2.1-1). If qd werfi given by (2.2.1), we would proceed quite similarly. Q.E.D.
220
VI. 2. 5
The density qá will be associated with the density
(?)                                   p=n/o(xi).
which obviously belongs to Hv Our aim is to establish the limiting distribution of
(■»)                                        s;=£«.v(Rr)signX,.
under qA.
Theorem. Let qNJ be given by (1), where fQ is symmetric about zero, /(/0) < ^ and (N, A) satisfy (2). Further assume that the f unctions aN(i + [«A']), 0 < « < 1. converge in quadratic mean to a square integrable function ip*(u). Then the statistics (4) are under qSÁ asymptotically normal (/ix, <tJ) with
(5) and
(6)
ftN = 4NCV+(u)<p*(ulf0)au
*ž-I
■ I MOľ - N \\ť(u)Y du . '='               Jo

Remark. Recall that <p*(u,f0) = <p(4 + £u,/0).
Proof. Following the pattern of the proof of Theorem 2.1, we can show that
w
IogLVd ~ á 2 <P*(Ut,fo) signAT, - ±i»2
i=l
where t/,^ = F*([iT||). The rest follows from SÉJ ~ rWi 7;v given by (V.l.7.5). Application 1. Put öiV(/) = Ojjfŕ,/),/symmetric about zero, /(/) < co. Then
(8)                                          Sj - S fli(f,/) sign Xt is under qSd asymptotically normal (/iv, crj) with
(9)                                   ft» =.V1 ľ <p*(u,f)<p+(uJo)du ,
(10)                                        o* =* N C[<p+(u,j)]2 du .

221
VI. 2. 6
Note that
jyM*w°)H/B,^u,/o)du'
Application 2. Put aA.(i) «* (p(i + iŕ/(/C + l),/), where / is symmetric about zero, l(/) < <x>, and <?(»,/) is a finite sum of monotone and square intcgrable functions. Then the statistics
"    '- +7^-7,/) signX
given by (9) and (10) respectively.
(11)                            ■"•-£A2'N + 1
arc
tóyD1p,oucaUy normal W 4) w^"»andffS
do not have at out disposal »us-
iese statis
we shov
IZlte following result: Consi
«.   RanU static fo, ír«*-*^!K:-.**-í:
lank statistic» ■<*• —i. factory methods for treating the distribution of these statistics uuuw ».._ -. Heuristic considerations suggest, however, that we should obtain, possibly under some additional assumpti        *    " *'-"-:-" wmlt: Consider
(i)
with (2)
r
«-n**"'í
,(x,v) = j"^-^^-^^'
(see Section II .4.11). Assume that the variance corresponding to Af(s) is finite and
positive, and that
(3)                             NJ*i(/o) i(9o) - b1 ,   0 < b2 < co .
Further, introduce the statistic
and suppose that
(5)
and
i=i
ÍÍ-* JO
lim(W> + ^-wľdu = 0-