V. 1. 2
J 53
TISTICS
! distributions B correspond B denote the ed However, idexed by jV. s *»: and « of
m £ 1, dexed by real «f our study
frj, cT =
l _-der some »c-: that the I »ach that
land say that
_ asymp-
:, Bvergonce
is asymptotic
t > 0 there

The reader may find it useful to keep this alternative interpretation in mind throughout the present chapter.
1.2. A special central limit theorem. Let A be the set of all non-vanishing real vectors a = (ait..., aN) of all finite dimensions N ^ 1. We shall say that the statistics Ta, a e A, are asymptotically normal (/(„, o*) for
(1)                                                 £<i?/max<i2 - co,
i-i      isís.v
if (1) entails
(2)           f(r.g/i.+ xff„)-»(2n)-*r   «p(-±/)dj,   -
CO < X < op .
Thus asymptotic normality (/ia, a2) is equivalent to convergence in distribution of (7^ — pa)ja„ to a standardized normal random variable. Obviously, asymptotic normality (pm, o2) is equivalent to the same property with (/i*. a*2), if
(3)
»:/*.-i. w-*.)/».-><>.
Theorem. Let V,, Y2, ... be independent copies of a random variables with finite expectation p and finite variance a2. Put
(4)                                            r.-Eajy,,   «eX.
(-1
Then, for (1), ííie statistics T„ are asymptotically normal (//„, fffl) with
(5) (6)
/*« = p Z "f
J-l
2           2 V     2
Proof. The Lindcberg condition (Loěve (1955), p. 295) takes on the form
(7)                     a;2 Í   f       x2áP(al(Yi-p)^x)-^0,
'•-"J|«l>«u
where o\ is given by (6). Upon substituting aj> for x, wc obtain
(8)
x2 dP^y, - /.) g x) = aj f           y2 dP(y, - p S j.)
|»I>w,                                                       J|jiJi|>e«.
^ a? ľ       y3 dŕ(y, - /i á *)
154
V. I. 3
where
,v
v- = S fl?/ max o* ■
/-I        lgi'g.V
Consequently, Ihe Y,'s having the same distribution,
(9)    o;2 £ ľ        x2 dPiaM -n) í x) * e~2 [        y2 dP(Y1 -jiíy).
However, the variance of 7, is supposed finite and va -* co in view of (l), so that
a'1
|jr|>ra.
which, in accordance with (9), entails (7). Q.E.D.
Remark. The above theorem could be reformulated as follows: For every a > 0 there exists an n„ such that
(10)                                             J>f/max «J > «0 entails
(11)                            sup \P( £ «,V, Š& + *rj - *(*)| < £ -where $ denotes the standardized normal distribution function.
1.3.    A convergence theorem
Theorem. Let (Q, j/, /í) be a measure space with a c-finite measure ft. Consider a sequence {ft,} of square integrable functions converging almost everywhere to a square integrable f unction ft. Assume that
h2 du,
(1)                                        lim sup   ft; d/í £
*-»   J
Then
(2)                                          Urn f(ft¥-ft)2d/I = 0.
v-aj J
Proof. Fatou's lemma together with (l) implies
(3)                                           Iim  ľft2d;(= jVd/i.
VJJ________________________                           155
Furthermore, the Schwartz inequality yields
so thai
lim sup f (Midi* á U*d/i.
Consequently, according to Theorem 11.4.2
(5)                                     Um [môu= f li' d|í.
Now (3) and (S) imply (2).
1.4.   Further preliminaries. Consider a probability space (O, j/, P) and a sequence of sub o-fields ^cijc.ca'. Denote by _FW the smallest c-field containing
CO
the field U^V For cverVeVcl11 ^ ~ &*>an^ 6verVE > ® l^ere cx'sts an ^ ant* ^ e ■^•v i
such that
01                                           P(4 + i4N)<e,
where 4- denotes the symmetric difference. Actually, the assertion h trivially true
for A e U-^.v and the events havingihis property obviously consist a cr-ficld. Denoting
i by lA and 1All the respective indicators, (1) may be rewritten as follows:
(2)                    ífi-íjl<«.
If y is a .^-measurable function such that EY2 < co, then there exists for every N a ^ ..--measurable random variable Ys such that
(3)                                        E(YN - V)2 š 6(1* - V)2
for any other ^"„-measurable random variable Y*. It is well-known that this property is possessed by the conditional expectation with respect to fs,
(4)                          "                 Y, = E(Y|^,).
If &n is generated by a statistic Ts, then E(Y| ^N) = i//(TN), where ijr(r_) »« = E(Y| TK == ts). Now, if /* = E(íx \ && then (2), (3) and (4) imply that
(5)                                                E(I_ - ij)2 < e
156                                     ______   _________________________v. l. 4
for JV sufficiently large, and, consequently,
(6)                                             limE(/^-7j)2 = 0.
Before generalizing ihe relation (6) to all random variables with finite variance, let us recall that
(7)                                                     £Yt S Er'
for Ys given by (4).
Lemma a. Let Y be a &^-measurable random variable such that EY2 < ce. and let Yy be given by (4). Then
(8)                                             limE(v-v- y)2 = 0
and
(9)                                                IimE>;2 = EV2.
JV-ro
n
Proof. Fix an c > Oand find a &^measurable simple funciion Xc.'^such that
Í-1
Denoting /*, = B(lAi \ FN) and noting (7), we have
E(v-„. - y)2 í 3E(yÄ - Sd/jy +
.--i
+ 3E(ľ - £c,UJ + 3E[Xc,.(/„-;>',)]'g
i-1                             f-1
1*1                        í-1      i=l
<e + 3 Jef J£(/.,, -Jjj».
i-1      1=1
Since, in view of (6), the last sum converges to 0 as .V -» co, we conclude that
E(yv - Y)2 < e for N sufficiently large. This proves (8).(9) follows from the well-known relation
(io)                                  E(yiV- y)2 = zy2-by2,
holding for any conditional expectation. Q.E.D.
V.l. 4
137
Now Jet Uv U2, •■• be independent random variables, each uniformly distributed over (0, l). Let RNi denote the rank of V„ 1 <; l á N, in the partial sequence U,,... ..., U/,-. Let <?(»)• 0 < u < 1, be some square integrable function
(11)                                              í <p2(u) du < co ,
and put
(12)                      flj(i) = E[>(U,) | Kffl - i],   lSiáN<M.
Theorem a. Under assumption (11),
(13)                                    KmE[flS(ÄÄI)-fl<í/.)]2-0,
Jí—to
holds, where tfJS(») is defined by (12). Proof. Let ^>- be the sub tf-field generated by (RNl,..., RAiV). Note that &x <=
co
c #*w+, c ... and recall that &^ denotes the smallest cr-field containing \J^S.
i We first show that <p{U^) is equivalent to a .^ ^-measurable random variable. In view
of (ÍI. 1.2.12), we have
■(*-A-J-i*K*-i±-JM-
■ Ižvarl/y-iž     ^-/ + 1>    <I,
WA        "      W ;-i (N + l)2 (iV + 2)     N
so that
Hm J*!!*!- „ y
y-«N, + 1
with probability 1 for some properly chosen subsequence {Nv}. Consequently Ult and hence also <p(Ut)t is equivalent to a ^"„-measurable random variable. Now it remains to apply the above lemma with ^(V,) = ľ"and a£(Rvl) = YK. The proof is thus concluded.
Lemma b. (D.K. Faddeev.) Let the functions fN(t, u), N ^ 1, 0 < t, u < 1, be densities in t for each fixed u, such that for every s > 0
(14)                           lim I    fs(t, u) at = 1,   0 < « < 1.
158
V. 1.4
Moreover, assume that
(15)                         Ml, u) S tfA-(i, '<) ■   A' š 1. 0 < r. » < 1 ,
where the functions gs(t, u) are increasing in t e (O, u) and decreasing in t e (u, I) for every fixed N > 1 and 0 < u < I, and
fl
(16)                              sup    g Jí, ») dt < co ,   0 < u < 1.
» Jo ' TVieii /or eeery imegrable function <p(u)
(17)                                      lim C<p(t)Mt,u)út = <p(u)
ä-«Jo
in almost all points u e (0,1).
Proof. See I. P. Natanson (1957), Theorem 3, §2, Chapter X, and Theorem 5, §4, Chapter IX.
Theorem b. Let <p(u), 0 < u < i, be square integrable and let alj(i) be given fry (12). Then
(18)                          lim
.v-»
[oS(l + [aJV]) - <p(u)]2 ou=0.
with QiiiVJ denoting the largest integer not exceeding uN. Proof. Since, in accordance with (7), where y = <p(t/,),
(19)                              í '0S(1 + [ujV])]2 du á fV(«)d«, Jo                                 Jo
it suffices to prove that
(20)                                        lim «35(1 + [hAF]) = tp{u)
almost everywhere and then apply Theorem 1.3. Now (20) follows from Lemma b, if we put
(21)       f A, u) = N (N ~ l\ rHl - if"1,   — ú « < - , 0 < t < 1 i - 1 /                             Ar              N
V. 1.5
159
and
i-i
= /a-('» u) .   otherwise . Actually, then (see (IU.2.10))
4(1 + [«*]) - E v(C/Jp) = j%(0 W ŕ* ~ M r" '(1 - í)""1 dt =
í:
«KOAÍMOdf, ^j-á«<~.
while (15) is satisfied since fN(t, u) is unimodal with mode at (i — l)/(iV — 1), which lies within the interval ((ŕ - \)JN, i/N). Also (16) holds true, since
Thus (17) holds, which is equivalent to (20). Q.E.D.
1.5.   Locally optimum rank-test statistics for H<}. Now we are prepared to prove easily all the theorems needed. Consider real vectors c = (ct,..., c,v) such that
(O                                    i(ct~čy>o
i*i                                                                        ——-----------
where
(2)                                    £ = líC"
Let C be the set of real vectors of all finite dimensions JV ž 1, satisfying (l). We shall consider limiting distributions of statistics indexed by c g C for
£fe-«ř
(3)                                            -^-,-------3-».
max {ct - cy
Take a square integrablc function <?(")> 0 < w < 1, and denote by aJJ(i) the scores associated with </> by (1.4.12). Put
(4)                               $,-£*! 4(4«) >    ceC
i
160
V. 1.5
where RiVi is the rank of X, in a set of N independent observations Xt, ...,XS, each with density/. If U, = F(Xt), F(x) = Jíw/Q')dy, then the random variables Ut will be uniformly distributed and RSi may be interpreted as the rank of (/,- in the set U,,...,UN as well. As we know from §11. 4, the test statistics generating locally most powerful rank tests are just of the type (4).
Theorem a. Let the scores a$(i) be associated with a square integrable function <p(u)by{lAA2).Put<p = J*ô <p(u)áu and assume fô [<p(") — ^>]2dw > 0. Assume II0, Then, for (3), the statistics (4) are asymptotically normal (u(, <?*) with
(5)                                         *-«j>«(0
and
(6)                    aJ.[E(c(-2)2]fw«)-flad«,
'='             Jo
or o\ = var S(.
Proof. Rewrite Sc in the following form:
(7)                         Sc = £(c,-čH(RA,) + č£>s(í).
i-1                                        i-i
Introduce
(8)                          Tc = £(c,-č)^C/() + č£^(i),
i-i                                    1=1
where t/j = F(Xi), I S i S W, Now drop N in RiV(> and recall that the distribution of (K„..., Rs) is independent oft/10. Consequently, by (11.3.1.23), we obtain
(9)                    E{(TC - s,)! | u<> = „<->} =
= E{I(c1-č)(a>(R,)-V(u'»"))}; =
N —   I i-1              fml
1
fl                     W
s ~ I («=.-a)í I, [«.vO)-í>(«,',,)]! = —-: í (c, - čf £{[<..,(«,) - «-(u,)]21 u" = u»}.
iV — 1 1=1
Consequently
(10)              E(rc - S«)' S -ÍL- Í(c, - čf E[fl,(R,) - 9(c/,)]2
W — 1 i-i
V. I. 5
161
and
(11)    e p^-j2 s j~zi(J'M") - *ľd")~' EM«.) - *PäF ■
On the other hand
(12)                                            -tli-------_ £ Af
max [ci — c)
HUN
so that (3) enlails N -* co. This fact, together with Theorem 1.4.a and (11). implies
(13)                                   limEf^l^Y = 0,
and a fortiori
/It _ ^
>el = 0,   £ >0.
(14)                        limpfp—-c
Now we know from Theorem 1.2 that the random variables Te arc asymptotically normal with parameters given by (5) and (6). Furthermore,
(15)                 s^ = w£ + sL-ni
ae             oc             ae
where the last term converges to 0 in probability according to (14). Thus asymptotic normality (0,1) of (Tc — /ic)/ffc implies the same for (Sc — Pc)fffc>m v'ew °*"a weH" known lemma (see Cramer (1945), Section 20.6).
Now a] given by (6) equals var Te, and (13) implies var Sř/var Te -* 1, since |(var Sc)4 - (var tJ*| <; [E(rc - Sf)2.]*. Consequently, we may put o\ = var Se as well. Q.E.D.
In the two-sample problem we consider statistics
(i6)                                 sM = j;u^..i)
i-l
and we are concerned with their limiting properties for (17)                                         min (m, n) -* co .
Theorem b. Let the scores a*(i) be associated with a square integrable function <p(u) by (1.4.12) and assume JJ |V(u) - <p\z du > 0.
11 — Hálek-Sidák: Theory of Raok Tc»»