ASYMPTOTIC NORMALITY OF STATISTICS BASED ON THE CONVEX MINORANTS OF EMPIRICAL DISTRIBUTION FUNCTIONS BY PIET GROENEBOOM AND RONAL.DPYKE TECHNICAL REPORT NO. 5 JULY 1981 THIS RESEARCH IS BASED UPON WORK PARTIALLY SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION UNDER GRANT MCS-78-09858 DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE) WASHINGTON 98195 DEPARTMENT STAT
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ASYMPTOTIC NORMALITY OF STATISTICS BASED ON THE CONVEX MINORANTSOF EMPIRICAL DISTRIBUTION FUNCTIONS
BY
PIET GROENEBOOM
AND
RONAL.DPYKE
TECHNICAL REPORT NO. 5JULY 1981
THIS RESEARCH IS BASED UPONWORK PARTIALLY SUPPORTED BY THE
NATIONAL SCIENCE FOUNDATIONUNDER GRANT MCS-78-09858
DEPARTMENT OF STATISTICSUNIVERSITY OF WASHINGTON
SEATTLE) WASHINGTON 98195
DEPARTMENT STAT
ASYMPTOTIC NORMALITY OF STATISTICS BASED ON THE CONVEX MINORANTS
OF EMPIRICAL DISTRIBUTION FUNCTIONS
by
Piet Groeneboom1 and Ronald Pyke2
University of Washington
Abstract
A
Let Fn be the Uniform empirical distribution function. Write FnA
for the (least) concave majorant of Fn, and let fn denote the corres-
ponding density. It is shown that n!~(fn(t)-1)2dt is asymptotically
standard normal when centered at log n and normalized by (3 log n)~. AA
similar result is obtained in the 2-samp1e case in which fn is replaced byF -1the slope of the convex minorant of r m= FmoHN .
FOOTNOTES
1. This work was done while this author was on leave from the Mathematical
Centre, Amsterdam, as a Visiting Professor in the Departments of
Statistics and Mathematics at the University of Washington.
2. The research of this author was supported in part by the National
Science Foundation, Grant MCS~78-09858.
AMS 1970 Subject Classification. Primary: 62E20
Secondary: 62G99, 60J65
Key words and phrases: empirical distribution function, concave majorant,
This shows that U -U ~> 0 as desired. The proof is complete. 0n m
20..
4. ~-norm of slopes of convex minorants of truncated Brownian bridges.We shall prove the following result.
Theorem 4.1. Let B= {B(t):tE [O,l]} be (standard) Brownian bridge on [0,1],let Bt,u =B.l [t,u]' where 1[t,u] is the indicator of the interval [t,u]and let gt,u be a version of the slope of the convex minorant of Bt,u on(0,1). Then
(4.1) {f~ g~/n,1_1/n(U)dU-109 n}/1310g n h. Z,
where Z is a standard normal random variable.
Theorem 4.1 will be used in section 5 to derive the asymptotic
from Theorem 3.1 by using strong approximation of the empirical process byversions of Brownian bridges in Komlifs et al(1975).
The following class of functions will playa fundamental role inthe sequel.
Definition 4.1. U is the set of right-continuous and nondecreasingstep-functions J:[O,lJ +m, which have only finitely many jumps and satisfyf~ J(u)du =0 and f~ J2(u)du =1.
Notice that all functions J€ Msatisfy the inequalities
() -k (-k4.2 -u 2~J(U)~ l-u) 2, UE(O,l).
The class Mis also considered in Behnen(1975) and Scho1z(1981)(with slight modifications). It can be used to give a convenientrepresentation of the L2-norm of the slope of the convex minorant ofbounded real-valued functions on [O,lJ, which satisfy certain regularityconditions near the boundary of the interval [O,lJ. The representationof the L2-norm of the slope of the convex minorant by means of functionsin 1.4 has been studie.d by F. Scholz, and the following lemma is a generalizationof results in Scholz(1981).
21-
Leoma 4.1. Let G:[O,lJ +R be a bounded function such that G(O) =G(1) =0
and
(4.3) 1imt.J-
ot-~-oG(t) = limt.J- ot-~-oG(1-t) = 0,
for some 0 > O. Then 119'112 < 00 and
(4.4) IIgl~= -infJ, M 1(0,1) G(u)dJ(u),
where 9 is a version of the slope of the convex minorant Gof G.
Proof. First suppose that G is a step-function which only has jumps at the
points t 1< ••• < tn' where t 1 > 0 and t n < 1. It follows from (4.3) that in this
case G(t) = 0, if t < t 1 or t> Since G~G, we have
(4.5)
Integration by parts and the Cauchy-Schwarz inequality give
Since G(O) = G(1) = 0, we also have 6(0) =6(1) = 0 and hence 16 g(u)du = o.
Suppose 119'112 > O. Without loss of generality we may take a ri ght-conti nuous- -vers i on 9 of the slope of G and in thi s case the functi on J =g/ 119'112 belongs
to M. Hence the upper bound in (4.6) is attained for J=g/ 119'112,
Combining (4.5) and (4.6) we get
(4.7) -infJt: M 1(0,1) G dJ~ -/(0,1) GdJ=IIgI12·
--Let 0 be the set of discontinuity points of J, then 0 is not
empty , since otherwi seG :: 0 and hence Ilgl~ = O. The set 0 is a subset of
the set {t1, ... ,tn} of discontinuity points of G. Let H:[O,l] +R be the
function defined by_ {G(t-)I\ G(t+), if t e 0 and G(t) > G(t-)AG(t+)
H(t) - G(t), otherwise.
Then H(t) ='G(t), if t s 0 and hence, since j is a step-function which
only has jumps in 0,
(4.8) 1(0,1) H dr= I (0, 1) 'G dJ.
(4.9)
22..
It is clear that the integral 1(0,1) H dJ can be approximated arbitrarily
close by integrals 1(0,1) 6 dd, with JE M (move the points t, D, where
6(t) > 6(t-)" 6(t+) a bit to the right or left and consider functions J~ At
which have jumps of approximately the same height as J at the shifted
points instead of the original points). Relation (4.4) now follows from
(4.7) and (4.8).- - (If IIgI12=0, then 6=0, and hence 6~0. In this case 4.4) also holds,
since 1(0,1) 6 dJ = 0 for any function Jf'M such that J is constant on the
intervals [O,t) and [t,1), with tE(0,t1) .
Now consider an arbitrary bounded function 6:[0,1] +:R such that
G(t) = 0, if t~a or t~ l-a, where a E (O,~). Define for each n the intervals
[ - n ( ) - n) n [ - n ]. z"Ik,n by Ik,n= k2 , k+l 2 , k=0,1, .... ,2 -2, Ik,n= k2 ,1 ~ tf k= -1,
Gn(t) = inf E I G(u}, iftE Ik n ' k = 0,1, ... ,2n_l.
u k,n '
Fix e> O. Let P be the set of finitely discrete probabil ity
measures on [0,1]. Then, if 'iT is the convex minorant of a function H:[O,l] +:R,
we have for each t e [0,1],,...H(t) = i nf{f [O,l]H (u) dP:1[0,1]udP = t , PeP}
(see e.g. Rockafellar(1970), p. 36). Thus there exist positive constants
c1,n' .... 'cm(n),n and points tl,n' .... 'tm(n),n belonging to m(n) disjointintervals Ik such that, for each n and fixed t E [0,1],,n
m( n) _ m( n) _ - () m( n) ( ) _l:i=l ci,n -1, l:i=l ci ,nti,n - t and 6n t > l:i=l ci ,n6n ti,n e ,
This implies that there are points t ' , with It~ n-t1' n l ~2-n, such that1,n 1"
1m( n) I 1 -n "'" () m( n) (I) 2t-z'; 1 c. t ; < 2 and G t > r: 1 c. 6 t : - e1= 1,n 1,n - n 1= 1,n 1,n
(let t. and t~ belong to the same interval Ik ,and use the definition of G ).1,n 1,n ,n nThe sequence lGn} is increasing and hence limn +
ClO"trn(t) exists (and is~O).
The convex minorant Gof G is continuous on [0,1], since 6 is bounded on
[at l-aJ and zero outside this interval. Hence by (4.9), G(t) ~ lim n +ClO
Gn(t) + 2e.
We also have G(t)2.Gn(t), for all n, .and thus limn +ClO
Gn(t) ='G(t).
Since the sequence {G }converges pointwise to G, the right~cont~nuous_ n _
slopes 9n of Gn converge to the right-continuous slope 9 of G, except
possibly at countably many points of [O,lJ (see e.g. Roberts &Varberg(1973),
23-
Problem C(9), p. 20). The functions Gn and G are uniformly boundedbelow on (a,l-a) and zero outside this interval. This implies that theslopes gn and g are uniformly bounded on (0,1). Hence, by dominatedconvergence, 1imn+ co lI'9n-g1l2 =O.
Choose nO such that 119nll2 > 11'9112- e , for n~nO' By the firstpart of the proof there exists for each n a step-function JnE M, such that
-/(O,l)GndJ> lI'9nlk- e:, where the points of jump of In, say ul,n, ... ,up(n),n'belong to disjoint intervals Ik,n and are contained in [a,l-a]. By thedefinition of Gn there exist points u',n' •.•. 'up(n),n such thatG(ui ,n) < Gn(u i ,n) + e and ui,n and ui,n belong to the same interval Ik,n'Furthermore, let J~ be the right-continuous step-function which has thesame J'umos as In, but at the ooints u: instead of u, (note that in. . 1,n 1,ngenera1J~ .M).· Then,by (4.21,wehave for n~nO'
Thus, for n sufficiently large we can find a J~( M, obtained from J~ by,making slight adjustments of mass, which satisfies
1
-/(O,1)GdJ~ >1/9112- 3e:- 2e:a-'2.
Therefore -infJ EM1(0,1) GdJ ~ 1I'9lk· Since -infJ E M1(0,1) GdJ2,
2, infJE M1(0,1) GndJ = 119n1l2' for each n, relation (4.4) now follows.
Finally, let G be an arbitrary bounded function, such that G(O)=G(l)=Oand (4.3) is satisfied. By (4.3) and the boundedness of G there exists aconstant c >0, such that
(4.10) IG(t) 1 ~ c.min{t¥O ,(l_t)~+o}, tE [0,1].
--Thus, if 9 is the right-continuous slope of the convex minorant G of G,we have
24-
(4.11)
This implies
(4.12)
Define for each t e (O,~) the function Gt by
G (u)= {G(uL if u([t,l-t],to, otherwise.
By (4.10), (4".2) and integration by parts, we have for all JE M
Remark 4.1. It is clear that condition (4.3) can be somewhat weakenedand we mainly chose (4.3) for convenience.
Proof of Theorem 4.1. Let Un be the empirical process defined byUn(t)=Irl(Fn(t)-tL tE'[O,l], where Fn is the empirical df of a uniformdistribution on [0,1]. With probability one all observations are containedin the open interval (0,1) and hence Un satisfies almost surely theconditions of Lemma 4.1. Let un be a version of the slope of the convex
".,
minorant Un of Un' Then, by Lemma 4.1,
IlunU2=...infj, JJ f(O,l)UndJ.
Fix £ >0 and let an = (log n)4/n, bn=l-an. There exists 0> 0 suchthat P[suPt E (0,1) IUn(t) I/It(l-t) 2. Mil 092n] < e for all n2.3, where1092n= loglog n (this follows from the law of the iterated logarithmfor the empirical process). If IUn(t)I.=:.M/t 1092n and JE M, we have
for some constant c independent of n. A similar upper bound holds for
f[bn,l-o/nJ IUnldJ.
Since sUPJE:M f(O,o/n] t dJ(t)+O~ and similarlysUPJ~ Mf[1-o/n,1) (l-t)dJ(t) +0, as n+ oo , there exists a constant k suchthat, for all large n, P[sUPJc:M f(o,an]v[bn,l) IUnldJ2.k 1092n] <2£.
By Theorem 3.1 and Lemma 4.1,
({infJ EMf(O,l) UndJ}2-1 0gn)/1.3109 n kZ,
where Z is a standard normal random variable. Furthermore, since
infJc:Mf(O,l) UndJ-infJ EH f(an,l-an)
UndJ= 0(1092n) on a set
of probability> 1-2£, and since £>0 was arbitrarily chosen, we have
(4.16) ({infJ(; Mf(an,bn)
UndJ}2_10g n)/1310g n k Z.
26..
By Komlos et al(1975), there are versions of Brownian bridges Bnsuch that SUPtE (o,l)IUn(t) -Bn(t)1 =O«1og n)/In) with probability one.Hence, by (4.2),
linfJEMf(a b) UndJ-infJEMf(a b) B dJI=O(suPJEMf(a b )n-~109 n dJ)n' n n' n n n' n
~ 0(1/10g n},
almost surely. This implies
(4.17) ({infJ E}.{ f(a b ) BndJ}2- log n)/1310g n ~ t:n' n
By the law of the iterated logarithm for Brownian bridge thereexists a constant k >0 such that
sUPJEMf[l/n,anlIBnl dJ <k(10g2n)-~f[1/n,anJt-3/2IBn( t) Idt =0(1092n)
and similarly sUPJE MfEb ,1_1/nJIBnldJ=0(1092 n) on a set of probability> l-e:.n
Thus we can replace an by l/n and bn by 1-1/n in (4.17). By Lemma 4.1 we
have -infJ~M f[1/n,1-1/nJ BndJ=lIg1/n,1-1/n ll2' where gl/n,l-l/n is a versionof the slope of the convex minorant of Bn.1[1/n,1-1/nJ. Since the distributionof Ilgl/n,l-l/nl~ will be the same for any version of the Brownian bridge Bn,the result now follows. 0
5.' Asymptotic normality of a statistic proposed by Behnen.Let X1, ... ,Xm and Y1' ... 'Yn be two independent samples from a
uniform distribution on [O,lJ, let Fm (Gn) be the empirical df of the first(second) sample and let HN be the empirical df of the combined sample. Withprobability one, all observations in the combined sample are different andcontained in the open interval (0,1). Thus, on a set of probability one,we can define the inverse HN
1 of HN as the right-continuous df such thatHN(H N
1(k/N)) = kiN and HN1(u) =HN
1(kiN) ,k/(N+l) 2 u < (k+l)/(N+l), k=0, ... ,N.
In the sequel we will restrict our attention to the set where HN1 is
well-defined and we shall omit the expressdon "wt th probabtlity one": vie 'definethe (random) dfs Fmand Gn by
- -1 - -1Fm= Fmo HN and Gn =Gno HNNote that by our definition of HN
1 these dfs are right-continuous.
27...
Behnen(1975) considered the statistic
(5.1) TN=suPJEU f(O~l) J(u)dFm(U)
(actually he considered slightly different versions, but this will
make no difference for the limiting behavior). By integration by parts
with Z standard normal, _i f AN stays bounded away from 0 and 1, as N~ co.To see this, note that LNO again represents a Brownian bridge (as a sum
of two independent Brownian bridges), but that the variance is {l-AN)/ANtimes the variance of the standard Brownian bridge on [0,1]. Furthermore,
it was shown in the proof of Theorem 4.1 that replacing [aN,bN] by [l/N,1-1/N]le~ds to the same limiting (normal) distribution.
The asymptotic (standard) normality of the statistic
{{NAN/(l-AN»T~- log m}/.t310g m,
withTNdefined by (S.2) (or,:quivalent1y, (S.1», will now follow if we
can show thafsuPJ E Ml(o,aN]
FmdJll10gmandsuPJ E Ml[bN,1)(T-~m)dJII1()9 rn
tend to zero in probability (with a similar statement for the functional
with Fm replaced by Gn) . First, by our definition of HN1, we have tm(t) =0,
if t < (N+l )-1. Second, for fixed e > 0, there exists b = b(e:) such that
P[Fm(t) ~Fm(bt), all t€ [0,1] ]~ l-€.
(see Lemma 2.5, p.761, Pyke &Shorack(1968); our interval for t is [0,1]
rather than [1/N, 1], because of our defi ni ti on of HN1) . There exi sts M> 0
such that P[supt £ {0,1)IUm(bt)INt~MI10g2m]<€, for all large m. Thus,
P[sUPJ€M 1[l/(N+1),aN]
1m FmdJ~k 1092m]<€, if m is large,
for some constant k > 0 (see the proof of Theorem 4. 1). Similar arguments
ho1d for I [bN,
1) IID'( 1-t m) dJ. We have proved
Theorem 5.1. Let TN=suPJ E M1(0,1) J(u)dtm(u). Then TN=lIfm,N-1112' where.- -1fm,N is a version of the slope of the convex minorant of FmoHN ' andthe statistic {(NAN/{l-AN»T~- log m}/1.3 log m tends in law to a standardnormal distribution, if AN stays bounded away fromO and 1, as N~co.
30-
6. Concluding Remarks
Both limit theorems involve non-negative random variables, namely,
square L2-norms. As such, one possible guide to the rate of convergence
is the sample size required before zero is 3 standard deviations from the
mean under the approximating Normal distribution. In the one-sample case,
this requires log n =3(3 log n)~ or n > 5 x 101~ For 2 standard
deviations, one requires n ~ 162,755. The results are similar for the
2-samp1e statistic. By this, one sees the extreme slowness of the conver-
c::nIIIA..,~rl norms
find functions of the statistics for which the convergence is much improved.
Behnen (1974) used the L2-norm itself, that is, the square-root transformation,
for his Monte Carlo simulations. Here, the asymptotic variance is constant
and the corresponding sample sizes are 854 and 20, respectively.
Monte Carlo simulations of sample sizes n = 4(1)10 (20,000 replications)
and 50 (5,000 replications) for the log transformation have been carried out
by Scholz (personal communication). They indicate tails that are still too
heavy for n =50. Behnen (1974) had earlier provided simulations for the
two-sample statistic for selected sample sizes up to m=n =100. Although
the convergence is slow, the fit was sufficiently close to suggest the
asymptotic normality of the statistic.
It is possible to generalize the representation approach used for
Theorem 3.1 to obtain an alternate proof of the two-sample result, Theorem
5.1. The only difficulty is in defining a suitable I randomization' of the
coincidences that can now occur in order that the resultant distribution of
31..
heights remain the same as in (2,2), The coincidences enter because fm•
unlike Fmt has its jumps occurring at the equi-distant points' {i/N}.
One approach is to affix small (continuous) random perturbations to these
points to prevent ties among the slopes of the segments of the concave
majorant without changing ~ignificantly the value of the statistic. Once
this is done t one uses Negative Binomial rather than Gamma random variables
for the {Sj'i}'
Acknowledgement
The authors are grateful to Dr. F.W. Scholz of Boeing Computer
Services for introducing us to the problem and some of the relevant
literature. We also appreciate the extensive computations which he
provided during our research,
33-
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