Vietnam J. Math. (2014) 42,233-245 DOI 10 1007/sl OOI 3-014-0074-2
On Weak Convergence of the Bootstrap General Empirical Process witli Random Resample Size
Nguyen Van Toan
Received- 4 June 2012 /Accepted: 3 December 2013 / Published online: 3 June 2014
© Vietnam Academy of Science and Technology (VAST) and Sprmger Science+Busmess Media Singapore 2014
Abstract It is proved that Che cenCral limic theorem for general bootstrap empirical process with random resample size indexed by a class of functions T and based on a probability measure P holds a.s. '\f T e CLT(P). / p - ( / P < oc. ||P„ - P | | ^ ^ a s 0 and the random resample size (A'„| satisfies ^ ^p f, where P — s u p y g ^ | / | , Pn is die empincal measure, D is a positive random variable, G =^ T V T U T . T". and T'- denote the classes of squared functions and squared differences of functions from J, respectively. The bootstrap general empirical process wich random resample size is also considered in ihe case where the resample size is independent of che original sample and of the bootstrap sample Keywords Bootstrap • Central limit theorem • Empirical process • Random resample size Mathematics Subject Classillcation (2010) Primary 60B12 • 60F17 • Secondary 60FO5 • 62E20
1 Introduction
Let X\ .Xi,.. be an independent, identically disti-ibuted (lid) sequence from the probabil- ity space (A. B, P), and we take the sequence to be defined on Che canonical probability space
{n,S.Pfp) = (A.B.P)^- Let Pn(iD) be die empirical measure for w G S2, i.e..
p„(^)^p^=-x;^^(-
N VanTojn (.r_)
University of Technical Education—Ho Chi Minh Citv. 01 Vo Van Ngan Strcei, Thu Due Dismci, Ho Chl IHinhCiiy, Vietnam
e-mail: nvtoan@hcmute,edu vn
fi Spnnger
let X^p . . , , X^„ be an iid sample from the empurical measure P„ (a)) and let P„m (eo) be the empirical measure based on IX^ \ , i.e..
Let (9 = e ( P ) be a parameter of mterest, and let ^n = e „ ( X i , . . . , X„) be an estimator of 9. The bootsfrap technique was introduced by Efron {4, 5] as a method for estimating the sampling distribution of a statistic. The bootstrap principle is to estimate the unknown distribution of 9n b-y9„ where 9„ is distributed as^nC^ni. • - • • ^nn)-
Often, flie quanlity V"(^n - ^) to be bootsttapped can be expressed as a function of the empirical process which is defined as
A„ = V ? ( P „ - P ) ,
The bootstrap empirical measure and process for the bootstrap resample of size m are respectively P^^ and
= vS(P™
In the modem theory of empirical processes, it is customary to identify P, P„, and X^ wifli the mappings given by / H^ / fdP ^ Pf, f \^ j fdP„ = ^ Yl"=i f(^i) = P " / ^nd / H^ f fdXn = •j^Y,"^i(f(Xi) - Pf) = X „ ( / ) respectively. Here, / g ,7^, and 7" is a class of measurable functions on (A, B). In this way, X„ becomes a random element of l'^(J^), the space of bounded real functions on T.
A P-Brownian bridge process Gp is a 0-mean Gaussian process indexed by T with covariance function
Cov(G/.(/). Gp(g)) = Pfg'PfPg, figeJ'.
Let pp be the pseudometric on LiiP) given by
pUfi 8) - Var(/(X) - g{X)) = P ( ( / - g)^) - [P(f - g)f.
If there exists a version Gp of a P-Brownian bridge, indexed by J^, which has bounded and p-uniformly continuous sample paths, we say that J- is P-pregaussian, We say that T is P-Donsker or J^ G C L T ( P ) if J^is P-pregaussian and
^ ( P „ -P)^ Gp.
This convergence is convergence in distribution in l°°(T) in the sense of Hoffmann- J0rgensen {10] (see Andersen {1], Dudley {3], Van der Vaart and Welhier {24] (p, 81), or Gine and Zinn {9]) for an explanation.
We now let
J^' = [f-g:f,ge7}, r^ = iif ^g)^:f,ge J"), and for a pseudomeUic d on L^(P) and S > 0,
r(8. d) = {(f.g)eTxT: d(f, g) < 5 } . Then, for any real-valued function tp on J^,
||i^||.F= sup | 0 ( / ) | , \mri.&.d)^ sup \<p(f) - e}}{g)\.
f^^ (.f.si^ri.&.d) We assume that T possesses enough measurability for randomization with iid multiphers to be possible; such a set of conditions is ,?^ G NLDM(P) and T^, J^'^ g NLSM(P) fi Springer
Weak Convergence of the Bootstrap General Empirical Process
in the terminology of Gine and Zinn {7-9]. When all of these conditions hold, we wnte T e M(P). These conditions, spelled out in [7], are diat T be nearly linearly supremum measurable for P(NLSM(P)) if there exists oJ^ C J^ such that
(a) the quantities
sup y ] f l , / ( X , ) - f e P / , Oi, & € R , n G f
/eo-F 1 ^ I
are Pr/>-completion measurable;
(b) supye^jr | / ( j ) | < oo for all j e A;
(c) for all n > 0
Pr*p sup | X „ ( / ) | ^ sup | X „ ( / ) | = 0,
and Chac T be nearly linearly deviation measurable for P(NLDM(P)) if there exists oT C T such that both o^ and oJ^ for all 5 > 0, where o^i — 1 / — 5 - /- g e 0J", P ( / - gf < 8], satisfy (a) and (b), and for every 5 > 0 and n > 1
Pr* sup | X „ ( / ) 1 ^ sup | X „ ( / ) | In {9], Gind and Zinn have proved the following theorem.
Theorem 1 ({9], Theorem 2.4) Suppose that T e M(P). Then, the following are equivalent:
(i) P(P^) <OQandT € CLT(P).
(ii) ?rp-a.s., X^„ =J- Gp in l°°(T).
PraesCgaard and Wellner {14] considered Che Efron bootstrap wifli arbitrary bootstrap resample size and proved Che following theorem.
Theorem2 ({14], CoroIIary2.1) J^ e CLT(P) and P(F^) < oo imply that X"„ ^ Gp inl'^(F) a.s. asm An -^ cc.
CsSrgd {2] and Van der Vaart and Wellner {24, Sect, 3.5] considered the direct analogue of X„ for a random sample size, i.e.,
Xw„-yw;(p^„-p).
where {N„,n > 1} denotes a sequence of non-negative integer-valued random variables.
Suppose dial (A^n, n > 11 is a positive integer-valued stochastic process saDsfying
*P V as « -* oo. U) where u is a positive random variable.
Klaassen and Wellner (11] proved die following theorem for die general empirical process with random sample size.
Theorem3 ({11], Theorem 4) Suppose that 7 e CLT(P). //{M,l satisfiesd). then XA,„ => Gp in l'^(T) as n-^ oc
fl Springer
Fyke {15], Fernandez [6] (for v = 1), Csorgo [2], and Sen [17] (for v > 0 arbitrary) have proved Theorem 3 m tiie case where A = [0, 1], B is tiie a-algebra of Borel sets, and J" ^ {l|0,ii : 0 < r < 1). In this case, Oie bootsU-ap with random resample size was considered by Mammen {12], Rao et al. {16], and Toan {19-23]. In {21], it was proved that the bootsU-ap works for the empirical process if the random bootstirap sample size A'n is such tiiat (1) holds.
Klaassen and Wellner {11] have studied the Poissonized, or Kac, empirical process
rXP"-"')'
" / v „ — ,—
y/n
where Nn ~ Poisson(n) is independent of the X,'s, and showed that if the class F is P-Donsker with finite envelope function and such that ||P||j^ < co, then
KN„ =*• 5p in l°°(T) as n ^ - oo,
where Sp denotes the so-called P-Brownian motion process, which is a zero-mean Gaussian process with Cov{Sp(f), Sp(g)) = Pfg.
Klaassen and Wellner {11], Van der Vaart and Wellner {24, Sect 3.6], considered the Poissonized bootstrap empirical process K*^ defined by
"VS
E^K
where Nn ~ Poisson(«) is independent of the X, 's and of the X^ 's. The authors showed that if J^ G M(P),F is P-Donsker and such tiiat P(F^) < oo, where F with P(x) = sup tg_p \f(x)\, X e A, is the envelope function of J^, then
K ^ =^ Sp m l^(D a.s. as « -> oo. (2) Our goal in this paper is bootstrapping the general empincal process with random resam-
ple size. First, we study XJ^^ where A'n is random and possibly dependent on the Xj's and on che X^-'s, and we establish sufficient conditions on the random resample size and flie classes F for the cenfral limit theorem to hold for the bootstrap general empirical process with random resample size almost surely. In the case where A'n is independent of the X, 's and of the X " 's, we show that the centi-al limit theorem to hold for the bootstrap general empirical process with random resample size if
A'n - > j J CO a s H - > o o . Finally, we extend (2) to more general sequences [NA-
2 Main Results
For a> G Q. n — 1.2. . . . X'^y... , X^„,,... are row independent and identically dis- tributed with distribution P ^ on (A. B). We take the resulting triangular array to be defined on a common probability space
(£2,5,Pr^) s (A,B,P^)^ x -•• x ( A , 6 , P„<")^ x ••• ,
^ Sprmger
Weak Convergence of the Bootstrap General Empincal Process
Let {Af", n > 1} be a sequence of positive integer-valued random variables, such that r,. K
Prp-ae CO, >p^ v'" a s « - * o o . (3) where v'^ is a positive random variable defined on the same probability space (£2, S, Pr^),
i.e.,
Pr"(i;'" > 0} = 1. (4) The following is our fnst result to the general bootstrap empirical process wifli random
resample size.
Theorem 4 Suppose that (i) J ^ G C L T ( P ) ; (ii) P(F^) < co;
(iu) IJP^ -P\\l -^a.s- Qasn^oo. where G = TVF^[J T"^: and (iv) {iV^I and i;-" satisfy (3) and (4).
Then,
X^^ ^ Gp in l'^(7') a.s. as n -^ oo.
where'k'^^ ^ X^^^.
When Nn is independent of the X, 's and of che X'" "s, we obtain the following theorem.
Theorems Suppose that assumptions (i) — (iu) of Theorem A are satisfied and let [Nn] be a sequence of positive integer-valued random variables such that
(a) N„ IS independent of Ihe X, 's and ofthe X'^ 's;
(b) N„ -^prp oo as n ^f oo.
X^ ^ Gp m l'^(F) a.s. as n -•• oc.
Fmally, note dial if N„ - Poisson(H) then ^ | ^ "* T ^ ^Pt^. L and ^ ^ ^ ^ " - Z - N(0.1) in distribution (^A as /i ^ oc Let ENn = p„. VarM, = cr„-. We define
J:5^...-P„K
The following flieorem can also be viewed as an extension of (2) to more general sequences [Kl
Theorem 6 Suppose thai assumptions ('[) and (i'i) of Theorem 4 are sansfied and let {Nn]
be a sequence of positive integer-valued random variables such that (A) N„ IS independent ofthe X, 's and ofthe X^j 's:
(B) foralln = i.2 0 < ENn = M-i- VarA',, = CT„' < x. and hm u„ = oc. lim - ^ = /) > 0: and
" - ° o "--^ (in
fl Spnnger
(C) either N„ is a degenerate random variable for all norcr^ > 0,
— ^^p\ and ^ ' ' ~ ^ " ^ r i Z - A ' ( 0 , l ) asn^oo.
S%^ =f. 5^ in l°°(D a.s. as n ^ OQ, where Sp is Gaussian, centered, and with
C o . ( S « ( / ) . s f (S)) = PfS - (1 - fi)''fPg-
3 Proofs
For flie proof of our theorems, we will need the following results.
Lemma 1 ({18, Lemma 3.1]) Suppose that S = (aij) and t, = (dij) are twod x d covari- ance matrices on W' and let N(0, T ) , iV(0, E) be the coresponding mean zero Gaussian laws. Let 71 and dsi' denote the Prohorov and bounded Lipschitz metrics for probability measures. Then,
Tt (N(0, S), A'(0, t)) < Mrf IIE - SII l^"* < Mrf IIE - S f ^ " , and
dBL- (A'(0, S), N(0, t)) < Cd lis - Sll^" < Cj W^^tf^, where Md, Mj, Cd, Cd are constants depending only on d. and
d
| | £ - S|L ^ max Y ^ ICT,-, - o-,j|, ||E - E I L = max ICT,, -CTUI .
" "' \<\^d^-^ " '"^ l<i<d By the proof of Corollary 2.1 in {14], we have
Lemma 2 Suppose that (i) and (ii) of Theorem 4 are satisfied. Then,
For proving finite-dimensional convergence in distribution, we use tiie following lemma.
Lemma 3 Suppose that the assumptions of Theorem 4 are satisfied. Let Td = ( / i , . -, /j}
he any finite subset of F Let F^^ denote the /aw o / f e „ ( / l ) , . . , , X ^ „ ( / d ) ] on IR'' and let FNI,O,Z) denote the Nd(0. T.)lawonW', Where's ^ (Covp(f,, fj)). Then.
^bm^Ji (^w„' ^WCO.Z)} ^ 0 a.s,, where F^^^ = F^^_^.
Proof We first prove that F is a.s. {P^|-uniformly square integrable. Since P(F^) < oo, it follows that
hm P^(F^1[F>A)) = P(P'^lir>il) a.s. (5)
^ S p r u
Weak ConvCTgence of the Bootstr^ General Empirical Process
for any countable collection of X's and the right side -> 0 as X -> oo by the strong law of large numbers. Let
£2f = {<u e fi : F is (P^}-unifonnly square mtegrable}, f ^n „ \ SJAT = 1 £0 e S2 : >-p^ V as n -*• o o ) , nc = {(oe^: | | P ^ - P | | * ^ „ j . O a s n - * oo],
^FNC = QFT\QNC\^C- Then,
Prpi^FNc) = 1 (6) by (iii), (3), and (5),
By Theorem 5.2 in {18] for every a> s ^fNG< there exist random variables {Cl(>l " ^ K' {^n,}i>i "^ with a Gaussian law Nd (0, E^), where F ^ is die law of ((s^,o - P ^ ) ( / i ) , . . . , (s^^ - P^\ (fd)\ , S ^ = {Covpo,(f, / j ) ) , aU defined on some common probabihty space and, for every s > 0,
lim supPrp;^ | — max 5 Z ^"'^ ~ X ! ^^' P " ^ 1 ^ *^- f^' Lel i > 0. By (4), choose 0 < a < f o < c o s o that
Pr^(a < v'^ < b) > i - /..
Wilh 0 < f] < (3, w e h a v e
I
. I /^-i Nl I •)-T= y" T" - y" z'". > £
^/^|tr ^1 I I
<Pr^l\^-v'"\>q]+Pr^{v"'i(a.b]]
Ec-Ez",
(V(i> +1)"to-i)""'~<»+i)"|;r;'"' ^ * " ' l " ' ' ' + ' 1 Now. (7) implies that
lim ;„ = 0.
(3) iinplies that
[ IN*^ I 1 lim Pr" H - - V ' " >i? = 0 , and consequently
^ " ^ TSK^''"""? ^"'
fl Spnnger
Since A > 0 is arbitraiy. the left side of (9) equals 0:
1 1 1 " " "• I 1
„i™„Pn^T^ E ' - . i - E ^ " ' > ^ ="• ('»)
By (5) and (iii),
However. Nj(0. Z^) is the law of - ^ Tfj:, Z " . f " is the law of-i-V"-, Y". Hence, itfoIlowsfrom(lO), (11), and Lemma 1 thatTr (F^^, Nd(0, E ) l ^ 0 asn -»- oo, andflie
lemma follows from (6). • Lemma 4 Suppose that the assumptions of Theorem 5 are satisfied. Let Td, F ^ , and
FNIO.I.', be as in Lemma 3. TTi^w,
J ^ ^ (^w„' PNio.-E)) = 0 a.s,.
Proof This can be proved in the same way as in the proof of Lemma 3, Let QpQ — ^p DQG-
Then,
Prp(QpG)^l.
For every cu e E^FG, the requisite analog of (8) is
I
, I «n ' ^ " 1 1^f;prp{M, = mjPrp.\-^\J2y:.-i:z:\>4 [f'y(-)]
1^..-.
<Prp{A'„ <k]-\-Jnk, where
Jnk - X! p^'-i'V" ^ '"ip^pif -^ f" c - y z,rJ > 4 .
Let ,5 > 0 is given. Then, by (7), there exists k = k(5) > 0 such that for m > k, supPrp^. 1 ^ X C - y ^ C , > £ <8, and hence
J„k < ^- Now, (b) implies that
hm f^p[Nn < k] = 0.
and consequently
lim Pn-t I — : = V y*" - V Z""
Since 5 > 0 is arbitrary, we obtain (10)
^ S p r m g e r
Weak Convergence ofthe Bootstrap General Empincal Process
Lemma 5 Suppose that the assumptions (i) and (ii) of Theorem 4 are satisfied and let [Nn) be a sequence of positive integer-valued random variables such that
(A) hotds;
(B ) for all n — 1 , 2 , . . , , 0 < FA'n = fJ-n < oo, lim„_,ooMn = °^' ^nd Nn -*Prp
X'j!/ — y^l'u„] ~*Pr^* "J ' " l^(T) a.s. as n —* oo, where [x] is the integer part of x, fl«dX[^j = X^^^^,
Proof Let
S2s = |w e £2 : X^„ ^ Gp in / ^ ( J ^ as m /\n^ oc).
Then
Prp(£25) - 1 by Theorem 2. Let w e £2^ and
^^(-•/) = ;;^Efe,-'^)<^>-
By the Slutsky's theorem (see Van der Vaart and Wellner {24], p. 32). the sequence Z^
converges in distribution in /°^({0, 2] x JP) to a Kiefer-Muller process Z, From (B'), it follows thac che sequence (Z^, N„/pn) converges in disCribution to (Z. 1) in Ihe space /•^({0,2] X ,7^) x M by Che Slutsky's lemma (see Van der Vaart and Wellner {24]. p 32).
Thus, by the continuous mapping theorem (see {24], p. 20),
z'^(N„ip„.•) -z^(i,•) ->z;;(i.•) -z;;'(i, ) = o mi^(T).
Convergence in distribution and in outer probability to a degenerate limit are equivalent. D Proof of Theorem 4 To prove Theorem 4, it suffices by Pollard ({13), Theorem 10.2) to show lhal flie following two conditions are satisfied.
A2, (X^„(/i), • - . X N . ^fi)) ^ ( G p ( / i ) Gp(fA). V./1 fd^T. a,s..
Proof of AL Let
and
\li^f.j ^ fi£ n f i w Then,
Prp(fi£A') = l (12) by Lemma 2 and (iv). Let w e fifjv- For any X > 0. we can. by (4). find 0 < a < i' < oc
such that
Pr"|a <v'" <b\> 1 - A . (13) fi Spnnger
d then, for 0 < ?? < a
< p i " l
< P l " ( | — - v l >r,\-^Pt"(«fla.b]]
II n I I
-(- P r ^ I max XJf,,, > e L
^ l<a-'j)'i<m<{fc+^)n II "™ d J^'(5./)p) J By assumption (iv),
llN"^ I 1 hm P I " M ' - v " > )) = 0.
(14)
" l(o-'l)n<m<(i'+'7)'' V(a - 1 ) " V S x ^ , : P r } . j max I s / S x ^ ^ l > 8 7 ( 0 - i,)n 1 , it follows that
PrL,l
{by Ottaviani's inequality] f^g\
<-LprJ.(||x:J| > - / ^ ^ !
= [(6 + i|)n].
c, = max Prf. j IvS^X^T^ - VtX;;'J| > ' ' ^ ' ' ' ~ " ' " 1
%-.?.-Hii^-ii...„,>l/^
I . . I 1 (""l
SmaxPrJ. X^J > ^ . ^ - ^
< m a x ? / ^ £ | x „ " J |
fi Springer
Weak Convergence ofthe Bootstrap General Empincal Process Smee ea € fif, it follows that
max - , / £ X„J < -
for every e > 0 and for n sufficiently large.Combining (13)-(17) yields
and hence, Al holds since X > 0 is arbitrary and (12). D Proof of A2. Now, let Td = {/i fd] be any finiCe subset of T and leC F ^
denote the law of ( x ^ „ ( / i ) , . . , X ^ „ ( / r f ) ) on R''. Then, F^,^ F„^^^ is flie law of (X^ (/]) X ^ ifd)) on W'. Let JT denotes the Prohorov metric for probability mea- sures and A'd(0, S ) be the law of ( G p ( / i ) , . . , G p ( / d ) ) , where E ^ ( C o v p ( / , , / ; ) ) . It follows from Lemma 3 fliat 7r(F;i;;__, Nd(0, E)) ^ Oa s. as/i - * oo, that is,
( x ^ , , ( / i ) , ...,X%^ (fd)) ^ ( G p ( / i ) Gp(fA) a,s.,
which proves A2 and hence also Theorem 4, • Proof of Theorem 5 We can now proceed analogously lo the proof of Theorem 4. Lel w G
fi^- By assumption (a).
^ ^ P r p l A ' n - m l P r p s . ||;
< Prp{A'„ <I^]+J2 P^''''^" = ""'^"^ I ||^^™ll:F'(i
< Prp{A'„ <k] + -J2 P'P^^" = ' " 1 ^ P'"'" 1^,,^ ^^,
{by Markov's inequality].
Let k> 0. Since co e fi£. there exists .So - &o(.A such fliat
for alio <S <So.
LetO < .5 < 5o be given. From (18), il follows dial there is a/; = *:(/, 5) such that
for all m,n > k. Hence, we have, for n >k.
>^^IF».|..,..„.^=I-'"'"'" ^"'
and consequentiy
by assumption (b).
fi Spnnger
From what has already been proved, it may be concluded that lim lim sup PrL. | \\'X% > e> = 0, s^o „ ^ ^ ^ *^ |ti '^"WriS.pp) \ and A l is proved.
Finally, we apply Lemma 4 to get A2. Q Proof Theorem 6 With the same notation as in the proof of Lemma 5, we have
SS. = y ^ {K - XE,.,) + ( / ^ - 1 ) *i".i + Xi^.i + sAlI ( ^ - i) C- Now, by Lemma 5, by assumption (C) and by the Slutsky's theorem,
[N^/' - \
V ~t^ ^^^" ~ ^^""^^ ^ ^ as /I - * o o . Since w G fis, it follows that
^ \-
1 X[,..i => 0 as n ^ oo, by assumptions (B), (C), and again the Slutslcy's theorem.
Finally, X[;,„] and 7/7;^ ( ^ - 1 j P ^ are independent. X[^„] ^ Gp in view of (B) and CO € Qs- From (B) and (C). we have
VST 1 = ^ — - ^ ^ i ^ f i l : KP-,: I (J-„ VJf„ " ^ so we get
Since Z is independent of G , and both V ? Z P and C p are Gaussian and centered, xJ is Gaussian, centered, and with
C o v ( s J ( / ) . 5 j ( j ) ) = C o v ( G , ( / ) + y ^ Z f / . C , ( s ) - ) - y ; 6 Z / ' s )
= C o v ( G f ( / ) . C f ( / s ) ) H - j 3 C o v ( Z P / . Z P g )
= Pfg-i\-e)PfPe.
This completes the proof. • From Theorem 6, we can obtain the following interesting particular cases:
(i) Nn=n:fi = 0. S%^ ^ Gp a.s. (Theorem 1).
(ii) A'n-Poisson(Mn) fi = i. S%^=^Sp a.s.
This case corresponds to a generalization of (2).
(iii) Nn'-Bin(n.p):fi^i-p.S%^^s'p-'' a.s..
This case is of particular interest for the wide applicability of the binominal distribution.
Finally, we remark thac 5^ is a generalization of the P-Brownian bridge process Gp, to which ic reduces for fi — 0.
fi Spnnger
Weak Convergence of the Bootstrap General Empirical Process
Acknowledgments The author is grateful lo the referee for carefully reading the manuscript and for his valuable comments and suggestions which improved this paper
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