SEQUENTIAL POINT ESTIMATION OF ESTIMABLE PAJUU~ETERS

BASED ON U-STATISTICS

by

Pranab Kumar SenDepartment of Biostatistics

University of North Carolina, Chapel Hill

and

Malay GhoshIowa State University

Institute of· Statistics Mimeo Series No. 1236

June 1979

SEQUENTIAL POINT ESTIMATION OF ESTIMABLE PARAMETERSBASED ON U-STATISTICS

by

PRANAB KUMAR SEN 1

UIdversity of North Carolina

MALAY GHOSH 2Iowa State University

ABSTRACT

Asymptotically risk-efficient sequential point estimation of

regular functionals of distribution functions based on U-statistics

is considered under appropriate regularity conditions. Some

auxiliary results on U-statistics are also considered in this

context.

fu~S Subject Classification: 62L12, 62LlS, 62G99.

Key Words &Phrases: Asymptotic normality, asymptotic risk-efficienc~

. estimable parameter, risk function, sequential estimation, stopping

times, U-statistics.

lWork supported partially by the National Institutes of Health,Contract No. NIH-NHLBI-71-2243 from the National Heart, Lung andBlood Institute.

2Work supported by the Army Research Office, Durham GrantNumber DAAG29-76-G-OOS7.

-2-

1. INTRODUCTION

Robbins (1959) initiated the study of the l~equentiaZ point

estimation of the mean of a normal distribution and this was later

extended by Starr (1966) and Starr and Woodroofe (1969). Sequential

point estimation of the scale parameter of a gamma distribution was

considered by Starr and Woodroofe (1972), while the case of the multi­

normal mean vector was treated in Ghosh, Sinha and Mukhopadhyay (1976).

The relevant properties of the normal and gamma distributions were

fully exploited in the above papers.

Recently, Ghosh and Mukhopadhyay (1979) have proposed a sequential

procedure for the point estimation of the mean of an unspecified

distribution (admitting finite eight moments) and established its

asymptotic risk-efficiency (to be defined in Section 2). Their non­

parametric appraoch provides clues for further generalizations

embracing a broader class of statistics and requiring less stringent

conditions.

The object of the present investigation is to study nonparametric

sequential point estimation of an estimabZe parameter based on U­

statistics. In this context, the moment-condition of Ghosh and

Mukhopadhyay (1979) is relaxed considerably and their results are

extended to a broad class of U-statistics. Along with the preliminary

notions, the main theorems are presented in Section 2. Two relevant

theorems on U-statistics are also considered in this section. Section 3

is devoted to the proofs of the main theorems. Section 4 deals with

some generalizations of these theorems along with some general remarks.

The Appendix deals with the proofs of the theorems on U-statistics.

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2 • THE lviAIN RESULTS

Let {X., i ~l} be a sequence of independent and identically1

distributed random variables (i.i.d.r.v.) with a distribution function

(d.f.) F defined on the real q-plane for some q~l. Let

~(Xl"",Xm)' symmetric in its m(~ 1) arguments, be a Borel

measurable kerne~ of degree m and consider the estimabZe parameter

(a functional of the d.f. F)

6(F) = E~(Xl"",Xm)

= J... f<P (Xl' ... , Xm)dF (xl) ... dF (Xm), F E F ,Rqm

where F ={F: 16(F) I <co}. Then, for n ~m, the U-statistic

(2.1)

U ,n

corresponding to 6(F), is defined by [c.f. Hoeffding (1948)]

U =(n)-\ <p(X. , ... ,X.); C ={il, ... ,i: 1 ~il<"'<im~n}n m LCn,m 1 1 1m n,m m

(2.2)

Note that Un is symmetric in Xl'" "Xn and is unbiased for 6(F).

Let then ~d(xl"",xd) =E~(xl"",xd' Xd+l, ... ,Xm), 0 ~d ~m and let

Then, whenever

2 2E<Pd (Xl' ... , Xd) - 6 (F),

n ~m and E~2 <co

o ~d ~m (1;0 =0). (2.3)

(2.4 )

Note that by the reverse martingale property of {U, n ~m}n

a~ -a~+l = Var(Un -Un+l ) ~O, so that a~ is ~ in n(~m).

To motivate the sequential procedure, suppose that the Zoss

incurred in estimating 6(F) by Un is

L = a [U - 6 (F) ] 2 + cn ; a >0, C >0 ,n n (2.5)

where a and c (cost per unit sample) are specified constants. The

object is the minimize the risk (for given a, c)

-4-

2R (n; a, F) = EL = aa +cn,c n n

by a proper choice of n. Towards this, we have the following

(2.6)

Lemma 2.1.. For every a >0, C >0 and m ~ 1, whenever E<j>2

<00

Rc(n; a, F) is a convex function of n.

The proof of the lemma is given in the Appendix. Note that by

Lemma 2.1, there exists an n* (= n*(a, c; F)),c such that

R (n*; a, F) =min R (n; a, F)c c n c

= aa~* +cn~ , (2 .7)c

where in (2.7), the minimization is restricted to integers n ~m and

thereby, n* is also an integerc (~ m), though it need not be

unique (there may be two ~onsecutive values of n* for whichc

(2.7) holds).

From (2.4), (2.6) and (2.7), it follows that n*cdepend-'t on

a, c, m as well as ~d' 1 sd sm, where the later parameters are

all (unknown) functionals of the (unspecified) d.f. F. Thus, in the

absence of knowledge of these ~d' 1 sd sm, no fixed sample size

minimizes the risk simultaneously for all ~d) 1 sd sm, and hence,

a sequential procedure may be desirable to achieve this goal.

We assume that 8(F) is stationary of order 0 [viz. Hoeffding

(1948)], so that

1o < ~l s - ~ <00 •m m

Note that by (2.4), (2.8) and Theorem 5.2 of Hoeffding (1948),

(2.8)

2 2-1an = m n l';l + ~ (n) where -2

~(n) =O(n ) (2.9)

and nt.:(n) + 0 as n t 00. Suppose that in (2.6), we neglect (for

the contribution of t.:(n) and, then in (2.7), we denote the

-5-

resulting solution by on ,c so that we have for small c,

o 0R (n " a, F) - 2cn ,c c c (2.10)

where g(c) -h(c) means that g(c)/h(c) ~l as c ~O. Using (2.4)

tOO -i= 1..' ld. (n*) +cn*,

1= 1 C ·c222

so that c(n~) -a m 1',;1.

possible to write R (n*; a, F)c c

Loo • * -i-land . lId. (n ) - c,1= 1 C

and (2.7), it is

2where dl =am 1',;1

Hence,

lim n*/no =1c-l-O c c

and limR(n*; a,F)/R (no; a, F)=l,c~O c c c c

(2.11)

whenever (2.8) holds. Hence, in the sequel, we shall occasionally

interchange n* and nOc c for the convenience of our manipulations.

For the proposed sequential procedure, we proceed to estimate 1',;1

first. As in Sen (1960, 1977), we let U(i)n-l be the U-statistic

based on (Xl, ... ,X. l' X. l"'.'X), for i =l, ... ,n (~m +1). Let1- 1+ n

then

(2.12)

2m 1',;1is a (strongly) consistent estimator of

s2 = (n _l)-lt~ [U(i) -U ]2 1n 1..1 =1 n-l n ' n ~m + .2Whenever I: <00 s

'1Il ' n

(= lim n02). Motivated by (2.7), (2.5), (2.10) and (2.11), wen~ n

propose the following sequential procedure:

Let nO (~ m +1) be an initial sample size and define the

stopping number N (= N (a)) byc c!,,; y

Nc =min{n ~nO: n ~ (a/c) 2(sn +n- )}, (2.13)

where y (> 0) is a suitable constant, to be defined later on.

Our proposed (sequential point) estimator of 8(F) is UN and thec

risk for the proposed procedure is

(2.14)·2

= aE{UN -8(F)} +cENc

.c

is the following

R~(a) = ELNc

The main theorem of the paper

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Theorem l. If (i) 8(F) is stationary of order zero .. (ii) EI4>1 4+

0 <00

forsome 0>0 and (iii) in (2.13), yE(O, (2+0)/4), then

lim 'OR*(a)/R (n*; a, F) = 1 . (2.15)CT c c C

The proof of the theorem is deferred to Section 3. It may be

remarked that (2.15) [in the sense of Starr (1966)] asserts that the

risk involved in the sequential procedure is asymptotkally (as

c -I- 0) equivalent to the risk involved in the corresponding "optimal"

fixed-sample size procedure and hence, the sequential procedure is

asymptotically risk-efficient.. for all F satisfying (i) and (ii)

of Theorem 1. Also, it may be mentioned that Ghosh and Mukhopadhyay

(1979) have considered the case of the population mean (which

corresponds to m=1) and obtain (2.15) under oE 14> I <00 and

assuming that in (2.13) y E(O, ~). In our present setup, m (~l)

is arbitrary, y E (0, (2 +6)/4) (note (2 +0)/4) >~) and we need

that EI4>14

+0 < 00 for some 0 >0. The relaxation of the regularity

conditions is achieved here by using some reverse martingale properties

of {U} and the components ofn

Further, results weaker than

(2.15) can be obtained even without assuming that EI4>1 4+

0 <00 for

some 0 >0. In fact, we have the following:

Theorem 2. Under (2.8), and for EI4>1 2+

0 <00 for some 0 >0,

limC-l-OE(N/n~)k=1, V k E [0, 1], (2.16)

V[UN -8(F)]/On* -+- N(O, 1), as c -1-0 , (2.17)

c c

and in (2.17), 0n* may also be replaced by m(s1/Nc)~c

-7-

It is of natural interest to study the asymptotic distribution

of N (if it eXists). We shall see later on that under E~4 <00,c

V(s2) =v2/n +0(n-2), V n ~2mn

where v2 depends on F. Then, we have the following

(2.18)

Theorem ;5. If (i) E I~ 14

+0 <00 for some 0 >0 and (ii) in (2.13),

~ <y < (2 +0)/4, then as c-l-O

2m2l;;1 (N

c- n~)/ (}n~)~ ~ N(O, 1) •

The proofs of these theorems are presented in Section 3.

(2.19)

For an integer k(~ 1), moment-inequality for U-statistics have

been considered by Funk (1970) and Grams and Serf1ing (1973), while

Sen (1974) studied the LP-convergence of U-statistics, when p ~1.

In the following lemma, we derive a moment in equality for U-statistisc

valid for any power bigger than 1.

Lemma 2.2. Asswne that EI~ Ir <00 for some r >1 . Then, there exists

a positive aonstant Kr « 00), suah that

(2.20)

where

s = {r_1,. if

miner -1,

1 <r s2;

k), if 2(k -1) <r s2kj k~2, (2.21)

and K does not depend on n.r

The proof of the lemma is considered in the Appendix. In the

remaining of this section, we consider the representation of

(2.12) in terms of a set of U-statistics, due to Sproule (1969). For

each d(=O,l, ... ,m), let

-8-

~ }

-1

<j>(.d)(X1,,,,,X

2d) = (2m ..;d)12 L(d)<j>(X , ... ,X )<P(Xa , ... ,X

Q)

m- t ((m _d) 1) 0.1 am 1 IJm

(2.22)

where the summation L(d) extends over all combinations of (distinct)

a1, ... ,am (a1, ... ,am) from (1, ... ,2m -d) with exactly d of the

being common with the a .. Let then (for n ~2m)1

(d) (n) -1 (d)Un = 2m-d l.c <j> (Xi "",Xi ), 0 ~d ~m.

n,2m-d 1 2m-d(2.23)

Then, by (2.12), (2.22) and (2.23), we have (by some routine steps)

where, for some positive constants K1 and K2 (independent of n),

(2.25)

(2.26)

This representation plays a vital role in the proof of the main

theorems.

3. PROOFS OF THE MAIN nIEOREMS

First, we consider the following lemma, which is crucial in the

proofs to follow.

Lemma 3. Z. If EI<j> \2r <00., for some r ~ 1 and (2.8) hoZds., then.,

for every E E (0, 1),

P{N ~n*(l -E)} =0(Cs/ 2(1+y )), as c -1-0 , (3.1)c c

where s is defined by (2.21).

PROOF: Note that by (2.24) - (2.26),

s~ -m21;1 =m2(U~l) - u~O) - 1;1) + L:=l endu~d) , (3.2)

-9-

and, by (2.13), N ~bl/(l+Y), with probability 1, where b =(a/c)~.c

Let then n = [b 1/ (1 +Y)] and n =n* (1 - £) . choo se c so smalllc 2c c '

that nlc sn2c (otherwise, there is no need to prove (3.1)). Then,

by (2.13),

P{Nc Sn~(l -£)} =P{Nc sn2c } SP{sn <b-ln, for some nlc sn sn2c }

< p{ 2 <b- 2 2 f < < }- S - n2 or some nl - n - n2n c c c222 2

SP{sn-ml;;l sml;;l{(l-£) -I}, for some nlc snsn2c }

S p{ Is~ -m2I:ll/m2I:l ~£(2 -E), for some nlc sn sn2c }

(3.3)

By (2.25), (3.2) and (3.3), we have

P{N Sn* (1 - £)} Sp{ max \u (1) - U(0) -I;; I· ~ £I:l

}c c < < n n 1nlc-n_n2c

+ p{ maxnlcsnsn2c

,m \u (d) I > Kn }L.d=O n - lc

where K (> 0) does not depend on c (but depends on £).

(3.4)

Let F be the a-field generated by the ordered collection ofn

Xl, ... ,Xn and by X .,n+J

Then, {U, F ; n ~m} andn n

j ~l (so that F is nondecreasing in n).n

{V (d) F· n ~ 2m - d} for everyn ' n' ,

d =0, l, ... ,m are reverse martingales, and hence {D(l) _U(O) -1:1

, F .n n n'

n ~2m} is also a reverse martingale, so that for nlc ~2m, by the

Kolmogorov-Hajek-Renyi-Chow inequality for reverse martingales and

our Lemma 2.2,

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(3.5)

where s is defined by (2.21), and,

S (m +l)K nir { max Elu~d) Ir } , (3.6)r c Osdsm lc

where EI<t>1 2r <00='> Elu~d) Ir <00, V 0 sd Sm, n ~2m7 (3.1) then

follows from (3.4), (3.5) and (3.6) after noting that by (2.21),

s <r and rilc

= [bl/(l+Y)] =0(c l / 2 (1+Y)) as c +0. Q.E.D.

Now, by virtue of (2.24) - (2.26) and Lemma 2.2, we have for

some r ~ 1,

I 2 2 Ir -sE s - m 1':;1 S K n ,n r (3.7)

where s is defined in (2.21). Using (3.7) one can follow the lines

of proof in part (d) of the lemma of Ghosh and Mukhopadhyay (1979) to

conclude that

E(N /n*)k -~ 1 as c +0, V k <s,C C

(3.8)

where s is defined in (2.21). In particular, if in (3.7), we let

r =1 +0/2, 0 >0, then s >1, so that (3.8) holds for every

o sk sl. This proves (2.16). Moreover, by (2.13),

(3.9)

all converge a.s. to their expectations

222E'" <00 ---> S ~ m r a s as'I' n "'I ..,

bSN SNc snO +b(sN -1 + (Nc -l)-Y)c c

where b2 =a/c. Since by the Convergence Theorem for reverse

martingales, u(d) 0 sd Smn '

as n ~OO, by (3.2), we claim that

n -+00.

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Hence, dividing all sides of (3.9) by n* and letting c-l-Oc

(i. e., n~ -+00), we obtain that

N /n* -+ 1 a.s., as c -l-O;c c (3.10)

(2.17) follows then by using (3.10) and the results in Section 5 of

Miller and Sen (1972). In fact, for (2.17), E~2 <00 suffices. This

completes the proof of Theorem 2.

We proceed now to prove Theorem 1. First, note that by (2.16)

(EN )/n* -+ 1 as c -l-O. Hence, by virtue of (2.6) and (2.14), toc c

prove (2.15), it suffices to show that

lim aE{UN - 6(F)}2 / (cn~) =1.c-l-O c

Let us write E{UN _6(F)}2 =E{Un* _6(F)}2 +E{UN

-Un

*}2 +c c c c

2E(U * -6(F» (UN -U *) and note that by (2.9) and (2.11),n nc c c

(3.11)

2E{Un* - 6(F)} =

c(m2Z;1/n*) +0(n*-2), so that

c c

lim aE{U *-6(F)}2/(cn*) = 1.c-l-O nc c

(3.12)

Hence, to prove (3.11), it suffices to show that

lim n*E{UN -U *}2 = 0 .ctO c c nc

Using the definition of the ~d' prior to (2.2), we may write

U =mU(l) +U*· V n ~m,'n n n'

(3.13)

(3.14)

(1) -lInU =n . l~l(X.),n ~= ~

EU* = o·n '

(3.15)

(3.16)

where C1 and C2 are positive and finite constants, independent of n.

Also, we have by (3.14),

(3.17)

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where by (3.16),

En*U*2 *-1~ Clnc ~ 0 asc n*c

c i- 0 . (3.18)

Further,

= n* '\ I(N =k)u*2c L c kk~nlc

~ n* I I(N =k)Uk*2 + (sup U*2)I(N >n2

)c k~ c n >n2 n c cnlc~ n2c c

(3.19)

where nlc and n2c are defined after (3.2). Now

(3.20)

~ n*/P (N ~n )c c 2c

-+ 0 as c ~O, by (3.1), (3.19) and definition n1c .

Also, by the Doob maximal inequality for (reverse) submartingales,

*2n*E{ sup U I(N >n2

)}c > n c cn n 2c

(3.21)c i- 0{ *2} ~ *2~ n*E sup U ~n*4E(U ) ~ 0 asc n c nn>n2c 2c+l

where the last step follows from (3.16) and the definition of n2c .

Hence, it suffices to show that

lim n*E{uN(l) _U(~)}2 =0.ci-O c c nc

(3.22)

Now U(l) is a sample mean for all n ~m. If follows fromn

Anscombe's (1952) result and (3.13) that

(n*)J:2(u(l)_u(l))~O as 10c N n* c y •

c c(3.23)

-13-

We follow then the line of proof of Ghosh and Mukhopadhyay (1979)

[in view of our Lemma 2.2, which is stronger than the moment inequality

of Grams and Serfling (1973) (restricted to integer power), their

eighth moment condition is not needed here]. and obtain that

{n~(u~l) _u~:))2} is uniformly integrable in c scO'c c

(3.24)

for some cO~O. From (3.23) and (3.24), we conclude that (3.22)

holds and the proof of Theorem 1 is now complete.

To prove Theorem 3, first, note that from Sproule's (1974)

Theorem, one gets

(3.25)

where

using

2 2sN can also be replaced by sN_l

c cthe Mann-Wald theorem, we obtain that

in (3.25). Hence

(3.26)

where in (3.26), also, . sN may be replaced by sN -1' From (3.26)c c

and the definition of the stopping time in (2.13), one finds that

the sufficient condition in Theorem 3 of Ghosh and Mukhopadhyay (1979)

hold~with A direct appeal to this theorem

now yields (2.19). Q.E.D.

4. SOME ADDITIONAL REMARKS

Ghosh, Sinha and ~fukhopadhyay (1976) have considered sequential

point estimation of the multinormal mean vector with unknown covariance

matrix. If X., i ~l are LLd. random p(~ 1) - vectors with~1

E~l =].1 and V(~l) =1, positive definite (p.d.), then assuming the

loss function, based on Xl' ••• , X ,~ ""11

to be of the form

-14-

L = (X -11)' A(X - II) + cnn ""1l I::. ~ ""1l I::.(4.1)

where ~ is a known p.d. weight matpix, c is the cost per unit

sample and - -lInX =n . IX.,""1l 1.= 1.

n ~ 1, the risk function is given by

-1 \'Rn = n Tr (~i) +cn ,

so that for known L, the risk is minimized at

(4.2)

1 \' knO ={c- Tr(~f.)}2 . (4.3)

For unknown I, by analogy to (2.13), we define the stopping time

N =smallest positive integer n (~2) for which

n ~c-~{Tr(A S ) +n-Y} ,~ ""1l

-l\'nwhere S =(n - 1) L..' 1 (X. - X ) (X. - X )',""1l 1.= ~1. ""1l ~1. n

n ~2.

(4.4)

Now, Tr (A S ) is a~ ""1l

U-statistic and the results of the previous sections apply to yield

an "asymptotically risk efficient" sequential procedure. In this

context, the multinormality of the X.~1.

is no longer needed.

In the context of jackknifing, Sen (1977) has considered a class

of smooth functions of U-statistics. Under his assumed boundedness

conditions of the first and second order (partial) derivatives and

the conditions of Cramer (1946, p. 353) the results of Sections 2 and

3 can also be extended to such functions of U-statistics.

Finally, Robbins (1959), while considering the case of the

normal mean, proposed a slightly different loss function, namely,

L =alU - 8(F)1 + cn.n n (4.5)

Our asymptotically risk efficient procedure also holds for such a

loss function, provided in (2.6) through (2.10), we make the necessary

modifications. Note that as n +00,

-15-

so that the optimal value n*, in this case, is given byc

2 2 2 I;n~ ~ (a m 1:;1/ 21TC ) 3 as c ~ 0 ,

and analogous to (2.13), we define the stopping number by

(4.6)

(4.7)

(4.8)

where y and nO are defined as in Section 2. With these modifications,

the theorems in Section 2 extend directly to the case of the loss

function defined by (4.5). A similar case holds for L =a IU - e(F) Ib +cnn n

(5.2)

for some b >0, where for b ~2, we need to replace condition (ii)

of Theorem 1 by EI<P I(2+0) b < 00 for some 0 >o.

5. APPENDIX

Proof of Lemma 2.1

Note that by (2.6),

R (n +2; a, F)-2R (n +1; a, F) +Rc(n; a, F)c c222= a (a 2 - 2a 1 + a ),n+ n+ n

Hence, it suffices to show that for all

For this, define as in Hoeffding (1948),

where

n ~m,

2 < 2 ua l-a,vn~m.n+ n

62

R (n; a, F) ~O.c

d i (d)°d = Li=O(-l) i I:;d_i' (5.3)

so that 00

= So =0 . Then

I:;d = L~ -0 (~) 0d . = L~-0 (d d .) O.1- 1 -1 1- -1 1

From (2.4), (5.3) and (5.4), we have by some standard steps

( )-1 ( ) ( .)2 _ n m m n-1 0

an - m Li=l i m - i i

(5.4 )

(5.5)

-16-

so that by (5.2) and (5.5), we have

62Rc(n; a, F) .L~=1[~][:r[:=th~n (~:m~ :h+ 2

) - 2 n~\;1 +1]~o, (5.6)

as (n -i +l)(n -i +2) - 2(n +2)(n -i +1) +(n +l)(n +2) =i2 +1 >0, \f i ~1

and by Lemma 5.1 of Hoeffding (1948), ok ~O, \f k =0,1, ... ,m. Q.E.D.

Proof of Lemma 2.2

Let on =[n/m] , n ~m and let

oo -lrnT = (n ) 1</> (X ( 1) 1"" ,X ).n r= r- m+ rm (5.7)

Then, ETn =8(F), V n ~m and defining Fn as in after (3.4),

E(TnIFn) = Un' \f n ~m,

so that by the Jensen inequality for conditional expectations,

EIU - 8 (F) Irs EIT - 8 (F) Ir, for any r ~ 1 .n n

(5.8)

(5.9)

On the other hand, oT is an average of n i.i.d.r.v.'s,n and hence,

Theorem 3 of Sen (1970) applies to the right hand side of (5.9) and

this yields (2.20) and (2.21). Q.E.D.

As noted already, the above generalizes and strengthens the results

of Funk (1970) and Grams and Serf1ing (1973), where they needed r to

be a positive integer. Also, our method of proof is elementary and ~

quite differept from the earlier ones.

REFERENCES

[1] ANSCOMBE, F.J. (1952). Large sample theory of sequentialestimation. Froc. Camb. PhiZ. Soc., ~, 600-607.

[2] CRAMER, H. (1946). f1athematicaZ Methods of Statistics.Princeton Univ. Press, New Jersey.

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[3] FUNK, G.M. (1970). The probabilities of moderate deviations ofU-statistics and excessive deviations of Ko1mogorov­Smirnov and Kuiper statistics. Ph.D. dissertation,Michigan State University.

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