Martingale inequalities and the Jackknife estimate of variance

Statistics & Probability Letters 4 (1986) 5-6 North-Holland

January 1986

MARTINGALE INEQUALITIES AND T H E JACKKNIFE E S T I M A T E OF VARIANCE

WanSoo T. RHEE Faculty of Management Sciences, The Ohio State University, 1775 College Roaa~ Columbus, OH 43210, USA

Michel TALAGRAND Equipe d'Analyse - Tour 46, University of Paris 1:1, 4 Place Jussieu, 75230 Paris 05, France Received May 1985 Revised October 1985

Abstract: We give a short proof of the Jackknife Estimate of Variance of Efron-Stein using martingale difference techniques. The method could be used to get information on other moments than the second moment. Keywords: jackknife, variance.

Let S be a measurable function of n - 1 variables, that is symmetrical, i.e. is invariant by any permutation of the variables. Let X1,..., X, be i.i.d. Consider S -- S ( X 1 , . . . , X,_a) and, for 1 ~< i ~
X/_ 1, X / + l , . . . ,

Xn).

Then S = S,. If S has a finite variance, the jackknife estimate of the variance of S is given by

approach through martingale differences is to give rise to a number of other inequalities. There is a lot of symmetry in the definition of VAR S. Surprisingly enough our proof will break that symmetry. For i >t 0, denote by Ei the o-algebra generated by Xx.... , Xi. So E0 is trivial, and S, is E,_l-measurable. So we have S ~ - E(S~)= ~ n Z l d i , where d i = E ( S , lED - E ( S , IE,-a). Note that d~ is Z;:measurable, and that E ( d i lEa-l) = 0. Such a sequence (d~) is called a martingale difference sequence (m.d.s.). It follows that n--1

B. Efron and C. Stein [2] have proved the following remarkable inequality: V ar S <<.E ( V"/~-#, S ) .

(1)

In other words, VAR is biased upwards. This inequality has been used by J.M. Steele [5] fol stochastic analysis of the Traveling Salesmar Problem. The authors have improved Steel's resul: using martingale difference techniques [4]. The purpose of this note is to give a very short proof of Eron-Stein inequality (1) using martingale difference techniques. The main advantage of the

Var s.= E

(2)

i=l

On the other hand, straightforward computation shows that

e(v-Xg s) -- E e ( s , - sj)2/. i
= ( n - 1 ) E ( S 1 - $2)2/2.

(3)

Fix 1 ~
0167-7152/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

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Volume 4, Number 4

OPERATIONS RESEARCH LETTERS

dent of )(1 and )(2. We have

With the same notations as above, we have

)

E Is, - S~l ~ >1E ( I E ( s ~ - s2 IE,+I) I")

=EIU-VI

E([ E( s, lz,+,) - e( =E(V-v)L where U = E ( S I I E / + I ) - E ( S l l A) and V = E(S2 [Zi+l) - E(S2IA). Since U and V are conditionaUy independent with respect to A, we have E ( U - V) 2 = E ( U 2) + E(V2). By symmetry, we have E ( U 2 ) = E ( V 2 ) = E ( d 2 ) . Thus, we have E(S1 - $2) 2 >/2E(d2). So inequality (1) follows from (2) and (3). An obvious application of this method is to get information on other moments than the second moment. The proper tool is Burkholder's inequality [1]: For each p > 0, there is a constant Cp such that, for each m.d.s. (di)~ ~ ,,,

(where 11f lip = (EIfIP)I/P) • Unfortunately the best constant Cp does not appear to be known. At the present time, this will limit applications to statistics. We have 2 \ 1/2

E d,)

i<~n--1

,,--,

E d,

i<~n--1

" p/2

~1/2

<

~

p.

For p>~2, using the fact that E [ X + Y I p>/ E [ X[ p + E [ Y [ P, when X and Y are conditionally independent given A, we get 11Sl - S 211 p ~ 21/pE [d i [ P So,

II an - E & IIp < 2-1/P( n - 1)1/2Cp II S1 -- S2 II p" In the case I ~ p < 2, we get E [ U - V [ p >i E [ U I P, SO

II Sn - e s n II p < (n - 1)'/2Cp II S1 - S2 II p.

Other inequalities can be obtained using the various m.d.s, inequalities of [1] or [3].

11 E di [[p ~ Cp H( E d 2 ) 1 / 2 []P i<~n i<~n "

,

January 1986

Ild~llp/2j

i <~n--1 < (n -- 1) 1/2 S u p II di II pi~n--1

References Burkholder, D.L. (1973), Distribution function inequalities for martingales, Ann. Prob. I (1), 19-42. Efron, B. and C. Stein (1981), The jackknife estimate of variance, Ann. Statist. 9, 586-596. Garcis, S.M. (1973), Martingale Inequalities, Seminar Notes on Recent Progresses (W.A. Benjamin). Rhee, W. and M. Talagrand (1985), Martingale inequalities and NP-complete problems, to appear in Math. Oper. Res. Steele, J.M. (1981), Complete convergence of short paths and Karp's algorithm for the TSP, Math. Oper. Res. 6 (3) 374-378.