An improved Bennett s inequality

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1 COMMUNICATIONS IN STATISTICS THEORY AND METHODS 017,VOL.0,NO.0,1 8 hps://do.org/ / A mproved Bee s equly Sogfeg Zheg Deprme of Mhemcs, Mssour Se Uversy, Sprgfeld, MO, USA ABSTRACT Ths pper gves mproveme o Bee s equly for l probbly of sum of depede rdom vrbles, whou mposg y ddol codo. The mproved verso hs closed form expresso. Usg refed rhmec-geomerc me equly, we furher mprove he obed equly. Numercl comprsos show h he proposed equles ofe mprove he upper boud sgfcly he fr l re, d hese mprovemes ge more prome for lrger smple sze. ARTICLE HISTORY Receved 19 Jue 017 Acceped 10 Augus 017 KEYWORDS Arhmec-geomerc me equly; Bee s equly; Lmber W fuco; l probbly. MATHEMATICS SUBJECT CLASSIFICATION 60E15; 6E17 1. Iroduco Bee s equly Bee 196provdes upper boud o he probbly h he sum of depede rdom vrbles deves from s expeced vlue. For coveece of referece, we gve he full seme of Bee s equly d he skeched proof below. Theorem 1 Bee. Assume X 1,...,X re depede rdom vrbles, d EX = 0, EX = σ, X < M lmos surely. The, for y 0 < M, P X exp σ M M h B σ O 1 where hx = 1 + x log1 + x xdσ = σ. Proof. Sce X 1,...,X re depede, by expoel Mrkov s equly, for y λ>0, here s P X where EexpλX = E E exp λx = E expλx = EexpλX e λ e λ e λ k=0 = 1 + σ M λ k X k = 1 + k! λ k M k k= k! k= λ k EX X k 1 + k! k= λ k σ Mk k! = 1 + σ M eλm 1 λm CONTACT Sogfeg Zheg SogfegZheg@MssourSe.edu Deprme of Mhemcs, Mssour Se Uversy, Sprgfeld MO 65897, USA. 017 Tylor & Frcs Group, LLC

2 S. ZHENG σ exp M eλm 1 λm 3 The, we hve P X e λ EexpλX 4 σ e λ exp M eλm 1 λm σ = e λ exp M eλm 1 λm 5 6 The proof he proceeds by mmzg boud 6 wh respec o llλ >0. There re some effors ryg o refe he Bee s equly. I F, Grm, d Lu 015, mssg fcor of order 1/ s dded o Bee s equly uder he Berse s codo; Pels 014, uder he codo mposed o he hrd order momes, he uhor developed shrp mproveme o Bee s equly. However, hey dd o cosder he dffereces mog he vrces of he rdom vrbles. Ths pper derves mproved verso o Equo 1, uder he sme codos. The derved mproveme hs closed form expresso. Usg refed rhmec-geomerc meequly,wefurhermprovehedervedboud.wewllcomprehedervedew bouds o he orgl Bee s equly by grphcl plos.. Improved Bee s equly We frs prese lemm whch wll be used hs pper Lemm. For < 0 d b > 1,heresuqueposvesoluooheequoe = + b, whch s gve by = W 1 exp b/ b/ where W s he Lmber W fuco Corless e l Proof. Le f = e b, he f 0 = 1 b < 0sceb > 1, d s, f > 0. The dervve f = e > 0becuse < 0. Thus, f s moooclly cresgfuco.byheermedevlueheorem,heresuqueposvesoluo o he equo f = e b = 0. The gve equo s e = + b = + b h s exp + b b = exp + b

3 COMMUNICATIONS IN STATISTICS THEORY AND METHODS 3 or equvlely, [ + b ] [ exp + b ] = 1 exp b By he defo of Lmber W fuco Corless e l [ + b ] = W 1 exp b Cosequely, he soluo s = W 1 exp b b We hve o show h he soluo s posve, h s W 1 exp b > b Sce he Lmber W fuco W x s cresg whe x > 0, < 0db > 1, here s W 1 exp b < W b exp b = b 7 8 where he equly follows from he dey W xe x = x. Clerly,Equos8 d7 re equvle. I he proof of Bee s equly skeched Sec. 1, weusedheequly1+ x e x o ge Equo 3,whchsofelooserelxo.Ised,wecproceeddreclywh Equo, h s, EexpλX 1 + σ M eλm 1 λm 9 dhswllgveusgherboud.wehvehefollowgheorem. Theorem 3. Wh he sme ssumpos s Theorem 1,le A = M σ + M 1 d B = M 1 d = A W Be A,whereW s he Lmber W fuco. The, for y 0 < M, P X exp /M + log [1 + σ ] M e 1 B I 10 Proof. As he proof of Theorem 1, wecgeequo4, d pplyg Equo 9, here s P X e λ EexpλX e [1 λ + σ ] M eλm 1 λm 11 [ e λ 1 + σ ] M eλm 1 λm 1

4 4 S. ZHENG = exp λ + log [1 + σ ] M eλm 1 λm 13 where Equo 1 follows from he rhmec-geomerc me equly whch ses h for posve x 1, x,...,x, 1 1/ x x By he elemery equly 1 + x e x,heres [1 + σ ] σ M eλm 1 λm exp M eλm 1 λm Thus, comprg Equos 1 o6, 1 provdes beer boud. Cosequely, we could ge gher upper boud for he l probbly by mmzg he boud 1or13wh respec o λ>0. Leg gλ = λ + log [1 + σ ] M eλm 1 λm Equo 13becomes P X expgλ Togeghboud,weeedommzehefucogλ wh respec o λ>0. For hs purpose, clculg g λ = + σ Me λm M M 1 + σ e M λm 1 λm d seg g λ = 0, here s σ M MeλM M = + σ M eλm 1 λm or σ M M eλm = σ M λm + σ M + 15 M Sce < M,wele d b = σ M = σ M = M < 0 M σ M M + σ M M = 1 + M σ Usg he defed d b,equo15couldbewres e λm = λm + b M > 1 14

5 COMMUNICATIONS IN STATISTICS THEORY AND METHODS 5 By Lemm,wehve where λm = W = M σ 1 exp b b + M M 1 W M 1 exp σ + M 1 = A W Be A 16 A = M σ + M 1 d B = M 1 We le = A W Be A. Flly, subsug λm.e., oequo13, we ob he upper boud for he l probbly s P X exp /M + log [1 + σ ] M e 1 whch s wh o be proved. 3. Furher mproveme I he proof of Theorem 3, he rhmec-geomerc me equly Equo 14 ws used s mjor grede. There s refed verso of he rhmec-geomerc me equly Crwrgh d Feld 1978 whch mkes possble o furher mprove he resul Theorem 3. Lemm 4 Refed Arhmec-Geomerc me equly Crwrgh d Feld Suppose h x k [, b] wh > 0, p k 0 for k = 1,,...,, d furher ssume h p k = 1.Le x = p kx k.wehve 1 b p k x k x x x p k k 1 p k x k x 17 The proof of Lemm 4 sgvecrwrghdfeld1978. We re eresed specl cse whch ech p k = 1/,he p k x k x = 1 x k x = vrx where x = x 1, x,...,x,dvrx deoes he vrce 1 of he rry x = x 1, x,...,x. Thus, for hs specl cse, here s 1 b vrx 1 1/ x x k 1 vrx d cosequely x k 1 1 Alhough s o excly he vrce, cllg s vrce smplfes our oo grely. x 1 b vrx 18

6 6 S. ZHENG By comprg Equo 14 dequo18, clerly, Equo 18 provdesgher boud for he produc of he gve umbers by corporg he vrce formo. We ob Equos 1from11 by pplyg he rhmec-geomerc me equly Equo 14. By pplyg he refed rhmec-geomerc me equly Equo 18, s possble o furher mprove he boud. Le σm = mxσ 1,...,σ d V = σ σ /.Wehve,fory1, 1 + σ M eλm 1 λm 1 + σ M M eλm 1 λm M λ Applyg Equo 18, here s [1 + σ ] M eλm 1 λm [1 + σ ] M eλm 1 λm eλm 1 λm V 19 M 4 M λ SubsugEquo 19oEquo11heproofofTheorem 3,foryλ>0, here s P X e [1 λ + σ ] M eλm 1 λm [ e λ 1 + σ ] M eλm 1 λm eλm 1 λm V 0 M 4 M λ Idelly, o fd he bes upper boud, we would lke o choose vlue for λ o mmze he rgh hd sde of Equo 0, whch would yeld hgh order equo h s o esy o solve. We oe h Equo 0 holdsforyλ>0, hus, we c jus choose he λ whch mmzes Equo 1, h s, he λ whch s defed Equo 16. We summrze our dscusso s Theorem 5. Wh he sme ssumpos d oos s Theorems 1 d 3, leσm = mxσ1,...,σ d V = σ σ /, d M = 1 + σ M M e 1 For y 0 < M, here s [ P X e /M 1 + σ ] M e 1 e 1 V B M 4 F 1 M I s srghforwrd o verfy h he mproved Bee s equly Equo 10c be obed from Equo 1bysegV = 0,hs,gorghevrceofσ s. I hs sese, we see h Equo 10 wses pr of he formo coed σ s. We lso observe from Equo 1hlrgerV couldledogherboud. 4. Comprsos I hs seco, we umerclly compre he orgl Bee s equly B O gve Equo 1 d s ehced versos B I d B F gve Equos 10 d1, respecvely. Whou loss of geerly, we ssume M = 1, d le = pm wh 0 < p < 1. We ssume h he vrce σ = /,for = 1,,...,. For = 30, d for p eqully spced bewee 0.1 d 0.8, we clcule he ro B O /B I d B O /B F. We show he curves of he ros Fgure 1,dFgure 1b demosres

7 COMMUNICATIONS IN STATISTICS THEORY AND METHODS 7 Fgure 1. The ro bewee he orgl Bee boud B O d he mproved versos B I d B F. Sold curve s he ro B O /B F, whle he dshed curve s B O /B I. he ros whe 0.1 < p < 0.5formoredels.Weobservehhefrlrep lrge, he boud Equo 1 mproves he orgl Bee boud by 3 0 mes, whle he boudequo10 mproves 10 mes; for smll p h s, he er l re, he mprovemes re mor. We he choose = 50 d = 80, d he correspodg curves re gve Fgure 1c f. We observe he sme red s Fgure 1 d 1b, hs,equo1 gves beer boud hequo 10, whchs expecedbecuseequo 1 corpores more formo. For boh mproved versos, he mproveme s more sgfc he fr l re.e., for

8 8 S. ZHENG p lrge.comprghecurvesfordffere, weseehforlrge, he mproveme s more prome. 5. Cocluso d fuure work I he proof of Bee s equly, we employ gher upper boud for he mome geerg fuco, d hs ebles us o ob ew boud for l probbly for sum of depede rdom vrbles, whou mposg ddol codo. Ths pper furher mproves he obed upper boud by usg refed rhmec-geomerc me equly, whch corpores more vrce formo of he rdom vrbles. We compre he orgl Bee s upper boud d he mproved versos by plog he curves of he ros bewee dffere upper bouds, d he resuls show h he proposed upper bouds ofe mprove he orgl Bee s boud sgfcly he fr l re, d hs mproveme becomes more sgfc for lrge smple sze. Uder he followg Berse s codo: for posve cos ɛ, E X k 1 k!ɛk E X k for ll k dll [1, ] ws proved F, Grm, d Lu 015bhforllx > 0, where { P X x exp λx + log σ = 1 E X 1 + λ σ 1 λɛ } x/v d λ = xɛ/v xɛ/v Equo c be regrded s couerpr of Theorem 3 uder he Berse s codo. SmlroEquo, would be eresg o develop couerpr for Theorem 5 uder he Berse s codo d hs s our ex sep of work. Ackowledgmes The uhor would lke o exed hs scere grude o he oymous revewer for he suggesos, whch helped mprove he quly of hs pper. Ths work ws prlly suppored by Summer Fculy Reserch Fellowshp from Mssour Se Uversy. Refereces Bee, G Probbly equles for he sum of depede rdom vrbles. Jourl of he Amerc Sscl Assoco 57 97: Crwrgh, D. I., d M. J. Feld A refeme of he rhmec me-geomerc me equly. Proceedgs of he Amerc Mhemcl Socey 71: Corless, R. M., G. H. Goe, D. E. G. Hre, D. J. Jeffery, d D. E. Kuh O he Lmber W Fuco. Advces Compuol Mhemcs 5: F, X. Q., I. Grm, d Q. S. Lu Shrp lrge devo resuls for sums of depede rdom vrbles. Scece Ch Mhemcs 58 9: F, X. Q., I. Grm, d Q. S. Lu. 015b. Expoel equles for mrgles wh pplcos. Elecroc Jourl of Probbly 0 1:1. Pels, I O he Bee-Hoeffdg equly. Ales de l Isu Her Pocré Probbly d Sscs 50 1:15 7.