Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric radom variables. We obtai some upper bouds o tail probabilities of self-ormalized deviatios 1k k ξ i / ξ i β 1/β x for x > 0 ad β > 1. Our boud is the best that ca be obtaied from the Berstei iequality uder the preset assumptio. A applicatio to Studet s t-statistic is also give. Keywords: Self-ormalized deviatios; Studet s t-statistic; expoetial iequalities 000 MSC: rimary 60E15; 60F10 1. Itroductio Let ξ i i 1 be a sequece of idepedet, cetered ad odegeerate real-valued radom variables r.v.s. Deote by S = ξ i ad V β = ξ i β 1/β, β > 1. The study of the tail probabilities S /V β x certaily has attracted some particular attetios. I the case where r.v.s ξ i i 1 are idetically distributed ad E ξ 1 β =, β > 1, Shao [9 proved the followig deep large deviatio priciple LD result: for ay x > 0, lim S 1/ [ { 1 V β 1 1/β x = sup if E exp t cx x c 0 t 0 β X β + β 1 β cβ/β 1. The related moderate deviatio priciples MD are also give by Shao [9 ad Jig, Liag ad Zhou [6. However, the LD ad MD results do ot dimiish the eed for tail probability iequalities valid for give. Such iequalities have bee obtaied i particular by Wag ad Jig [10. For istace, they proved that if the r.v.s ξ i i 1 are symmetric aroud 0, the for all x > 0, S V x exp { x. 1 Correspodig author. E-mail: faxiequa@hotmail.com X. Fa. reprit submitted to Elsevier March 9, 017
This boud is rather tight for moderate x s. Ideed, as showed by the MD result of Shao [9 cf. Theorem 3.1, for certai class of r.v.s it holds that x S l V x = 1, for x ad x = o as. See also Theorem.1 of Jig, Liag ad Zhou [6 for o idetically distributed r.v.s. I Fa, Grama ad Liu [3, iequality 1 has bee further exteded to the case of partial imum: for all x > 0, 1k V x exp { x. 3 O the other had, by the Cauchy-Schwarz iequality, it is easy to see that S V. Therefore, for all x >, S V x 1k V x = 0, 4 which caot be deduced from 1 ad 3. Hece, the iequalities 1 ad 3 are ot tight eough for large x s. I this paper we give a improvemet o iequality 3; see iequality 5. Our iequality coicides with 4. More geeral, we establish a upper boud o tail probabilities 1k /V β x, x > 0, for symmetric r.v.s ξ i i 1. I particular, we show that our iequality is the best that ca be obtaied from the classical Berstei iequality: X > x if λ>0 E[e λx x. A applicatio to Studet s t-statistic is also give. The paper is orgaized as follows. Our mai results ad applicatios are stated ad discussed i Sectio. roofs are deferred to Sectio 3.. Mai results I the followig theorem, we give a self-ormalized deviatio iequality for idepedet ad symmetric radom variables. Theorem.1. Assume that ξ i i 1 is a sequece of idepedet, symmetric ad odegeerate radom variables. Deote by V β = ξ i β 1/β, β 1,. The for all x > 0, 1k V β x B β, x := 1 t 1 + t 1 1/βx 1 t {x β 1/β, 5 where t = β 1/β + x β 1/β x
with the covetio that B β, β 1/β =. Moreover, B, x is icreasig i ad for ay x > 0, lim B, x = exp { x. 6 Hölder s iequality implies that S V β β 1/β. Thus whe x > β 1/β, it holds that 1k S k V β x = 0. 7 This feature coicides with the fact that B β, x = 0 for all x > β 1/β. Notice that boud 5 is the best that ca be obtaied from the followig Berstei iequality S [ V β x if E S λ e x Vβ. 8 λ 0 Ideed, if ξ i = ±a, a > 0, with probabilities 1/, the it holds for all 0 < x <, [ if E e λ S x [ Vβ = if E S λ e x a 1/β = if λ 0 λ 0 λ 0 e λx cosh λ 1/β = B β, x. Moreover, whe x β 1/β, boud 5 teds to, which is the best possible at x = β 1/β. Ideed, for the ξ i s metioed above, it holds S k 1k V β β 1/β = ξ i = a for all i [1, = 1. By 6, it holds lim B, x = e x / e /, x 1/, 9 which seems to be cotracted with B, x as x 1/. I fact, 6 holds for ay fixed x > 0, but it does t hold for x = c 1/, where c is a positive absolute costat. For example, cosider the followig fuctio { B, x = exp x l 1 x4, x > 0. The the last fuctio satisfies 9 ad lim x 1/ B, x =. Sice the r.v.s ξ i i 1 are symmetric, it is obvious that for all x > 0, 1k V β x B β, x, where B β, x is defied by 5. Whe β 1,, iequality 5 implies the followig boud. 3
Corollary.1. Assume coditio of Theorem.1. If β 1,, the for all x > 0, 1k V β x exp { x β 1. 10 I particular, the last iequality implies that for ay β 1,, i probability. S 0,, V β For β 1,, iequality 10 implies the followig upper boud of LD: 1 lim sup l 1k V β β 1/β x x, x 0, 1. 11 It also implies the followig upper boud of MD: for ay α β β, β 1 β, lim sup 1 α+ β l 1 1k V β α x x, x 0,. 1 With certai regularity coditios o tail probabilities of ξ i, the LD ad MD results are allowed to be established. We refer to Shao [9 ad Jig, Liag ad Zhou [6. Wag ad Jig [10 proved that for all x > 0, S V x exp { x. 13 A earlier result similar to 13 ca be foud i [5, where Hitczeko has obtaied the same upper boud o tail probabilities S x V. Whe β =, iequality 10 reduces to the followig iequality of Fa et al. [3: for all x > 0, 1k V x exp { x. 14 Thus iequality 10 ca be regarded as a geeralizatio of 13 ad 14. Moreover, sice boud 5 is less tha boud 14, our iequality 5 improves o 14. If ξ i i 1 have + δth momets with 0 < δ 1, the iequalities 13 ad 14 ca be further improved. For istace, Jig, Shao ad Wag [7 proved the followig Cramér type large deviatios for i.i.d. ot ecessarily symmetric r.v.s: S xv = 1 Φx 1 + o1,, 15 uiformly for all 0 x = o δ/4+δ. Similarly, Liu, Shao ad Wag [8 proved the followig result for the imum of sums: xv = 1 Φx 1 + o1,, 16 1k 4
uiformly for all 0 x = o δ/4+δ. Moreover, these asymptotic estimatios are also more precise tha 5 for moderate x s. Let Y i i 1 be a sequece of idepedet odegeerate r.v.s, ad d i i 1 be a sequece of idepedet Rademacher r.v.s, i.e. d i = ±1 = 1. Let ξ i = d i Y i. Assume that Y i i 1 ad d i i 1 are idepedet. The we ow have S = d i Y i, V β = Y i β 1/β, β > 1. The followig result easily follows from Theorem.1 ad Corollary.1. Corollary.. Let ξ i = d i Y i for all i 1. If β 1,, the for all 0 < x β 1/β, 1k V β x B β, x exp { x β 1. I particular, the last iequality implies that for ay β 1,, i probability. where Cosider Studet s t-statistic T defied by It is kow that for all x > 0, ξ = S T x = S 0,, V β T = ξ / σ, ad σ = S x [S ξ i ξ. 1 + x 1 1/ ; 1/ see Efro [1. Notice that for all x > 0, it holds that 0 < x +x 1 < 1/. Usig iequality 5, we have the followig expoetial boud for Studet s t-statistic. Theorem.. Assume that ξ i i 1 is a sequece of idepedet, symmetric ad odegeerate radom variables. The for all x > 0, where B, x is defied by 5. T x B, x + x 1 1/, 17 5
3. roofs of Theorem.1 ad Corollary.1 The proof of Theorem.1 is based o a method called chage of probability measure for martigales. The method is developed by Grama ad Haeusler [4. See also Fa, Grama ad Liu [. roof of Theorem.1. For ay i 1, set η i = ξ i V β, F 0 = σ ξ j, 1 j ad F i = σ ξ k, 1 k i, ξ j, 1 j. 18 Sice ξ i i 1 are idepedet ad symmetric, the [ [ E[ξ i > y F i 1 = E ξ i > y ξ i = E ξ i > y ξ i = E[ ξ i > y F i 1. Thus η i, F i,..., is a sequece of coditioally symmetric martigale differeces, i.e. E[η i > y F i 1 = E[ η i > y F i 1. It is easy to see that S V β = η i 19 is a sum of martigale differeces, ad that η i, F i,..., satisfies η i β = For ay x > 0, defie the stoppig time T : { T x = mi k [1, : with the covetio that mi = 0. The it follows that ξ i β V β β = 1. k η i x, 1 { 1k /V β x = 1 {T x=k. For ay oegative umber λ, defie the martigale Mλ = M k λ, F k k=0,...,, where M k λ = k k=1 exp {λη i E [exp {λη i F i 1, M 0λ = 1. Sice T is a stoppig time, the M T k λ, λ > 0, is also a martigale. Defie the cojugate probability measure λ o Ω, F: d λ = M T λd. 0 Deote by E λ the expectatio with respect to λ. Usig the chage of probability measure 0, we have for all x > 0, S [ k 1k V β x = E λ M T λ 1 1 { 1k /V β x [ { k = E λ exp λ η i + Ψ k λ 1 {T x=k, 1 k=1 6
where Ψ k λ = k [ log E exp {λη i F i 1. Sice η i, F i,..., is coditioally symmetric, oe has ad thus it holds Sice E [exp {λη i F i 1 = E [exp { λη i F i 1, E [exp {λη i F i 1 = E [coshλη i F i 1. coshx = k=0 1 k! xk is a eve fuctio, the coshλη i is F i 1 measurable. Thus implies that E[ exp {λη i F i 1 = coshλη i. Notice that the fuctio gx = log coshx is eve ad covex i x R ad icreasig i x [0,. Sice η i 1 1/β η i β 1/β = 1 1/β, it holds Ψ k λ Ψ λ = 1 g λη i g λ λη i g 1/β. By the fact k η i x o the set {T x = k, iequality 1 implies that for all x > 0, 1k V β x The last iequality attais its miimum at k=1 exp { λ E λ [exp λx + g { λ λx + g { exp λx + g 1/β 1 {T =k [ E 1/β λ 1 {T =k k=1 λ. 3 1/β λ = λx = 1/β β 1/β log + x β 1/β, x 0, β 1/β. x Substitutig λ = λx i 3, we obtai the desired iequality 5 for all x 0, β 1/β. Whe x = β 1/β, we have 1k V β β 1/β = lim x β 1/β 1k V β x lim B β, x =. x β 1/β 7
Whe x > β 1/β, the desired iequality follows from 7. Notice that the fuctio hx = g x is covex ad icreasig i x [0,. Therefor g x/x is icreasig i x, ad g λ //λ / is decreasig i. Thus { λ B, x = if exp λx + g λ 0 1/ is icreasig i. Sice g λ 1/ λ /,, we obtai lim B, x = sup B, x = if λ 0 exp { λx + λ = exp { x. This completes the proof of Theorem.1. roof of Corollary.1. Sice coshx exp{x /, we have λ g 1/β for all λ > 0. Thus, from 3, for all x > 0, 1k V β x λ 1 if exp λ>0 if exp λ>0 = exp β 4 { λ λx + g { λx + λ { x β 1 1 β 1/β, 5 which gives the desired iequality 10. Ackowledgemets The author would like to thak the aoymous referee for valuable commets ad remarks. The work has bee partially supported by the Natioal Natural Sciece Foudatio of Chia Grats o. 11601375 ad o. 116650. Refereces [1 B. Efro, Studet s t-test uder symmetry coditios, J. Amer. Statist. Assoc. 64 381969 178-130. [ X. Fa, I. Grama, Q. Liu, Hoeffdig s iequality for supermartigales, Stochastic rocess. Appl. 1 01 3545 3559. [3 X. Fa, I. Grama, Q. Liu, Expoetial iequalities for martigales with applicatios, Electro. J. robab. 0, o. 1 015 1. [4 I. Grama, E. Haeusler, Large deviatios for martigales via Cramér s method, Stochastic rocess. Appl. 85 000 79 93. [5. Hitczeko, Upper bouds for the L p -orms of martigales, robab. Theory Related Fields 86 1990 5 38. 8
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