期刊名称：International Journal of Statistics and Probability
印刷版ISSN：1927-7032
电子版ISSN：1927-7040
出版年度：2021
卷号：10
期号：5
页码：27-30
语种：English
出版社：Canadian Center of Science and Education
摘要：The purpose of this article is to improve Hoeffding’s lemma and consequently Hoeffding’s tail bounds. The improvement pertains to left skewed zero mean random variables X ∈ [a, b], where a b. The proof of Hoeffding’s improved lemma uses Taylor’s expansion, the convexity of exp(sx), s ∈ R, and an unnoticed observation since Hoeffding’s publication in 1963 that for −a > b the maximum of the intermediate function τ(1 − τ) appearing in Hoeffding’s proof is attained at an endpoint rather than at τ = 0.5 as in the case b > −a. Using Hoeffding’s improved lemma we obtain one sided and two sided tail bounds for P(S n ≥ t) and P( S n ≥ t), respectively, where S n = Pn i=1 Xi and the Xi ∈ [ai , bi], i = 1, ..., n are independent zero mean random variables (not necessarily identically distributed). It is interesting to note that we could also improve Hoeffding’s two sided bound for all {Xi : −ai , bi , i = 1, ..., n}. This is so because here the one sided bound should be increased by P(−S n ≥ t), wherein the left skewed intervals become right skewed and vice versa.
