Bounds on Tail Probabilities in Exponential families

  title={Bounds on Tail Probabilities in Exponential families},
  author={P. Harremoes},
In this paper we present various new inequalities for tail proabilities for distributions that are elements of the most improtant exponential families. These families include the Poisson distributions, the Gamma distributions, the binomial distributions, the negative binomial distributions and the inverse Gaussian distributions. All these exponential families have simple variance functions and the variance functions play an important role in the exposition. All the inequalities presented in… Expand

Figures and Tables from this paper

Tail bounds for empirically standardized sums
Exponential tail bounds for sums play an important role in statistics, but the example of the t-statistic shows that the exponential tail decay may be lost when population parameters need to beExpand
Almost Optimal Anytime Algorithm for Batched Multi-Armed Bandits
An anytime algorithm is proposed that achieves the asymptotically optimal regret for exponential families of reward distributions with O(log log T · ilog(T ))1 batches, where α ∈ OT (1). Expand
Thompson Sampling Algorithms for Mean-Variance Bandits
Thompson Sampling-style algorithms for mean-variance MAB and comprehensive regret analyses for Gaussian and Bernoulli bandits with fewer assumptions are developed and shown to significantly outperform existing LCB-based algorithms for all risk tolerances. Expand


A Complete Proof of Universal Inequalities for the Distribution Function of the Binomial Law
We present a new form and a short complete proof of explicit two-sided estimates for the distribution function $F_{n,p}(k)$ of the binomial law with parameters $n,p$ from [D. Alfers and H. Dinges, Z.Expand
Natural Real Exponential Families with Cubic Variance Functions
Pursuing the classification initiated by Morris (1982), we describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree lessExpand
Information divergence is more χ2-distributed than the χ2-statistics
  • P. Harremoës, G. Tusnády
  • Mathematics, Computer Science
  • 2012 IEEE International Symposium on Information Theory Proceedings
  • 2012
For random variables the authors introduce a new transformation that transform several important distributions into new random variables that are almost Gaussian and formulate a general conjecture about how close their transform are to the Gaussian. Expand
A sharpening of Tusnády’s inequality
Let ε1, . . . , εm be i.i.d. random variables with P (εi = 1) = P (εi = −1) = 1/2, and Xm = ∑m i=1 εi. Let Ym be a normal random variable with the same first two moments as that of Xm. There is aExpand
Some Approximations to the Binomial Distribution Function
Let p be given, 0 n (k) = ∑ n r=k ( n r )p r q n-r , where q = 1 - p. It is shown that B n (k) = [( n k ) p k ,q -k ] qF(n + 1, 1; k + 1; p), where F is the hypergeometric function. ThisExpand
Limiting distribution of the G statistics
The G statistic and its local version have been used extensively in spatial data analysis. The paper proves the asymptotic normality of the G statistic. Theorems in this paper imply that the regularExpand
Saddlepoint Approximations in Statistics
Gaussian approximation of large deviation probabilities, unpublished
  • 2012
Natural Exponential Families with Quadratic Variance Functions