A Complete Proof of Universal Inequalities for the Distribution Function of the Binomial Law

Zubkov, A. M.; Serov, Alexander Aleksandrovich

doi:10.1137/s0040585x97986138

Cited by 44 publications

(35 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A proof of this conjecture was recently given by Serov and Zubkov [8]. Were we will prove that the intersection results for binomial distributions and Poisson distributions follows from more general results for waiting times.…”

Section: Familiesmentioning

confidence: 69%

Mutual information of contingency tables and related inequalities

Harremoës¹

2014

2014 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

Abstract-For testing independence it is very popular to use either the χ 2 -statistic or G 2 -statistics (mutual information). Asymptotically both are χ 2 -distributed so an obvious question is which of the two statistics that has a distribution that is closest to the χ 2 -distribution. Surprisingly the distribution of mutual information is much better approximated by a χ 2 -distribution than the χ 2 -statistic. For technical reasons we shall focus on the simplest case with one degree of freedom. We introduce the signed log-likelihood and demonstrate that its distribution function can be related to the distribution function of a standard Gaussian by inequalities. For the hypergeometric distribution we formulate a general conjecture about how close the signed loglikelihood is to a standard Gaussian, and this conjecture gives much more accurate estimates of the tail probabilities of this type of distribution than previously published results. The conjecture has been proved numerically in all cases relevant for testing independence and further evidence of its validity is given. I. CHOICE OF STATISTICWe consider the problem of testing independence in a discrete setting. Here we shall follow the classic approach to this problem as developed by Pearson, Neuman and Fisher. The question is whether a sample with observation counts (X ij ) has been generated by the distribution Q = (q ij ) where q ij = s i ·t j for some probability vectors (s i ) and (t j ) reflecting independence. We introduce the marginal counts R i = j X ij and S j = i X ij and the sample size N = ij X ij . The maximum likelihood estimates of probability vectors (s i ) and (t j ) are given by s

show abstract

Section: Familiesmentioning

confidence: 69%

Mutual information of contingency tables and related inequalities

Harremoës¹

2014

2014 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

show abstract

“…Although it was not explicitly denoted as such in [5], it is easy to see that ( , ) represents the Kullback-Leibler ( KL ) divergence between two Bernoulli variables with respective probabilities of success and ; hence, ( , ) represents the KL divergence of summed pairs of such variables. This observation allows the relatively straightforward extension of the above result to the case of a Poisson distributed random variable, which is given in Theorem 2.…”

Section: Sign( ) Is the Usual Signum Function With Argument φ( ) Ismentioning

confidence: 99%

“…Now, suppose we increase by one and reduce such that the constraint = still holds. Again, this variable still satisfies inequality (5). Now, incrementally repeat this procedure for increasing under the constraint that = indefinitely; in the limit as → ∞ one has the following:…”

Section: Sign( ) Is the Usual Signum Function With Argument φ( ) Ismentioning

confidence: 99%

“…Although quickly applied and potentially very accurate for large input parameters (due to the central limit theorem), the sign of the approximation errors is typically unknown. Exceptions are the inequality of Bohman (see [1, page 169]), which always overestimates the true Poisson probability, and the expressions recently proposed by Zubkov and Serov for the Binomial distribution [5]. Methods to obtain provable bounds with known error signs (typically one would require to underestimate (1) and (2), whilst overestimating (3) and (4) in most engineering and computer science applications) principally include the Bernstein/Chernoff/Hoeffding-type exponential probability inequalities and their close variants [1,2,6,7].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, the recent work on the categorization of the Binomial with respect to the Normal distribution in [5] is first extended to obtain universal inequalities (with known error signs) relating the Poisson and Normal distribution functions. Section 3 obtains asymptotically near-exact analytic inequalities relating both the Poisson and Binomial upper tail quantiles to the Normal quantiles.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions

Short

2013

ISRN Probability and Statistics

View full text Add to dashboard Cite

The exact evaluation of the Poisson and Binomial cumulative distribution and inverse (quantile) functions may be too challenging or unnecessary for some applications, and simpler solutions (typically obtained by applying Normal approximations or exponential inequalities) may be desired in some situations. Although Normal distribution approximations are easy to apply and potentially very accurate, error signs are typically unknown; error signs are typically known for exponential inequalities at the expense of some pessimism. In this paper, recent work describing universal inequalities relating the Normal and Binomial distribution functions is extended to cover the Poisson distribution function; new quantile function inequalities are then obtained for both distributions. Exponential bounds-which improve upon the Chernoff-Hoeffding inequalities by a factor of at least two-are also obtained for both distributions.

show abstract

Efficient random graph matching via degree profiles

Ding

et al. 2020

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

A Complete Proof of Universal Inequalities for the Distribution Function of the Binomial Law

Cited by 44 publications

References 4 publications

Mutual information of contingency tables and related inequalities

Mutual information of contingency tables and related inequalities

Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions

Efficient random graph matching via degree profiles

Contact Info

Product

Resources

About