# Some Approximations to the Binomial Distribution Function

@article{Bahadur1960SomeAT, title={Some Approximations to the Binomial Distribution Function}, author={R. R. Bahadur}, journal={Annals of Mathematical Statistics}, year={1960}, volume={31}, pages={43-54} }

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. This representation seems useful for numerical and theoretical investigations of small tail probabilities. The representation yields, in particular, the result that, with A n (k) = [( n k )p k q n-k+I J [(k + 1)/(k + 1 - (n + l)p)], we have 1 ≤ A n (k)/B n (k) ≤ 1 + x -2 , where x = (k - np)/(npq) t . Next, let N n…

## 62 Citations

### Sharp asymptotics of large deviations in ℝd

- Mathematics
- 1995

AbstractGiven a sequence {X1}i=1,2,3,... of i.i.d. random variables taking values in ℝd,d≥2, letSn=Σi=1nXt=1. For Λ a Borel set in ℝd having smooth boundary, witha=infx∈ΛI(x) the minimal value of the…

### M ay 2 01 2 Some Refinements of Large Deviation Tail Probabilities

- Mathematics
- 2012

If μ is close to the mean ofX1 one would usually approximate Pn,μ by a tail probability of a Gaussian random variable. If μ is far from the mean of X1 the tail probability can be estimated using…

### Binomial Mixture Model With U-shape Constraint

- Mathematics
- 2021

In this article, we study the binomial mixture model under the regime that the binomial size m can be relatively large compared to the sample size n. This project is motivated by the GeneFishing…

### Some Numerical Comparisons of Several Approximations to the Binomial Distribution

- Mathematics
- 1969

Abstract Several approximations to the binomial distribution were compared in 1956 by Raff [9], including the normal, arcsine, normal Gram-Charlier, Camp-Paulson, Poisson, and Poisson Gram-Charlier…

### A Normal Approximation for Binomial, F, Beta, and other Common, Related Tail Probabilities, II

- Mathematics
- 1968

Abstract This paper concerns a new Normal approximation to the beta distribution and its relatives, in particular, the binomial, Pascal, negative binomial, F, t, Poisson, gamma, and chi square…

### On estimation with elementary symmetric polynomials

- Mathematics
- 1998

In the paper we discuss some applications of the theory of infinite order symmetric statistics in estimation of an analytic function of an unknown population mean. Namely, if Sn is a normalized…

### Toward a Usable Theory of Chernoff Bounds for Heterogeneous and Partially Dependent Random Variables

- Mathematics
- 1995

Let X be a sum of real valued random variables and have a bounded mean E[X]. The generic Chernoff-Hoeffding estimate for large deviations of X is: P{X-E[X]<=a}=0}exp(-y(a+E[X]))E[exp(y X)], which…

### An empirical $G$-Wishart prior for sparse high-dimensional Gaussian graphical models

- Computer Science
- 2019

An empirical version of the $G-Wishart prior for sparse precision matrices, where the prior mode is informed by the data in a suitable way, and a marginal posterior distribution for the same is obtained that takes the form of a ratio of two $G$-W Wishart normalizing constants.

### On the Equivalence of the Binomial and Inverse Binomial Acceptance Sampling Plans and an Acknowledgement

- Mathematics
- 1963

The author (1957, 1960) has shown that the negative (inverse) binomial distribution function can be evaluated easily through certain identities by using available binomial or incomplete beta-function…

### Moderate deviations of subgraph counts in the Erdős-Rényi random graphs 𝐺(𝑛,𝑚) and 𝐺(𝑛,𝑝)

- MathematicsTransactions of the American Mathematical Society
- 2020

The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erdős-Rényi random graph
G
(
n
,
m
)
G(n,m)…