[Paper Review] A full proof of universal inequalities for the distribution function of the binomial law

A. M. Zubkov, Alexander Serov|arXiv (Cornell University)|Jul 16, 2012

Statistical Distribution Estimation and Applications4 references4 citations

TL;DR

This paper presents a concise, full proof of universal two-sided inequalities for the binomial distribution function $F_{n,p}(x)$, establishing tight bounds using the cumulative distribution function of the standard normal law. The key contribution is a new formulation of Alfers and Dinges' inequalities that are exact (upper bound for $F_{n,p}(k)$ equals lower bound for $F_{n,p}(k+1)$) and valid for all $n$, $p$, and $k$, enabling precise quantile estimation via adjacent integers.

ABSTRACT

We present a new form and a short full proof of explicit two-sided estimates for the distribution function F_{n,p}(x) of the binomial law from the paper published by D.Alfers and H.Dinges in 1984. These inequalities are universal (valid for all binomial distribution and all values of argument) and exact (namely, the upper bound for F_{n,p}(k) is the lower bound for F_{n,p}(k+1)). By means of such estimates it is possible to bound any quantile of the binomial law by 2 subsequent integers.

Motivation & Objective

• To provide a complete, accessible proof of universal two-sided inequalities for the binomial distribution function, which were previously known but difficult to verify due to lengthy and incomplete proofs.
• To resolve the practical challenge of estimating binomial tail probabilities, especially in large deviation regimes where standard approximations fail due to superexponential decay.
• To establish bounds that are exact in the sense that the upper bound for $F_{n,p}(k)$ coincides with the lower bound for $F_{n,p}(k+1)$, ensuring tight quantile estimation.
• To demonstrate that these bounds can be used to tightly bracket any quantile of the binomial law within two consecutive integers, offering a practical solution to the large deviation problem.

Proposed method

• The method uses an integral representation of the binomial cumulative distribution function via differentiation under the integral sign with respect to $p$, transforming the sum into an integral involving beta-type functions.
• It applies Stirling's approximation to the binomial coefficient, expressing the normalization factor in terms of $S_n^{k+1}$, a correction term involving logarithmic gamma function expansions.
• The proof introduces a function $B(z) = \alpha \ln(\alpha/z) + (1-\alpha)\ln((1-\alpha)/(1-z))$ with $\alpha = (k+1)/n$, which governs the exponential decay of the integrand.
• The integrand is re-expressed in terms of the standard normal density $\varphi$ and cumulative function $\Phi$, using a transformation $a(z)$ that links the binomial tail to the normal distribution.
• The proof establishes monotonicity and sign properties of the difference $\delta(p) = \mathbf{P}\{X_{n,p} \leq k\} - \Phi(a(p)\sqrt{n})$, showing $\delta(p) < 0$ for all $p \in (0,1)$, which yields the upper bound.
• By symmetry and duality using $1-p$, the lower bound is derived from the upper bound of the complementary distribution, ensuring tightness across the entire support.

Experimental results

Research questions

• RQ1 Can a concise and complete proof be given for the universal two-sided inequalities of Alfers and Dinges for the binomial distribution function?
• RQ2 How can the exactness of the bounds—where the upper bound for $F_{n,p}(k)$ matches the lower bound for $F_{n,p}(k+1)$—be rigorously established?
• RQ3 To what extent do these bounds remain effective in the large deviation regime, where normal approximations fail due to superexponential tail decay?
• RQ4 Can the bounds be sharpened analytically or numerically using the integral representation of the error term $\delta(p)$?
• RQ5 How can the resulting inequalities be used to bound any quantile of the binomial law within two consecutive integers?

Key findings

• The inequalities $C_{n,p}(k) \leq \mathbf{P}\{X_{n,p} \leq k\} \leq C_{n,p}(k+1)$ hold for all $n$, $p \in (0,1)$, and $k = 0,1,\dots,n-1$, with equality only at $k=0$ or $k=n-1$.
• The bounds are exact: $C_{n,p}(k)$ is the lower bound for $F_{n,p}(k)$ and $C_{n,p}(k+1)$ is the upper bound for $F_{n,p}(k)$, ensuring tightness at the transition between consecutive integers.
• The difference between the true probability and the lower bound is strictly less than the local probability $\mathbf{P}\{X_{n,p} = k\}$, confirming the bounds' precision.
• The upper bound is derived by showing $\delta(p) < 0$ for all $p \in (0,1)$, which implies $\mathbf{P}\{X_{n,p} \leq k\} < \Phi(a(p)\sqrt{n}) = C_{n,p}(k+1)$.
• The lower bound is obtained via duality: $\mathbf{P}\{X_{n,p} \leq k\} = 1 - \mathbf{P}\{X_{n,1-p} \leq n-k-1\}$, and applying the upper bound to the complementary distribution.
• The error term $\delta(p)$ is analytically bounded using monotonicity of the integrand components, allowing for numerical or analytic sharpening of the bounds in specific parameter regimes.