The Annals of Mathematical Statistics

Some Approximations to the Binomial Distribution Function

R. R. Bahadur

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Abstract

Let $p$ be given, $0 < p < 1$. Let $n$ and $k$ be positive integers such that $np \leqq k \leqq n$, and let $B_n(k) = \sum^n_{r=k} \binom{n}{r} p^rq^{n-r}$, where $q = 1 - p$. It is shown that $B_n(k) = \big\lbrack\binom{n}{k} p^kq^{n - k}\big\rbrack qF(n + 1, 1; k + 1; p),$ 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 $A_n(k) = \big\lbrack\binom{n}{k}p^kq^{n - k + 1}\big\rbrack \lbrack(k + 1)/(k + 1 - (n + 1)p)\rbrack$, we have $1 \leqq A_n(k)/B_n(k) \leqq 1 + x^{-2}$, where $x = (k - np)/(npq)^{\frac{1}{2}}$. Next, let $N_n(k)$ denote the normal approximation to $B_n(k)$, and let $C_n(k) = (x + \sqrt{q/np}) \sqrt{2\pi} \exp \lbrack x^2/2 \rbrack$. It is shown that $(A_nN_nC_n)/B_n \rightarrow 1$ as $n \rightarrow \infty$, provided only that $k$ varies with $n$ so that $x \geqq 0$ for each $n$. It follows hence that $A_n/B_n \rightarrow 1$ if and only if $x \rightarrow \infty$ (i.e. $B_n \rightarrow 0$). It also follows that $N_nN_n \rightarrow 1$ if and only if $A_nC_n \rightarrow 1$. This last condition reduces to $x = o(n^{1/6})$ for certain values of $p$, but is weaker for other values; in particular, there are values of $p$ for which $N_n/B_n$ can tend to one without even the requirement that $k/n$ tend to $p$.

Article information

Source
Ann. Math. Statist., Volume 31, Number 1 (1960), 43-54.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177705986

Digital Object Identifier
doi:10.1214/aoms/1177705986

Mathematical Reviews number (MathSciNet)
MR120675

Zentralblatt MATH identifier
0092.35203

JSTOR
links.jstor.org

Citation

Bahadur, R. R. Some Approximations to the Binomial Distribution Function. Ann. Math. Statist. 31 (1960), no. 1, 43--54. doi:10.1214/aoms/1177705986. https://projecteuclid.org/euclid.aoms/1177705986


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