On Littlewood's Estimate for the Binomial Distribution

We correct a theorem of J. E. Littlewood which gives an approximation for the tail of the binomial distribution. We also present several new approximations which are less accurate but have wider scope. One of them gives an estimate with relative error uniformly 0(11 a) over all values of all the parameters, where a is the standard deviation. For some types of probability calculations, the familiar DeMoivre-Laplace approximation to the binomial distribution (see Feller [2] for example), is insufficiently accurate. The most serious attack on this problem seems to have been that of Littlewood [3]. Unfortunately, Littlewood's Theorem 2 contains two typographic errors as well as an incorrect sign which can be traced to a clerical error in the proof. The main purpose of this note is to state Littlewood's theorem correctly. We also take the opportunity to give some other approximations which may be more convenient, though lesser in accuracy and scope in some cases. For the normal distribution, define ljJ(x) = e /1Viir, Q(x) = f: ljJ(u) du, and Y(x) = Q(x)1 ljJ(x). For the binomial distribution, define b(k) = b(k; n; p) = (:)pkqn-k and n B(k) = B(k; n, p) = 2: b(j; n, p), j=k where q = 1 p. The mean of this distribution is f.l = pn and the variance is a = npq. We begin with Littlewood's Theorem 2. With the errors corrected, we can state that theorem as follows. Theorem 1. Let p, 0 < p < 1, be fixed. Let t = t(n) be such that pn + t is an integer and o~ t ~ ~qn. Define x = t1a and p = q tin. Then B(pn + t; n, p) = Q(x) exp (AI +A 2 / v'p (1 p)n + O(nI)), where t 2 I ( t ) I ( t ) Al = -(pn + t 2) log 1 + (qn t + 2) log 1-2pqn pn qn and A 2 = ~(1 2p)«1 x 2)IY(x) + x 3 ) + ~(lIY(x) x). Proof. Littlewood's statement of this theorem has three errors: (i) The coefficient ~ should be ~ (it comes from c3 ) . (ii) The definition of p should be as in (his) Theorem 1. (iii) The sign of the O(n -112) term is wrong. Received 19 September 1988; revision received 13 December 1988. * Postal address: Computer Science Department, Australian National University, GPO Box 4, ACT 2601, Australia. 475 terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0001867800018668 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 20 Aug 2017 at 01:40:48, subject to the Cambridge Core 476 Letters to the editor where Errors (i) and (ii) are merely typos. Error (iii) can be traced to an incorrect sign change in passing from (22.5) to (22.6). Apart from this, Littlewood's proof appears valid. To guard against other gross errors, we have succesfully checked the theorem numerically for a wide range of values of the parameters. Since Y(x) has an asymptotic expansion of the form l/x 1/x + 0(1/x 5 ) , the coefficient A z is uniformly bounded over the range of validity of Theorem 1. In fact, 0 < A z < 0·532 and 0< (1 + x )A z < 1·084 for x ~ 0 and 0 ~ p ~ 1. Also, as Littlewood makes clear in his paper, the choice of ~qn as the upper limit for t is arbitrary; in fact the theorem and its proof are equally valid for 0 ~ t ~ aqn, where (X is any constant with (X < 1. Subject to this bound on t, the error term in Theorem 1 is uniform over nand t. However, it is not uniform over p or a. We note here that the slip leading to error (iii) also invalidates Littlewood's Theorem 1, but in that case we have not determined the correct form. Littlewood's method (but not his theorem) would doubtless work if p was not constant but instead decreased as some function of n. We have not attempted to modify his proof in this way, but have instead taken several alternative approaches. The first approach is suggested by Littlewood's theorem. The second and third terms of Al strongly suggest Stirling's approximation to log b(pn + t). Applying this approximation in reverse, and dropping A z, gives the estimate B(pn + t):::::: ab(pn + t)Y(x)(1 + t/(pn)). Amazingly, the error in this approximation turns out to be uniformly 0(1/ a) for all n, p and