Showing papers by "T. W. Anderson published in 1967"

Some inequalities among binomial and Poisson probabilities

Abstract: k (1.4) P(k; X) = E p(j; X), k = 0, 1, i-o for X = np. In this paper it is shown that the error of approximation of the binomial cumulative distribution function P(k; np) B(k; n, p) is positive if k < np np/(n + 1) and is negative if np < k. In fact, B(k; n, X/n) is monotonically increasing for all n (2 X) if k < X 1 and for all n > k/(X k) if X 1 < k < X, and is monotonically decreasing for all n (2 k) if X < k. Thus

Approximating the Upper Binomial Confidence Limit

Abstract: When a sample of size n with c “defectives” is drawn randomly from an infinite population with probability p of a defective, the upper binomial confidence limit can be approximated by two formulas based on , the upper confidence limit for the parameter m of a Poisson distribution. The approximations are easy to calculate and have precisely specified bounds on their error over a stated range of n and c/n. The error of the simpler formula is guaranteed to be within .1% (with trivial deviations) of the exact binomial confidence limit when n≥ 40 and c/n ≤ 1/10. The relative error of the second formula is not more than .1% (with no deviations) when n ≥ 20 and . Values of for small values of c are readily available from tables of the Poisson and chisquare distributions. For c ≥ 50, a simple formula permits approximating with no more than .045% relative error.