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

Abstract

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 tables. The identities involved have been established in various forms elsewhere by several authorst. The sum of the first few terms of the negative binomial expansion has been obtained by Pearson and Fieller (1933) as an incomplete beta-function by using Mclaurin Series. Some later references are Katz (1946) and Lieblein (1949). Somewhat related work is that of Finney (1947, 1949) where he discusses the relation between the binomial and the negative binomial in that tables giving confidence limits on p for the binomial can be adopted to the negative binomial. Bahadur$ (1960) obtains the identity between binomial and negative binomial distribution functions using probability argument. Rider (1962) has obtained the identity between the negative binomial and the incomplete-beta-function using the fact that the antiderivative of a derivative gives the original function up to a constant. As an application, the author (1960) showed the connection between the binomial and inverse binomial acceptance sampling plans as reflected in their parameters and suggested that tables and charts available to obtain binomial single-sample plans (BS) can be used to obtain inverse binomial single-sample plans (IBS) for given specifications. It is the purpose of this communication to show that the two types of acceptance sampling plans are equivalent and that when modified slightly they are in fact identical.