Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let S T,i be the stopped sum denoting the number of appearances of outcome 'i' in X 1, …, X T , 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, S T,0, S T,1, …, S T,m ). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

Recent advances in agricultural biotechnology have produced many new crop varieties with valuable transgenic traits. These varieties are being, and will continue to be, marketed alongside conventional non-transgenic varieties. As a result, seed purity in commercial seed lots is of particular importance to both seed consumers and seed producers. A key step in the seed production process is the design of sampling and testing procedures used to evaluate seed lot purity. However, due to uncertainties in such methods, there is always a risk of incorrectly rejecting or accepting a seed lot. This paper discusses factors that should be considered when designing and implementing seed purity testing procedures to manage this misclassification risk – especially with regard to the presence or absence of transgenic traits. Many sources of uncertainty in both seed lot sampling and in the assay methods are described, and recommendations for reducing their impact are provided. This paper also explains the statistical concepts of misclassification risk as it affects seed producers and seed consumers. Sampling plans and formulas for determining the sample sizes necessary to control these misclassification errors when accepting or rejecting seed lots are also provided. Both simple, single-stage testing plans and the, often more efficient, double-stage testing plans are described. Testing seed pools rather than individual seeds is introduced as another way of adding efficiency to the testing process. Formulas are given for determining, from a seed sample, the confidence limits for the actual purity level of a commercial lot.

A sampling system, MIL-STD-105D, used in quality control consists of three sampling plans with different acceptance probabilities used in turn for lot inspection. The decision to switch plan is based on the history of the lot acceptance records and a set of stopping rules. We derive the performance measure, average outgoing quality (AOQ), of this sampling system from a renewal process in which AOQ is expressed in terms of the moments of the stopping times. The renewal approach is simpler than that of the Markov chain generally used in computing AOQ for an infinite sequence of lots; it also provides a formula for AOQ for a finite sequence of lots.