Runs in superpositions of renewal processes with applications to discrimination

Wald and Wolfowitz [Ann. Math. Statist. 11 (1940) 147–162] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let (Sn)n⩾1(Sn)n⩾1 and (Tn)n⩾1(Tn)n⩾1 be the arrival processes for channel 1 and channel 2, respectively, and (Wn)n⩾1(Wn)n⩾1 their be superposition with counting process N(t)=defsup{n⩾1:Wn⩽t}. Let further Rn* be the number of runs in W1,…,WnW1,…,Wn and Rt=RN(t)* the number of runs observed up to time t  . We study the asymptotic behavior of Rn* and RtRt, first for the case where (Sn)n⩾1(Sn)n⩾1 and (Tn)n⩾1(Tn)n⩾1 have exponentially distributed increments with parameters λ1λ1 and λ2λ2, and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level αα tests for testing λ1=λ2λ1=λ2 against λ1≠λ2λ1≠λ2 in the first case, and for testing for equal scale parameters in the second.