Nonparametric change point tests for randomly stopped sequences

Abstract

Let X₁, X₂,...be a sequence of independent random variables. Assume that, for i = 1, 2, ⋯the random variable X_i has distribution function F_i. Let {N_n, n ≥ 1} be a sequence of non-negative integer-valued random variables which are not necessarily independent of the X's. Based on samples of the form N_n, X₁,...,X_Nn, we consider the problem of testing the null hypothesis of no change, H_ο : F_i = F,i = 1, 2, ⋯,N_n against the at most one change point alternative, H₁ : F_i = F, 1 ≤ i ≤ [λN_n] & F_i = G, i > [λN_n], where F ≠ G and 0 < λ < 1 are unknown.