Multivariate sequential analysis with linear boundaries

Abstract

Let {Sn = (Xn, Wn)} n≥0 be a random walk with Xn ∈ R and Wn ∈ R m . Let τ = τa = inf{n : Xn > a}. The main results presented are two term asymptotic expansions for the joint distribution of Sτ and τ and the marginal distribution of h(Sτ /a, τ /a) in the limit a → ∞. These results are used to study the distribution of t-statistics in sequential experiments with sample size τ , and to remove bias from confidence intervals based on Anscombe's theorem.

Corrected confidence intervals for secondary parameters following sequential tests

“…where c N,α/2 is the 100(α/2)-th percentile of the t distribution with N degrees of freedom. Note that the form of the above interval is similar to one obtained by Keener [10] using fixed θ expansions. However, his interval is only valid up to order o(1/ √ a) and only applicable to linear stopping boundaries.…”

“…With this Z N 1 and its corresponding mean and variance corrections, we obtain a renormalised pivot Z (0) N as in (10). Then, by Theorem 4, an asymptotic level 1 − α confidence interval for θ 2 is…”