Sequential Experimentation in Clinical Trials by Jay Bartroff Tze Leung Lai & Mei-Chiung Shih

Author:Jay Bartroff, Tze Leung Lai & Mei-Chiung Shih
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(5.25)

Lai [1995] showed that if and that if while

Hence, choosing b close to 1 (say b = 1. 1), there is little loss of efficiency in reducing the computational complexity of N G by its window-limited modification (5.25).

By using the likelihood ratio identity (3.3) and a change of measures, Lai [1998] proved an asymptotic lower-bound result for the detection delay subject to the ARL constraint E 0(T) ≥ γ, similar to Hoeffding’s lower bound for sequential testing, thereby proving the asymptotic optimality (5.17) of the CUSUM rule for nonindependent and multivariate observations, in which I(f 1, f 0) is defined as the limit (which exists by the strong law for ergodic sequences) of the average of the log-likelihood ratio statistics. Instead of the ARL constraint E 0(T) ≥ γ, he introduces a probability constraint of the form sup k ≥ 1 P 0{k ≤ T < k + m} ≤ m ∕ γ with m = o(γ) but m ∕ logγ → ∞. Then a similar change-of-measure argument gives an asymptotic lower bound for the detection delay of the form

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