Optimum Design 2000 pp 67-78 | Cite as

# Second-Order Optimal Sequential Tests

## Abstract

An asymptotic lower bound is derived involving a second additive term of order \(\sqrt {\left| {\ln \alpha } \right|} \) as *α* → 0 for the mean length of a controlled sequential strategy *s* for discrimination between two statistical models in a very general nonparametric setting. The parameter a is the maximal error probability of *s*.

A sequential strategy is constructed attaining (or almost attaining) this asymptotic bound uniformly over the distributions of models including those from the indifference zone. These results are extended for a general loss function *g*(*N*) with the power growth of the strategy length *N*.

Applications of these results to change-point detection and testing homogeneity are outlined.

## Keywords

change-point detection controlled experiments general risk second-order optimality sequential test## Preview

Unable to display preview. Download preview PDF.

## References

- Barron, A.R. and Sheu, C.-H. (1991). Approximation of density functions by sequences of exponential families.
*Ann. Statist.***19**, 1347–1369.MathSciNetzbMATHCrossRefGoogle Scholar - Centsov, N.N. (1982).
*Statistical Decision Rules and Optimal Inference*. Amer. Math. Soc. Transl.**53**. Providence RI.Google Scholar - Chernoff, H. (1959). Sequential design of experiments.
*Ann. Math. Statist.***30**, 755–770.MathSciNetzbMATHCrossRefGoogle Scholar - Chernoff, H. (1997).
*Sequential Analysis and Optimal Design*. Philadelphia: SIAM.Google Scholar - Keener, R. (1984). Second order efficiency in the sequential design of experiments.
*Ann. Statist.***12**, 510–532.MathSciNetzbMATHCrossRefGoogle Scholar - Lai, T.L. (1995). Sequential change-point detection in quality control and dynamical systems.
*J. Roy. Statist. Soc*. B**57**, 613–658.zbMATHGoogle Scholar - Lalley, S.P. and Lorden, G. (1986). A control problem arising in the sequential design of experiments.
*Ann. Probab.***14**, 136–172.MathSciNetzbMATHCrossRefGoogle Scholar - Lorden, G. (1971). Procedures for reacting to a change in distribution.
*Ann. Math. Statist.***42**, 1897–1908.MathSciNetzbMATHCrossRefGoogle Scholar - Malyutov, M.B. (1983). Lower bounds for the mean length of sequentially designed experiments.
*Soviet Math. (Izv. VUZ.)***27**, 21–47.zbMATHGoogle Scholar - Malyutov, M.B. and Tsitovich, I.I. (1997a). Asymptotically optimal sequential testing of hypotheses. In
*Proc. Internat. Conf. on Distributed Computer Communication Networks: Theory and Applications*, pp. 134–141. Tel-Aviv.Google Scholar - Malyutov, M.B. and Tsitovich, I.I. (1997b). Sequential search for significant variables of an unknown function.
*Problems of Inform. Transmiss.***33**, 88–107.MathSciNetGoogle Scholar - Malyutov, M.B. and Tsitovich, I.I. (2000). Asymptotically optimal sequential testing of hypotheses.
*Problems of Inform. Transmiss*. (To appear).Google Scholar - Schwarz, G. (1962). Asymptotic shapes of Bayes sequential testing regions.
*Ann. Math. Statist.***33**, 224–236.MathSciNetzbMATHCrossRefGoogle Scholar - Tsitovich, I.I. (1984). On sequential design of experiments for hypothesis testing.
*Theory Probab. and Appl.***29**, 778–781.MathSciNetzbMATHGoogle Scholar - Tsitovich, I.I. (1990). Sequential design and discrimination. In
*Models and Methods of Information Systems*, pp. 36–48. Moscow: Nauka (in Russian).Google Scholar - Tsitovich, I.I. (1993).
*Sequential Discrimination*. D.Sci. thesis. Moscow (in Russian).Google Scholar