Abstract
We propose a new approach to sequential testing which is an adaptive (on-line) extension of the (off-line) framework developed in [1]. It relies upon testing of pairs of hypotheses in the case where each hypothesis states that the vector of parameters underlying the distribution of observations belongs to a convex set. The nearly optimal under appropriate conditions test is yielded by a solution to an efficiently solvable convex optimization problem. The proposed methodology can be seen as a computationally friendly reformulation of the classical sequential testing.
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Goldenshluger, A., Juditski, A., and Nemirovski, A., Hypothesis Testing by Convex Optimization, arXiv preprint, arXiv:1311.6765, 2013.
Barnard, G.A., Sequential Tests in Industrial Statistics, Supplement J. Royal Statist. Soc., 1946, vol. 8, pp. 1–26.
Wald, A., Sequential Tests of Statistical Hypotheses, Ann. Math. Statist., 1945, vol. 16, no. 2, pp. 117–186.
Wald, A., and Wolfowitz, J., Optimum Character of the Sequential Probability Ratio Test, Ann. Math. Statist., 1948, vol. 19, no. 3, pp. 326–339.
Chernoff, H., Sequential Analysis and Optimal Design, SIAM, 1972, vol. 8.
Ghosh, B.K., A Brief History of Sequential Analysis, in Handbook of Sequential Analysis, New York: Marcel Dekker, 1991, pp. 1–19.
Bakeman, R., Observing Interaction: An Introduction to Sequential Analysis, Cambridge: Cambridge Univ. Press, 1997.
Lai, T.L., Sequential Analysis: Some Classical Problems and New Challenges, Statist. Sinica, 2001, vol. 11, no. 2, pp. 303–350.
Juditsky, A.B., and Nemirovski, A.S., Nonparametric Estimation by Convex Programming, Ann. Statist., 2009, vol. 37, no. 5a, pp. 2278–2300.
Burnashev, M., On the Minimax Detection of an Imperfectly Known Signal in a White Noise Background, Theory Probab. Appl., 1979, vol. 24, pp. 107–119.
Burnashev, M., Discrimination of Hypotheses for Gaussian Measures and a Geometric Characterization of the Gaussian Distribution, Math. Notes., 1982, vol. 32, pp. 757–761.
Ingster, Y., and Suslina, I.A., Nonparametric Goodness-of-fit Testing under Gaussian Models, Lecture Notes Statist., vol. 169, Berlin: Springer, 2002.
Le Cam, L., Convergence of Estimates under Dimensionality Restrictions, Ann. Statist., 1973, pp. 38–53.
Le Cam, L., On Local and Global Properties in the Theory of Asymptotic Normality of Experiments, Stochast. Processes Related Topics, 1975, vol. 1, pp. 13–54.
Birgé, L., Approximation dans les spaces métriques et théorie de l’estimation: Inégalités de Cràmer-Chernoff et théorie asymptotique des tests, PhD Dissertation, Université Paris VII, 1980.
Birgé, L., Sur un théor`eme de minimax et son application aux tests, Probab. Math. Stat., 1982, vol. 3, pp. 259–282.
Birgé, L., Robust Testing for Independent Non Identically Distributed Variables and Markov Chains, in Specif. Statist. Models, Berlin: Springer, 1983, pp. 134–162.
Le Cam, L., Asymptotic Methods in Statistical Decision Theory, Series in Statistics, Berlin: Springer, 1986.
Juditsky, A.B., and Nemirovski, A.S., On Sequential Hypotheses Testing via Convex Optimization, arXiv preprint, arXiv:1412.1605, 2014.
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Original Russian Text © A.B. Juditsky, A.S. Nemirovski, 2015, published in Avtomatika i Telemekhanika, 2015, No. 5, pp. 100–120.
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Juditsky, A.B., Nemirovski, A.S. On sequential hypotheses testing via convex optimization. Autom Remote Control 76, 809–825 (2015). https://doi.org/10.1134/S0005117915050070
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DOI: https://doi.org/10.1134/S0005117915050070