Bayesian Sequential Composite Hypothesis Testing in Discrete Time
ESAIM: Probability and Statistics, Tome 26 (2022), pp. 265-282

We study the sequential testing problem of two alternative hypotheses regarding an unknown parameter in an exponential family when observations are costly. In a Bayesian setting, the problem can be embedded in a Markovian framework. Using the conditional probability of one of the hypotheses as the underlying spatial variable, we show that the cost function is concave and that the posterior distribution becomes more concentrated as time goes on. Moreover, we study time monotonicity of the value function. For a large class of model specifications, the cost function is non-decreasing in time, and the optimal stopping boundaries are thus monotone.

DOI : 10.1051/ps/2022005
Classification : 62L10, 60G40, 62C10
Keywords: Sequential analysis, hypothesis testing, exponential family, optimal stopping 
@article{PS_2022__26_1_265_0,
     author = {Ekstr\"om, Erik and Wang, Yuqiong},
     title = {Bayesian {Sequential} {Composite} {Hypothesis} {Testing} in {Discrete} {Time}},
     journal = {ESAIM: Probability and Statistics},
     pages = {265--282},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {26},
     doi = {10.1051/ps/2022005},
     mrnumber = {4425000},
     zbl = {1493.62482},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2022005/}
}
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Ekström, Erik; Wang, Yuqiong. Bayesian Sequential Composite Hypothesis Testing in Discrete Time. ESAIM: Probability and Statistics, Tome 26 (2022), pp. 265-282. doi: 10.1051/ps/2022005

[1] M. Alvo, Bayesian sequential estimation. Ann. Statist. 5 (1977) 955–968. | MR | Zbl | DOI

[2] J.A. Bather, Bayes procedures for deciding the sign of a normal mean. Proc. Cambridge Philos. Soc. 58 (1962) 599–620. | MR | Zbl | DOI

[3] P.J. Bickel, On the asymptotic shape of Bayesian sequential tests of 0 ≤ 0 versus 0 > 0 for exponential families. Ann. Statist. 1 (1973) 231–240. | MR | Zbl | DOI

[4] L. Brown, Sufficient statistics in the case of independent random variables. Ann. Math. Statist. 35 (1964) 1456–1474. | MR | Zbl | DOI

[5] L. Brown, Fundamentals of statistical exponential families with applications in statistical decision theory. Institute of Mathematical Statistics, Hayward, CA (1986). | MR | Zbl | DOI

[6] P. Cabilio, Sequential estimation in Bernoulli trials. Ann. Statist. 5 (1977) 342–356. | MR | Zbl | DOI

[7] H. Chernoff, Sequential tests for the mean of a normal distribution. IV. (Discrete case). Ann. Math. Statist. 36 (1965) 55–68. | MR | Zbl | DOI

[8] S. Dayanik and S. Sezer, Multisource Bayesian sequential binary hypothesis testing problem. Ann. Oper. Res. 201 (2012) 99–130. | MR | Zbl | DOI

[9] E. Ekström, I. Karatzas and J. Vaicenavicius, Bayesian sequential least-squares estimation for the drift of a Wiener process. Stoch. Process. Appl. 145 (2022) 335–352. | MR | Zbl | DOI

[10] E. Ekström and J. Vaicenavicius, Bayesian sequential testing of the drift of a Brownian motion. ESAIM: PS. 19 (2015) 626–648. | MR | Zbl | Numdam | DOI

[11] E. Ekstrom and Y. Wang, Multi-dimensional sequential testing and detection. To appear Stochastics (2022). | MR | Zbl | DOI

[12] P.V. Gapeev and G. Peskir, The Wiener sequential testing problem with finite horizon. Stoch. Stoch. Rep. 76 (2004) 59–75. | MR | Zbl | DOI

[13] C. Hipp, Sufficient statistics and exponential families. Ann. Statist. 2 (1974) 1283–1292. | MR | Zbl | DOI

[14] D. Hobson, Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8 (1998) 193–205. | MR | Zbl | DOI

[15] S. Janson and J. Tysk, Volatility time and properties of option prices. Ann. Appl. Probab. 13 (2003) 890–913. | MR | Zbl | DOI

[16] T.L. Lai, Nearly optimal sequential tests of composite hypotheses. Ann. Statist. 16 (1988) 856–886. | MR | Zbl

[17] D. Lindley and B. Barnett, Sequential sampling: Two decision problems with linear losses for binomial and normal random variables. Biometrika 52 (1965) 507–532. | MR | Zbl | DOI

[18] S. Moriguti and H. Robbins, A Bayes test of “ p ≤ 1 / 2 ” v e r s u s “ p > 1 / 2 ”. Rep. Statist. Appl. Res. Un. Jpn. Sci. Eng. 9 (1962) 39–60. | MR | Zbl

[19] G. Peskir and A. Shiryaev, Sequential testing problems for Poisson processes. Ann. Statist. 28 (2000) 837–859. | MR | Zbl | DOI

[20] G. Schwarz, Asymptotic shapes of Bayes sequential testing regions. Ann. Math. Statist. 33 (1962) 224–236. | MR | Zbl | DOI

[21] A. Shiryayev, Optimal stopping rules. Applications of Mathematics, Vol. 8. Springer-Verlag, New York-Heidelberg (1978). | MR | Zbl

[22] A. Shiryaev, Two problems of sequential analysis. Cybernetics 3 (1967) 63–69 (1969). | MR | DOI

[23] M. Sobel, An essentially complete class of decision functions for certain standard sequential problems. Ann. Math. Stat. 24 (1953) 319–337. | MR | Zbl | DOI

[24] A. Wald and J. Wolfowitz, Bayes solutions of sequential decision problems. Ann. Math. Stat. 21 (1950) 82–99. | MR | Zbl | DOI

[25] M. Zhitlukhin and A.N. Shiryaev, A Bayesian sequential testing problem of three hypotheses for Brownian motion. Stat. Risk Model. 28 (2011) 227–249. | MR | Zbl | DOI

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The first author gratefully acknowledges support from Vetenskapsradet (The Swedish Research Council).