Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T18:07:02.440Z Has data issue: false hasContentIssue false
  • English
  • Français

The VPRT: A Sequential Testing Procedure Dominating the SPRT

Published online by Cambridge University Press:  11 February 2009

Noel Cressie  and
Peter B. Morgan
Noel Cressie
Affiliation:
Iowa State University
Peter B. Morgan
Affiliation:
SUNY at Buffalo
Get access Rights & Permissions [Opens in a new window]

Abstract

Under more general assumptions than those usually made in the sequential analysis literature, a variable-sample-size-sequential probability ratio test (VPRT) of two simple hypotheses is found that maximizes the expected net gain over all sequential decision procedures. In contrast, Wald and Wolfowitz [25] developed the sequential probability ratio test (SPRT) to minimize expected sample size, but their assumptions on the parameters of the decision problem were restrictive. In this article we show that the expected net-gain-maximizing VPRT also minimizes the expected (with respect to both data and prior) total sampling cost and that, under slightly more general conditions than those imposed by Wald and Wolfowitz, it reduces to the one-observation-at-a-time sequential probability ratio test (SPRT). The ways in which the size and power of the VPRT depend upon the parameters of the decision problem are also examined.

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 3 , June 1993 , pp. 431 - 450
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Arrow, K.J., Blackwell, D. & Girshick, M.A.. Bayes and minimax solutions of sequential decision problems. Econometrka 17 (1949): 213–244.CrossRefGoogle Scholar
2.Bellman, R.E.Dynamic Programming. Princeton, NJ: Princeton University Press, 1957.Google Scholar
3.Bellman, R.E. & Dreyfus, S.E.. Applied Dynamic Programming. Princeton, NJ: Princeton University Press, 1962.CrossRefGoogle ScholarPubMed
4.Berry, D.A. & Ho, C.H.. One-sided sequential stopping boundaries for clinical trials: A decision-theoretic approach. Biometrics 44 (1988): 219–227.CrossRefGoogle ScholarPubMed
5.Bhat, B.R. & Vaman, H. J.. Discount optimality of Wald SPRT for iid and certain Markov dependent sequences. Journal of the Indian Statistical Association 22 (1984): 13–21.Google Scholar
6.Cressie, N. & Morgan, P.B.. The VPRT: Optimal sequential and nonsequential testing. In Gupta, S.S. and Berger, J.O. (eds.), Statistical Decision Theory and Related Topics, IV, vol. 2, pp. 107–118. New York: Springer, 1988.Google Scholar
7.Cressie, N. & Morgan, P.B.. Design considerations for Neyman-Pearson and Wald hypothesis testing. Metrika 36 (1989): 317–325.Google Scholar
8.Easley, D. & Kiefer, N.M.. Controlling a stochastic process with unknown parameters. Econometrka 56 (1986): 1045–1064.CrossRefGoogle Scholar
9.Ehrenfeld, S.On group sequential sampling. Technometrics 14 (1972): 167–174.CrossRefGoogle Scholar
10.Ferguson, T.S.Mathematical Statistics: A Decision Theoretic Approach. New York: Academic Press, 1967.Google Scholar
11.Ghosh, B.K.Sequential Tests of Statistical Hypotheses. Reading, MA: Addison-Wesley, 1970.Google Scholar
12.Gupta, S.S. & Miescke, K.J.. Sequential selection procedures—A decision theoretic approach. The Annals of Statistics 12 (1984): 336–350.Google Scholar
13.Irle, A.Extended optimality of sequential probability ratio tests. The Annals of Statistics 12 (1984): 380–386.CrossRefGoogle Scholar
14.Kiefer, N.M. & Nyarko, Y.. Control of a linear regression process with unknown parameters. In Barnett, W.A., Berndt, E.R. & White, H. (eds.), Dynamic Economic Modeling: Proceedings of the Third Econometric Symposium in Economic Theory and Econometrics, pp. 105–120. Cambridge: Cambridge University Press, 1988.Google Scholar
15.Kiefer, N.M. & Nyarko, Y.. Optimal control of an unknown linear process with learning. International Economic Review 30 (1989): 571–586.CrossRefGoogle Scholar
16.Morgan, P.B. & Manning, R.. Optimal search. Econometrica 53 (1985): 923–944.CrossRefGoogle Scholar
17.Neyman, J. & Pearson, E.S.. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society, A 231 (1933): 140–185.Google Scholar
18.Olson, L.J.The search for a safe environment: The economics of screening and regulating environmental hazards. Journal of Environmental Economics and Management 19 (1990): 1–18.CrossRefGoogle Scholar
19.Pocock, S.J.Group sequential methods in the design and analysis of clinical trials. Biometrika 64 (1977): 191–199.CrossRefGoogle Scholar
20.Raiffa, H. & Schlaifer, R.. Applied Statistical Decision Theory. Cambridge: The M.I.T. Press, 1961.Google Scholar
21.Roberts, K. & Weitzman, M.. Funding criteria for research, development, and exploration projects. Econometrica 49 (1981): 1261–1288.CrossRefGoogle Scholar
22.Schmitz, N.Wald-Wolfowitz optimality of sequentially planned tests: Remarks and conjectures. Annals of Operations Research 32 (1991): 205–213.CrossRefGoogle Scholar
23.Spahn, M. & Ehrenfeld, S.. Optimal and suboptimal procedures in group sequential sampling. Naval Research Logistics Quarterly 21 (1974): 53–68.CrossRefGoogle Scholar
24.Wald, A.Sequential Analysis. New York: Wiley, 1947.Google Scholar
25.Wald, A. & Wolfowitz, J.. Optimum character of the sequential probability ratio test. The Annals of Mathematical Statistics 19 (1948): 326–339.Google Scholar
26.Whittle, P.Some general results in sequential design (with discussion). Journal of the Royal Statistical Society, B 27 (1965): 371–394.Google Scholar