The Annals of Statistics

Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families

Hock Peng Chan and Tze Leung Lai

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Abstract

Asymptotic approximations for the error probabilities of sequential tests of composite hypotheses in multiparameter exponential families are developed herein for a general class of test statistics, including generalized likelihood ratio statistics and other functions of the sufficient statistics. These results not only generalize previous approximations for Type I error probabilities of sequential generalized likelihood ratio tests, but also pro- vide a unified treatment of both sequential and fixed sample size tests and of Type I and Type II error probabilities. Geometric arguments involving integration over tubes play an important role in this unified theory.

Article information

Source
Ann. Statist., Volume 28, Number 6 (2000), 1638-1669.

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015957474

Digital Object Identifier
doi:10.1214/aos/1015957474

Mathematical Reviews number (MathSciNet)
MR1835035

Zentralblatt MATH identifier
1105.62367

Subjects
Primary: 62L10: Sequential analysis 62L15: Optimal stopping [See also 60G40, 91A60] 62E20: Asymptotic distribution theory
Secondary: 60F10: Large deviations 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]

Keywords
Sequential generalized likelihood ratio tests Bayes sequential tests multiparameter exponential families boundary crossing probabilities integration over tubes

Citation

Chan, Hock Peng; Lai, Tze Leung. Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families. Ann. Statist. 28 (2000), no. 6, 1638--1669. doi:10.1214/aos/1015957474. https://projecteuclid.org/euclid.aos/1015957474


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References

  • Bahadur, R. R. (1967). Rates of convergence of estimates and test statistics. Ann. Math. Statist. 38 303-324.
  • Barndorff-Nielsen, O. and Cox, D. R. (1979). Edgeworth and saddlepoint approximations with statistical applications (with discussion). J. Roy. Statist. Soc. Ser. B 41 279-312.
  • Borovkov, A. A. and Rogozin, B. A. (1965). On the multidimensional central limit theorem. Theory Probab. Appl. 10 55-62.
  • Chan, H. P. (1998). Boundary crossing theory in change-point detection and its applications. Ph.D. dissertation, Stanford Univ.
  • Chan, H. P. and Lai, T. L. (1999). Importance sampling for Monte Carlo evaluation of boundary crossing probabilities in hypothesis testing and changepoint detection. Technical report, Dept. Statistics Stanford Univ.
  • Chandra, T. K. (1985). Asymptotic expansion of perturbed chi-squared variables. Sanky¯a Ser. A 47 100-110.
  • Chandra, T. K. and Ghosh, J. K. (1979). Valid asymptotic expansions for the likelihood ratio statistic and the perturbed chi-square variables. Sanky¯a Ser. A 41 22-47.
  • Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493-507.
  • Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Groeneboom, P. (1980). Large Deviations and Asymptotic Efficiencies. Math. Centrum, Amsterdam.
  • Gray, A. (1990). Tubes. Addison-Wesley, Reading, MA.
  • Hirsch, M. (1976). Differential Topology. Springer, New York.
  • Hoeffding, W. (1965). Asymptotically optimal tests for multinomial distributions. Ann. Math. Statist. 36 369-405.
  • Hotelling, H. (1939). Tubes and spheres in n-spaces, and a class of statistical problems. Amer. J. Math. 61 440-460.
  • Hu, I. (1988). Repeated significance tests for exponential families. Ann. Statist. 16 1643-1666.
  • Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press, London.
  • Johansen, S. and Johnstone, I. (1990). Hotelling's theorem on the volume of tubes: Some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652-684.
  • Johnstone, I. and Siegmund, D. (1989). On Hotelling's formula for the volume of tubes and Naiman's inequality. Ann. Statist. 17 184-194.
  • Knowles, M. and Siegmund, D. (1989). On Hotelling's approach to testing for a nonlinear parameter in regression. Internat. Statist. Rev. 57 205-220. Lai, T. L. (1988a). Nearly optimal sequential tests of composite hypotheses. Ann. Statist. 16 856-886. Lai, T. L. (1988b). Boundary crossing problems for sample means. Ann. Probab. 16 375-396.
  • Lai, T. L. (1997). On optimal stopping problems in sequential hypothesis testing. Statist. Sinica 7 33-51.
  • Lai, T. L. and Siegmund, D. (1977). A nonlinear renewal theory with applications to sequential analysis I. Ann. Statist. 5 946-954. Lai, T. L. and Zhang, L. M. (1994a). Nearly optimal generalized sequential likelihood ratio tests in multivariate exponential families. In Multivariate Analysis and Its Applications (T. W. Anderson, K. T. Fang and I. Olkin, eds.). 331-346. IMS, Hayward, CA. Lai, T. L. and Zhang, L. M. (1994b). A modification of Schwarz's sequential likelihood ratio tests in multivariate sequential analysis. Sequential Anal. 13 79-96.
  • Lalley, S. P. (1983). Repeated likelihood ratio tests for curved exponential families. Z. Wahrsch. Verw. Gebiete 62 293-321.
  • Naiman, D. Q. (1986). Conservative confidence bands in curvilinear regression. Ann. Statist. 14 896-906.
  • Naiman, D. Q. (1990). Volumes of tubular neighborhoods of spherical polyhedra and statistical inference. Ann. Statist. 18 685-716.
  • Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York.
  • Shao, Q. (1997). Self-normalized large deviations. Ann. Probab. 25 285-328.
  • Siegmund, D. (1975). Error probabilities and average sample number of the sequential probability ratio test. J. Roy. Statist. Soc. Ser. B 37 394-401.
  • Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673-684.
  • Siegmund, D. (1985). Sequential Analysis. Springer, New York.
  • Siegmund, D. and Zhang, H. (1993). The expected number of local maxima of a random field and the volume of tubes. Ann. Statist. 21 1948-1966.
  • Spivak, M. (1965). Calculus on Manifolds. Benjamin, New York.
  • Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Statist. 36 546-551.
  • Wald, A. (1945). Sequential tests of statistical hypotheses. Ann. Math. Statist. 16 117-186.
  • Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472.
  • Woodroofe, M. (1976). A renewal theorem for curved boundaries and moments of estimation. Ann. Probab. 4 67-80.
  • Woodroofe, M. (1978). Large deviations of the likelihood ratio statistics with applications to sequential testing. Ann. Statist. 6 72-84.
  • Woodroofe, M. (1979). Repeated likelihood ratio tests. Biometrika 66 453-463.
  • Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia.