Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families. (English) Zbl 1105.62367

Summary: 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.


62L10 Sequential statistical analysis
62B05 Sufficient statistics and fields
Full Text: DOI


[1] Bahadur, R. R. (1967). Rates of convergence of estimates and test statistics. Ann. Math. Statist. 38 303-324. · Zbl 0201.52106
[2] 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. JSTOR: · Zbl 0424.62010
[3] Borovkov, A. A. and Rogozin, B. A. (1965). On the multidimensional central limit theorem. Theory Probab. Appl. 10 55-62. · Zbl 0139.35206
[4] Chan, H. P. (1998). Boundary crossing theory in change-point detection and its applications. Ph.D. dissertation, Stanford Univ.
[5] 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.
[6] Chandra, T. K. (1985). Asymptotic expansion of perturbed chi-squared variables. Sanky\?a Ser. A 47 100-110. · Zbl 0595.62010
[7] 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. · Zbl 0472.62028
[8] 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. · Zbl 0048.11804
[9] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications, 2nd ed. Springer, New York. · Zbl 0793.60030
[10] Groeneboom, P. (1980). Large Deviations and Asymptotic Efficiencies. Math. Centrum, Amsterdam. · Zbl 0421.62017
[11] Gray, A. (1990). Tubes. Addison-Wesley, Reading, MA. · Zbl 0692.53001
[12] Hirsch, M. (1976). Differential Topology. Springer, New York. · Zbl 0356.57001
[13] Hoeffding, W. (1965). Asymptotically optimal tests for multinomial distributions. Ann. Math. Statist. 36 369-405. · Zbl 0135.19706
[14] Hotelling, H. (1939). Tubes and spheres in n-spaces, and a class of statistical problems. Amer. J. Math. 61 440-460. JSTOR: · Zbl 0020.38302
[15] Hu, I. (1988). Repeated significance tests for exponential families. Ann. Statist. 16 1643-1666. · Zbl 0665.62081
[16] Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press, London. · Zbl 1274.62008
[17] 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. · Zbl 0723.62018
[18] Johnstone, I. and Siegmund, D. (1989). On Hotelling’s formula for the volume of tubes and Naiman’s inequality. Ann. Statist. 17 184-194. · Zbl 0678.62066
[19] 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. · Zbl 0707.62125
[20] Lai, T. L. (1997). On optimal stopping problems in sequential hypothesis testing. Statist. Sinica 7 33-51. · Zbl 0924.62085
[21] 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. · Zbl 0378.62069
[22] Lalley, S. P. (1983). Repeated likelihood ratio tests for curved exponential families. Z. Wahrsch. Verw. Gebiete 62 293-321. · Zbl 0509.62067
[23] Naiman, D. Q. (1986). Conservative confidence bands in curvilinear regression. Ann. Statist. 14 896-906. · Zbl 0607.62077
[24] Naiman, D. Q. (1990). Volumes of tubular neighborhoods of spherical polyhedra and statistical inference. Ann. Statist. 18 685-716. · Zbl 0723.62019
[25] Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York. · Zbl 0256.62002
[26] Shao, Q. (1997). Self-normalized large deviations. Ann. Probab. 25 285-328. · Zbl 0873.60017
[27] Siegmund, D. (1975). Error probabilities and average sample number of the sequential probability ratio test. J. Roy. Statist. Soc. Ser. B 37 394-401. JSTOR: · Zbl 0312.62063
[28] Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673-684. · Zbl 0353.62044
[29] Siegmund, D. (1985). Sequential Analysis. Springer, New York. · Zbl 0573.62071
[30] 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. · Zbl 0801.62087
[31] Spivak, M. (1965). Calculus on Manifolds. Benjamin, New York. · Zbl 0141.05403
[32] Stone, C. (1965). A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Statist. 36 546-551. · Zbl 0135.19204
[33] Wald, A. (1945). Sequential tests of statistical hypotheses. Ann. Math. Statist. 16 117-186. · Zbl 0060.30207
[34] Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472. JSTOR: · Zbl 0021.35503
[35] Woodroofe, M. (1976). A renewal theorem for curved boundaries and moments of estimation. Ann. Probab. 4 67-80. · Zbl 0368.60099
[36] Woodroofe, M. (1978). Large deviations of the likelihood ratio statistics with applications to sequential testing. Ann. Statist. 6 72-84. · Zbl 0386.62019
[37] Woodroofe, M. (1979). Repeated likelihood ratio tests. Biometrika 66 453-463. JSTOR: · Zbl 0418.62063
[38] Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia. · Zbl 0487.62062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.