Asymptotics and the theory of inference

Annals of Statistics

Volumn 31, Issue 6, 2003, Pages 1695-1731

  Reid, N ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

Ancillarity; Approximation; Bayesian inference; Conditioning; Laplace approximation; Likelihood; Matching priors; P*; P values; R*; Saddlepoint approximation; Tail area; Tangent exponential model

Indexed keywords

EID: 1442357279     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1074290325     Document Type: Review

References (119)

1. AMARI, S.-I. (1985). Differential-Geometrical Methods in Statistics. Lecture Notes in Statist. 28. Springer, New York.
2. BARNDORFF-NIELSEN, O. E. (1980). Conditionality resolutions. Biometrika 67 293-310.
3. BARNDORFF-NIELSEN, O. E. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika 70 343-365.
4. BARNDORFF-NIELSEN, O. E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73 307-322.
5. BARNDORFF-NIELSEN, O. E. (1990). Approximate interval probabilities. J. Roy. Statist. Soc. Ser. B 52 485-496.
6. BARNDORFF-NIELSEN, O. E. (1991). Modified signed log likelihood ratio. Biometrika 78 557-563.
7. BARNDORFF-NIELSEN, O. E. (1996). Two index asymptotics. In Frontiers in Pure and Applied Probability II (I. Melnikov, ed.) 9-20. TVP Science, Moscow.
8. BARNDORFF-NIELSEN, O. E. and CHAMBERLIN, S. (1991). An ancillary invariant modification of the signed log likelihood ratio. Scand. J. Statist. 18 341-352.
9. BARNDORFF-NIELSEN, O. E. and CHAMBERLIN, S. (1994). Stable and invariant adjusted directed likelihoods. Biometrika 81 485-499.
10. BARNDORFF-NIELSEN, O. E. and Cox, D. R. (1994). Inference and Asymptotics. Chapman and Hall, London.
11. BARNDORFF-NIELSEN, O. E., Cox, D. R. and REID, N. (1986). The role of differential geometry in statistical theory. Internat. Statist. Rev. 54 83-96.
12. BARNDORFF-NIELSEN, O. E. and WOOD, A. T. A. (1998). On large deviations and choice of ancillary for p* and r*. Bernoulli 4 35-63.
13. BELLIO, R. (1999). Likelihood asymptotics: Applications in biostatistics. Ph.D. thesis, Dept. Statistics, Univ. Padua, Italy.
14. BELLIO, R. (2003). Likelihood methods for controlled calibration. Scand. J. Statist. 30 339-353.
15. BELLIO, R. and BRAZZALE, A. R. (2003). Higher-order asymptotics unleashed: Software design for nonlinear heteroscedastic regression. J. Comput. Graph. Statist. 12 682-697.
16. BELLIO, R., JENSEN, J. E. and SEIDEN, P. (2000). Applications of likelihood asymptotics for nonlinear regression in herbicide bioassays. Biometrics 56 1204-1212.
17. BENTKUS, V., GÖTZE, F. and VAN ZWET, W. R. (1997). An Edgeworth expansion for symmetric statistics. Ann. Statist. 25 851-896.
18. BERAN, J. (1994). Statistics for Long Memory Processes. Chapman and Hall, London.
19. BICKEL, P. and FREEDMAN, D. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196-1217.
20. BLOZNELIS, M. and GÖTZE, F. (2000). An Edgeworth expansion for finite-population U-statistics. Bernoulli 6 729-760.
21. BOX, G. E. P. and Cox, D. R. (1964). An analysis of transformations (with discussion). J. Roy. Statist. Soc. Ser. B 26 211-252.
22. BRAZZALE, A. (1999). Approximate conditional inference in logistic and loglinear models. J. Comp. Graph. Statist. 8 653-661.
23. BRAZZALE, A. (2000). Practical small-sample parametric inference. Ph.D. thesis, Dept. Mathematics, Swiss Federal Institute of Technology, Lausanne.
24. BUTLER, R. (2000). Reliabilities for feedback systems and their saddlepoint approximation. Statist. Sci. 15 279-298.
25. BUTLER, R. (2001). First passage time distributions in semi-Markov processes and their saddlepoint approximation. In Data Analysis from Statistical Foundations: A Festschrift in Honour of the 75th Birthday of D. A. S. Fraser (A. K. M. E. Saleh, ed.). Nova Science, Commach, NY.
26. BUTLER, R., HUZURBAZAR, S. and BOOTH, J. (1992a). Saddlepoint approximations for the generalized variance and Wilks' statistic. Biometrika 79 157-169.
27. BUTLER, R., HUZURBAZAR, S. and BOOTH, J. (1992b). Saddlepoint approximation in multivariate analysis for the Bartlett-Nanda-Pillai trace statistics. Biometrika 79 705-715.
28. BUTLER, R., HUZURBAZAR, S. and BOOTH, J. (1993). Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc. Ser. B 55 171-183.
29. CAKMAK, S., FRASER, D. A. S., MCDUNNOUGH, P., REID, N. and YUAN, X. (1998). Likelihood centered asymptotic model: Exponential and location model versions. J. Statist. Plann. Inference 66 211-222.
30. CHEAH, P. K., FRASER, D. A. S. and REID, N. (1995). Adjustments to likelihood and densities: Calculating significance. J. Statist. Res. 29 1-13.
31. CORCORAN, S., DAVISON, A. C. and SPADY, R. (1999). Reliable inference from empirical likelihood. Preprint.
32. COX, D. R. (1972). Regression models and life tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187-220.
33. COX, D. R. and HINKLEY, D. V. (1974). Theoretical Statistics. Chapman and Hall, London.
34. COX, D. R. and REID, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion). J. Roy. Statist. Soc. Ser. B 49 1-39.
35. DAVISON, A. C. and HINKLEY, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press.
36. DAWID, A. P. (1991). Fisherian inference in likelihood and prequential frames of reference (with discussion). J. Roy. Statist. Soc. Ser. B 53 79-109.
37. DICICCIO, T. J. and EFRON, B. (1992). More accurate confidence intervals in exponential families. Biometrika 79 231-245.
38. DICICCIO, T. J. and EFRON, B. (1996). Bootstrap confidence intervals (with discussion). Statist. Sci. 11 189-228.
39. DICICCIO, T. J., FIELD, C. A. and FRASER, D. A. S. (1990). Approximations of marginal tail probabilities and inference for scalar parameters. Biometrika 77 77-95.
40. DICICCIO, T. J., HALL, P. and ROMAND, J. (1991). Empirical likelihood is Bartlett-correctable. Ann. Statist. 19 1053-1061.
41. DICICCIO, T. J. and MARTIN, M. A. (1993). Simple modifications for signed roots of likelihood ratio statistics. J. Roy. Statist. Soc. Ser. B 55 305-316.
42. DINGES, H. (1986). Asymptotic normality and large deviations. In Proc. 10th Prague Conference on Information Theory, Statistical Decision Functions, and Random Processes. Academia, Prague.
43. EFRON, B. (1993). Bayes and likelihood calculations from confidence intervals. Biometrika 80 3-26.
44. EFRON, B. and HINKLEY, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information (with discussion). Biometrika 65 457-487.
45. FAN, J. (1993). Local linear regression smoothers and their minimax efficiencies. Ann. Statist. 21 196-216.
46. FISHER, R. A. (1920). A mathematical examination of the methods of determining the accuracy of an observation by the mean error, and by the mean square error. Monthly Notices Roy. Astronomical Soc. 80 758-770. [Reprinted (1950) in Contributions to Mathematical Statistics. Wiley, New York.]
47. FISHER, R. A. (1920). A mathematical examination of the methods of determining the accuracy of an observation by the mean error, and by the mean square error. Monthly Notices Roy. Astronomical Soc. 80 758-770. [Reprinted (1950) in Contributions to Mathematical Statistics. Wiley, New York.]
48. FISHER, R. A. (1934). Two new properties of mathematical likelihood. Proc. Roy. Soc. London Ser. A 144 285-307.
49. FRASER, D. A. S. (1968), The Structure of Inference. Wiley, New York.
50. FRASER, D. A. S. (1988). Normed likelihood as saddlepoint approximation. J. Multivariate Anal. 27 181-193.
51. FRASER, D. A. S. (1990). Tail probabilities from observed likelihoods. Biometrika 77 65-76.
52. FRASER, D. A. S. (1991). Statistical inference: Likelihood to significance. J. Amer. Statist. Assoc. 86 258-265.
53. FRASER, D. A. S. and MCDUNNOUGH, P. (1984). Further remarks on asymptotic normality of likelihood and conditional analyses. Canad. J. Statist. 12 183-190.
54. FRASER, D. A. S. and REID, N. (1993). Third order asymptotic models: Likelihood functions leading to accurate approximations for distribution functions. Statist. Sinica 3 67-82.
55. FRASER, D. A. S. and REID, N. (1995). Ancillaries and third-order significance. Utilitas Math. 47 33-53.
56. FRASER, D. A. S. and REID, N. (2002). Strong matching of frequentist and Bayesian parametric inference. J. Statist. Plann. Inference 103 263-285.
57. FRASER, D. A. S., REID, N. and WU, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86 249-264.
58. FRASER, D. A. S., WONG, A. and WU, J. (1999). Regression analysis, nonlinear or nonnormal: Simple and accurate p-values from likelihood analysis. J. Amer. Statist. Assoc. 94 1286-1295.
59. FREEDMAN, D. L. (1999). On the Bernstein-von Mises theorem with infinite-dimensional parameters. Ann. Statist. 27 1119-1140.
60. GHOSH, M. and MUKERJEE, R. (1998). Recent developments on probability matching priors. In Applied Statistical Science III (S. E. Ahmed, M. Ahsanullah and B. K. Sinha, eds.) 227-252. Nova Science, Commack, NY.
61. GUOLO, A. (2002). Second-order likelihood-based inference in mixed-effects models. Working paper 2002.11, Dept. Statistics, Univ. Padua, Italy.
62. HALL, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
63. HASTIE, T. and LOADER, C. (1993). Local regression: Automatic kernel carpentry (with discussion). Statist. Sci. 8 120-143.
64. HSU, J. (1995). Generalized Laplacian approximations in Bayesian inference. Canad. J. Statist. 23 399-410.
65. HUZURBAZAR, A. V. (2000). Modelling and analysis of engineering systems data using flowgraph models. Technometrics 42 300-306.
66. IBRAGIMOV, I. A. and HAS'MINSKII, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, New York.
67. JENSEN, J. L. (1992). The modified signed likelihood statistic and saddlepoint approximations. Biometrika 79 693-703.
68. JOHNSON, R. A. (1970). Asymptotic expansions associated with posterior distributions. Ann. Math. Statist. 41 851-864.
69. KASS, R., TIERNEY, L. J. and KADANE, J. (1988). Asymptotics in Bayesian computation. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 261-278. Clarendon Press, Oxford.
70. KOLASSA, J. and TANNER, M. (1994). Approximate conditional inference in exponential families via the Gibbs sampler. J. Amer. Statist. Assoc. 89 697-702.
71. LEHMANN, E. L. and CASELLA, G. (1998). Theory of Point Estimation, 2nd ed. Wiley, New York.
72. LUGANNANI, R. and RICE, S. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475-490.
73. LYONS, B. and PETERS, D. (2000). Applying Skovgaard's modified directed likelihood statistic to mixed linear models. J. Statist. Comput. Simulation 65 225-242.
74. MUKERJEE, R. and REID, N. (2000). On the Bayesian approach for frequentist computations. Braz. J. Probab. Statist. 14 159-166.
75. MURPHY, S. A. and VAN DER VAART, A. W. (2000). On profile likelihood (with discussion). J. Amer. Statist. Assoc. 95 449-485.
76. MYKLAND, P. A. (1995). Martingale expansions and second-order inference. Ann. Statist. 23 707-731.
77. MYKLAND, P. K. (2000). Bartlett identities and large deviations in likelihood theory. Ann. Statist. 27 1105-1117.
78. NEYMAN, J. and SCOTT, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica 16 1-32.
79. NICOLAOU, A. (1993). Bayesian intervals with good frequentist behaviour in the presence of nuisance parameters. J. Roy. Statist. Soc. Ser. B 55 377-390.
80. OWEN, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249.
81. OWEN, A. (2001). Empirical Likelihood. Chapman and Hall/CRC Press, Boca Raton, FL.
82. PEERS, H. W. (1965). On confidence points and Bayesian probability points in the case of several parameters. J. Roy. Statist. Soc. Ser. B 27 9-16.
83. PFANZAGL, J. (1985). Asymptotic Expansions for General Statistical Models. Lecture Notes in Statist. 31. Springer, New York.
84. PIERCE, D. A. and PETERS, D. L. (1992). Practical use of higher-order asymptotics for multiparameter exponential families (with discussion). J. Roy. Statist. Soc. Ser. B 54 701-737.
85. PIERCE, D. A. and PETERS, D. L. (1994). Higher-order asymptotics and the likelihood principle: One-parameter models. Biometrika 81 1-10.
86. 2/n is large. I. Consistency. Ann. Statist. 12 1298-1309.
87. 2/n is large. II. Normal approximation. Ann. Statist. 13 1403-1417.
88. REID, N. (1988). Saddlepoint methods and statistical inference (with discussion). Statist. Sci. 3 213-238.
89. REID, N. (1996). Likelihood and higher-order approximations to tail areas: A review and annotated bibliography. Canad. J. Statist. 24 141-166.
90. REID, N., MUKERJEE, R. and FRASER, D. A. S. (2003). Some aspects of matching priors. In Mathematical Statistics and Applications: A Festschrift for Constance van Eeden (M. Moore, S. Froda and C. Léger, eds.) 31-44. IMS, Bethesda, MD.
91. ROTNITZKY, A., COX, D. R., BOTTAI, M. and ROBINS, J. (2000). Likelihood based inference with a singular information matrix. Bernoulli 6 243-284.
92. SARTORI, N. (2003). Modified profile likelihoods in models with stratum nuisance parameters. Biometrika 90 533-549.
93. SEVERINI, T. A. (1999). An empirical adjustment to the likelihood ratio statistic. Biometrika 86 235-248.
94. SEVERINI, T. A. (2000). Likelihood Methods in Statistics. Oxford Univ. Press.
95. SINGH, K. (1981). On the asymptotic accuracy of Efron's bootstrap. Ann. Statist. 9 1187-1195.
96. SKOVGAARD, I. M. (1989). A review of higher-order likelihood inference. Bull. Internat. Statist. Inst. 53 331-351.
97. SKOVGAARD, I. M. (1990). On the density of minimum contrast estimators. Ann. Statist. 18 779-789.
98. SKOVGAARD, I. M. (1996). An explicit large-deviation approximation to one-parameter tests. Bernoulli 2 145-165.
99. SKOVGAARD, I. M. (2001). Likelihood asymptotics. Scand. J. Statist. 28 3-32.
100. SMITH, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika 72 67-90.
101. SMITH, R. L. (1989). A survey of nonregular problems. Bull. Internat. Statist. Inst. 53 353-372.
102. STIGLER, S. (1973). Studies in the history of probability and statistics. XXXII. Laplace, Fisher, and the discovery of the concept of sufficiency. Biometrika 60 439-445.
103. STONE, C. J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 1348-1360.
104. STONE, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040-1053.
105. SWEETING, T. J. (1995). A framework for Bayesian and likelihood approximations in statistics. Biometrika 8 1-23.
106. SWEETING, T. J. (2001). Coverage probability bias, objective Bayes and the likelihood principle. Biometrika 88 657-675.
107. TIBSHIRANI, R. J. (1989). Noninformative priors for one parameter of many. Biometrika 76 604-608.
108. TIERNEY, L. J. (1990). Lispstat. Wiley, New York.
109. TIERNEY, L. J. and KADANE, J. B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc. 81 82-86.
110. WALKER, A. M. (1969). On the asymptotic behaviour of posterior distributions. J. Roy. Statist. Soc. Ser. B 31 80-88.
111. WANG, S. (1993). Saddlepoint approximations in conditional inference. J. Appl. Probab. 30 397-404.
112. WASSERMAN, L. (2000). Asymptotic inference for mixture models using data-dependent priors. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 159-180.
113. WELCH, B. and PEERS, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihoods. J. Roy. Statist. Soc. Ser. B 25 318-329.
114. WONG, A. C. M. and WU, J. (2000). Practical small-sample asymptotics for distributions used in life-data analysis. Technometrics 42 149-155.
115. WONG, A. C. M. and WU, J. (2001). Approximate inference for the factor loading of a simple factor analysis model. Scand. J. Statist. 28 407-414.
116. WONG, A. C. M. and WU, J. (2002). Small sample asymptotic inference for coefficient of variation: Normal and non-normal models. J. Statist. Plann. Inference 104 73-82.
117. WU, J., JIANG, G. Y., WONG, A. C. M. and SUN, X. (2002). Likelihood analysis for the ratio of means of two independent lognormal distributions. Biometrics 58 463-469.
118. YAU, C. L. and HUZURBAZAR, A. V. (2002). Analysis of censored and incomplete survival data using flowgraph models. Statistics in Medicine 21 3727-3743.
119. YI, G. Y., WU, J. and LIU, Y. (2002). Implementation of higher-order asymptotics to S-PLUS. Comput. Statist. Data Anal. 40 775-800.