Accurate parametric inference for small samples

Statistical Science

Volumn 23, Issue 4, 2008, Pages 465-484

  Brazzale, Alessandra R ^a,c   Davison, Anthony C ^b  

^a UNIVERSITY OF MODENA AND REGGIO EMILIA   (Italy)

^b CNR   (Italy)

^c EPFL   (Switzerland)

Author keywords

Conditional inference; Heteroscedasticity; Logistic regression; Lugannani rice formula; Markov chain monte carlo; Nonlinear model; R, regression scale model; Saddlepoint approximation; Spline; Statistical computing

Indexed keywords

EID: 67649386185     PISSN: 08834237     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-STS273     Document Type: Article

References (104)

1. AESTI, A. (2002). Categorical Data Analysis, 2nd ed. Wiley, New York. MR1914507
2. AGRESTI,A. and CAFFO, B. (2000). Simple and effective conidence intervals for proportions and differences of proportions result from adding two successes and two failures. Amer. Statist. 4 280-288. MR1814845
3. AGRESTI,A. and COULL, B. A. (1998). Approximate is better han "exact" for interval estimation of binomial proportions. Amer. Statist. 52 119-126. MR1628435
4. ANDREWS,D. F. and HERZBERG, A. M. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker. Springer, New York.
5. IELSEN O. E. (1983). On a formula for the disribution of the maximum likelihood estimator. Biometrika 70 343-365. MR0712023
6. BARNDORFF-NIELSEN, O. E. (1983). On a formula for the disribution of the maximum likelihood estimator. Biometrika 70 343-365. MR0712023
7. BARNDORFF-NIELSEN, O. E. (1986). Inference on full or partial arameters based on the standardized signed log likelihood ratio. Biometrika 73 307-322. MR0855891
8. BARNDORFF-NIELSEN,O. E. and COX, D. R. (1979). Edge- worth and saddle-point approximations with statistical applications (with discussion). J. Roy. Statist. Soc. Ser. B 41 279-312.MR0557595
9. BRNDORFF-NIELSEN,O. E. and COX, D. R. (1994). Inference and Asymptotics. Chapman & Hall, London. MR1317097
10. ATES,D. M. and WATTS, D. G. (1988). Nonlinear Regression Analysis and Its Applications. Wiley, New York. MR1060528
11. BELLIO, R. (2000). Likelihood asymptotics: Applications in bio- statistics. Ph.D. thesis, Dept. Statistics, Univ. Padova, Italy.
12. BELLIO,R. and BRAZZALE, A. R. (1999). Higher-order asymptotics in nonlinear regression. In Proceedings of the 14th International Workshop on Statistical Modelling, Graz, July 19-23, 1999 (H. Friedl, A. Berghold and G. Kauermann, eds.) 440-443.
13. BELLIO,R.andBRAZZALE, A. R. (2003). Higher-order asymp- totics unleashed: Software design for nonlinear heteroscedastic models. J. Comput. Graph. Statist. 12 682-697. MR2005458
14. BELLIO,R., JENSEN,J. E. and SEIDEN, P. (2000). Applications of likelihood asymptotics for nonlinear regression in herbicide bioassays. Biometrics 56 1204-1212. MR1815621
15. ARTORI N. (2006). Practical use of modified maximum likelihoods for stratified data. Biom. J. 48 876-886.MR2291296
16. ELLIO,R. and SARTORI, N. (2006). Practical use of modified maximum likelihoods for stratified data. Biom. J. 48 876-886.MR2291296
17. BONDESSON, L. (1982). To reduce a composite hypothesis to a simple one by sampling from the structural or a conditional distribution. Scand. J. Statist. 9 129-138. MR0680908
18. BRAZZALE, A. R. (1999). Approximate conditional inference in logistic and loglinear models. J. Comput. Graph. Statist. 8 653661.
19. BRAZZALE, A. R. (2000). Practical small-sample parametric inference. Ph.D. thesis, Dept. Mathematics, Swiss Federal Institute of Technology Lausanne. Available at www.isib.cnr.it/~brazzale/lib.html
20. RAZZALE, A. R. (2005). hoa: An R package bundle for higher order likelihood inference. RNews 5 20-27. Available at
21. BRAZZALE,A. R., DAVISON,A. C. and REID, N. (2007). Applied Asymptotics: Case Studies in Small Sample Statistics. Cambridge Univ. Press. MR2342742
22. BROWN,L.D., CAI,T.T. and DASGUPTA, A. (2001). Interval estimation for a binomial proportion (with discussion). Statist. Sci. 16 101-133. MR1861069
23. BUTLER, R. W. (2007). Saddlepoint Approximations with Applications. Cambridge Univ. Press. MR2357347
24. ASELLA,G., WELLS,M. T. and TANNER, M. A. (1994). Using sampling-based calculations for pivotal inference. Technical Report BU-1178-M, Biometrics Unit, Cornell Univ., Ithaca, NY.
25. ASTILLO,J. D. and LOPEZ-RATERA, A. (2006). Saddlepoint approximation in exponential models with boundary points. Bernoulli 12 491-500. MR2232728
26. LOPPER,C. J. and PEARSON, E. S. (1934). The use of confidence interval or fiducial limits illustrated in the case of the binomial. Biometrika 26 404-413.
27. OX, D. R. (1958). The regression analysis of binary sequences (with discussion). J. Roy. Statist. Soc. Ser. B 20 215-242.MR0099097
28. OX, D. R. (1988). Some aspects of conditional and asymptoticinference: A review. SankhyaA 50 314-337. MR1065547
29. YTEL INC. (2007). StatXact/LogXact 8. Cytel Inc., Cambridge,Mas. Available at http://www.cytel.com.
30. ANIELS, H. E. (1954). Saddlepoint approximations in statistics. Ann. Math. Statist. 25 631-650. MR0066602
31. ANIELS, H. E. (1958). Discussion of "The regression analysis of binary sequences," by D. R. Cox. J. Roy. Statist. Soc. Ser. B 20236-238. MR0099097
32. ANIELS, H. E. (1987). Tail probability approximations. Internat. Statist. Rev. 54 37-48. MR0962940
33. AVISON, A. C. (1988). Approximate conditional inference in generalized linear models. J. Roy. Statist. Soc. Ser. B 50 445461. MR0970979
34. http://www.ams.org/mathscinet-getitem?mr=1998913 AVISON, A. C. (2003). Statistical Models. Cambridge Univ. Press. MR1998913
35. AVISON,A.C., FRASER,D. A.S. and REID, N. (2006). Improved likelihood inference for discrete data. J. Roy. Statist. Soc.Ser. B 68 495-508. MR2278337
36. AVISON,A.C. and HINKLEY, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press. MR1478673
37. AVISON,A.C. and SARTORI, N. (2008). The Banff challenge: Statistical detection of a noisy signal. Statist. Sci. 123 354-364.
38. AVISON,A.C.andWANG, S. (2002). Saddlepoint approximaions as smoothers. Biometrika 89 933-938. MR1946521
39. ENISON,D.G.T.,HOLMES,C.C.,MALLICK,B.K.and SMITH, A. F. M. (2002). Bayesian Methods for Nonlinear Classification and Regression. Wiley, New York. MR1962778
40. LACONIS,P. and STURMFELS, B. (1998). Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26363-397. MR1608156
41. LCLCCIO,T.J. and EFRON, B. (1996). Bootstrap confidence intervals (with discussion). Statist. Sci. 11 189-228. MR1436647
42. LCLCCIO,T. J. and FIELD, C. A. (1991). An accurate method for approximate conditional and Bayesian inference about linear regression models from censored data. Biometrika 78 903-910.MR1147027
43. http://www.ams.org/mathscinet-getitem?mr=1049410 LCLCCIO,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. MR1049410
44. LCLCCIO,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. MR1210437
45. LCLCCIO,T. J., MARTIN,M.A. and STERN, S. E. (2001). Simple and accurate one-sided inference from signed roots of likelihood ratios. Canad. J. Statist. 29 67-76. MR1834487
46. LCLCCIO,T.J. and YOUNG, G. A. (2008). Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families. Biometrika 95 747-758
47. URBIN, J. (1961). Some methods of constructing exact tests. Bioetrika 48 41-55. MR0126313
48. EFRON, B. (1979). Bootstrap methods: Another look at the jack- nife. Ann. Statist. 7 1-26. MR0515681
49. http://www.ams.org/mathscinet-getitem?mr=0521817 FRON,B. and HINKLEY, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected isher information (with discussion). Biometrika 65 457-487.MR0521817
50. EFRON,B. and TIBSHIRANI, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall, New York. MR1270903
51. http://www.ams.org/mathscinet-getitem?mr=1377843 ISHER, R. A. (1934). Two new properties of mathematical likelihood. Proc. R. Soc. Lond. Ser. A 144 285-307.FOR STER,J. J., MCDONALD,J. W. and SMITH, P. W. F. (1996). Monte Carlo exact conditional tests for log-linear and logistic models. J. Roy. Statist. Soc. Ser. B 58 445-453. MR1377843
52. FORSTER,J. J., MCDONALD,J. W. and SMITH, P. W. F. (2003). Markov chain Monte Carlo exact inference for binomial and multinomial regression models. Stat. Comput. 13 169-177.MR1963333
53. RASER, D. A. S. (1979). Inference and Linear Models.McGraw-ill, New York. MR0535612
54. RASER, D. A. S. (1990). Tail probabilities from observed likelihoods. Biometrika 77 65-76. MR1049409
55. RASER, D. A. S. (2003). Likelihood for component parameters. Biometrika 90 327-339. MR1986650
56. FRASER,D.A.S., ANDREWS,D. A. and WONG, A. (2005). Computation of distribution functions from likelihood information near observed data. J. Statist. Plann. Inference 134 180193. MR2146092
57. Ee H.S. and REID, N. (1990). Nonnormal linear regression: An example of significance levels in high diensions. Biometrika 77 333-341. MR1064805
58. RASER,D.A. S.,Lee,H.S. and REID, N. (1990). Nonnormal linear regression: An example of significance levels in high diensions. Biometrika 77 333-341. MR1064805
59. FRASER,D.A.S.andREID, N. (2009). Mean likelihood and her order approximations. Unpublished.
60. FRASER,D.A.S.,REID,N. andWU, J. (1999). A simple formula or tail probabilities for frequentist and Bayesian inference. Bio-metrika 86 249-264. MR1705367
61. RASER,D.A.S.,WONG,A. and Wu, J. (1999). Regression analysis, nonlinear or nonnormal: Simple and accurate p values rom likelihood analysis. J. Amer. Statist. Assoc. 94 1286-1295.MR1731490
62. GEYER,C. J. and MEEDEN, G. D. (2005). Fuzzy and randomized confidence intervals and P-values (with discussion). Statist. Sci. 0 358-387. MR2210225
63. GUOLO,A., BRAZZALE,A. R. and SALVAN, A. (2006). Improved inference on a scalar fixed effect of interest in nonlinear mixed-effects models. Comput. Statist. Data Anal. 51 16021613. MR2307530
64. ENSEN, J. L. (1992). The modified signed likelihood statistic and saddlepoint approximations. Biometrika 79 693-703.MR1209471
65. KAPPENMANN, R. F. (1975). Conditional confidence intervals for the double exponential distribution parameters. Technometrics 17 233-235. MR0370902
66. AWLESS, J. F. (1972). Conditional confidence interval procedures for the location and scale parameters of the Cauchy and logistic distributions. Biometrika 59 377-386. MR0334380
67. AWLESS, J. F. (1973). Conditional versus unconditional confidence intervals for the parameters of the Weibull distribution. J. Amer. Statist. Assoc. 68 655-669.
68. AWLESS, J. F. (1978). Confidence interval estimation for the Weibull and extreme value distributions. Technometrics 20 355368. MR0515992
69. EE,S.M.S.andYOUNG, G. A. (2005). Parametric bootstrapping with nuisance parameters. Statist. Probab. Lett. 71 143153. MR2126770
70. UGANNANI,R. and RICE, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab. 12 475-490. MR0569438
71. UNN,D.J.,THOMAS,A., BEST,N. and SPIEGELHALTER, D. (2000). WinBUGS-a Bayesian modelling framework: Concepts, structure, and extensibility. Stat. Comput. 10 325-337. Available at
72. EHTA,C. R. and PATEL, N. R. (1995). Exact logistic regression: Theory and examples. Stat. Med. 14 2143-2160.
73. EHTA,C. R., PATEL,N. R. and SENCHAUDHURI, P. (2000). Efficient Monte Carlo methods for conditional logistic regression. J. Amer. Statist. Assoc. 95 99-108.
74. ORGENTHALER,S. and TUKEY, J. W., eds. (1991). Configural Polysampling: A Route to Practical Robustness. Wiley, New York. MR1247031
75. PACE,L. and SALVAN, A. (1997). Principles of Statistical Inference from a Neo-Fisherian Perspective. World Scientific, Singapore. MR1476674
76. PACE,L., SALVAN,A. and VENTURA, L. (2004). The effects of rounding on likelihood procedures. J. Appl. Statist. 31 29-48.MR2041554
77. IERCE,D. A. and PETERS, D. (1992). Practical use of higher order asymptotics for multiparameter exponential families (with discussion). J. Roy. Statist. Soc. Ser. B 54 701-737. MR1185218
78. PIERCE,D. A. and PETERS, D. (1999). Improving on exact tests by approximate conditioning. Biometrika 86 265-277. MR1705363
79. R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. Available at http://www.R-project.org.
80. REID, N. (1988). Saddlepoint methods and statistical inference (with discussion). Statist. Sci. 3 213-238. MR0968390
81. REID, N. (1995). The roles of conditioning in inference (with discussion). Statist. Sci. 10 138-199. MR1368097
82. REID, N. (2003). Asymptotics and the theory of inference. Ann. Statist. 31 1695-1731. MR2036388
83. REID,N.,MUKERJEE,R.andFRASER, D. A. S. (2002). Someaspects of matching priors. In Mathematical Statistics and Applications: Festschrift for Constance van Eeden (M. Moore, S. Froda and C. Leger, eds.) 31-44. Hayward, CA. MR2138284
84. ROBERT,C.P.andCASELLA, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York. MR2080278
85. RONCHETTI,E.andVENTURA, L. (2000). Between stability and higher-order asymptotics. Stat. Comput. 11 67-73. MR1837146
86. S-PLUS (2007). S-PLUS (version 8). Insightful Corporation, Seattle, WA. Available at http://www.insightful.com.
87. SARTORI, N. (2003). Modified profile likelihoods in models with stratum nuisance parameters. Biometrika 90 533-549.MR2006833
88. SARTORI,N.,BELLIO,R.,PACE,L. andSALVAN, A. (1999). The directed modified profile likelihood in models with many nuisance parameters. Biometrika 86 735-742. MR1723792
89. SEBER,.A.F. and WILD, A. J. (1989). Nonlinear Regression.Wiley, New York. MR0986070
90. SEIDEN,P., KAPPEL,D. and STREIBIG, J. C. (1998). Response of Brassica napus L. tissue culture to metsulfuron methyl andchlorsulfuron. WeedResearch 38 221-228.
91. EVERINI, T. A. (1999). An empirical adjustment to the likelihoodatio statistic. Biometrika 86 235-247. MR1705371
92. EVERINI, T. A. (2000). Likelihood Methods in Statistics.Claren-don, Oxford. MR1854870
93. SKOVGAARD, I. M. (1987). Saddlepoint expansions for conditi onal distributions. J. Appl. Probab. 24 875-887. MR0913828
94. SKOVGAARD, I. M. (1996). An explicit large-deviation approximation to one-parameter tests. Bernoulli 2 145-166. MR1410135
95. SKOVGAARD, I. M. (2001). Likelihood asymptotics. Scand. J. Statist. 28 3-32. MR1844348
96. SMITH, P.W.F., FORSTER,J. J. and MCDONALD, J. W. (1996). Monte Carlo exact tests for square contingency tables. J. Roy.Statist. Soc. Ser. A159 309-321.
97. STRAWDERMAN,R.L.,CASELLA,G.andWELLS,M.T. (1996). Practical small-sample asymptotics for regression problems (with discussion). J. Amer. Statist. Assoc. 91 643-654.MR1395732
98. TIBSHIRANI, R. J. (1989). Noninformative priors for one parame- terofmany. Biometrika 76 604-608. MR1040654
99. TIERNEY, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762. MR1329166
100. TIERNEY,L. and KADANE, J. B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist.Assoc. 81 82-86. MR0830567
101. TIERNEY,L., KASS,R.E. and KADANE, J. B. (1989). Approximate marginal densities of nonlinear functions. Biometrika 76425-433. MR1040637
102. TROTTER,H. F. and TUKEY, J. W. (1956). Conditional Monte Carlo for normal samples. In Symposium on Monte Carlo Methods (H. Meyer, ed.) 64-79. Wiley, New York. MR0079825
103. ENTURA, L. (1997). Metodi asintotici, semiparametrici e robusti per l'inferenza in famiglie di gruppo. Ph.D. thesis, Dept. Statistics, Univ. Padova, Italy. (In Italian.)
104. ZAMAR,D., MCNENEY,B. and GRAHAM, J. (2007). elrm:Soft- ware implementing exact like inference for logistic regression models. J. Statist. Software 21, issue number 3. Available at http://www.jstatsoft.org/v21/i03.