Ancillary statistics: A review

Statistica Sinica

Volumn 20, Issue 4, 2010, Pages 1309-1332

  Ghosh, M ^a   Reid, N ^b   Fraser, D A S ^b  

^a University of Florida   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Ancillarity paradox; Approximate ancillary; Estimating functions; Hierarchical Bayes; Local ancillarity; Location; Location scale; Multiple ancillaries; Nuisance parameters; P ancillarity; P values; S ancillarity; Saddlepoint approximation

Indexed keywords

EID: 78349274371     PISSN: 10170405     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (63)

1. Barnard, G. A. (1982). Conditionality versus similarity in the analysis of 2×2 tables. In Statistics and Probability: Essays in Honor of C.R. Rao. (Edited by G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh). North Holland, Amsterdam.
2. Barnard, G. A. and Sprott, D. A. (1971). A note on Basu's examples of anamalous ancillary statistics. In Foundations of Statistical Inference. (Edited by V. P. Godambe and D. A. Sprott). Holt, Rinehart and Winston, Toronto 163-170.
3. Barndorff-Nielsen, O. E. (1980). Conditionality resolutions. Biometrika, 67, 293-310.
4. Barndorff-Nielsen, O. E. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika 70, 343-365.
5. Barndorff-Nielsen, O. E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73, 307-322.
6. Barndorff-Nielsen, O. E. and Cox, D.R. (1994). Inference and Asymptotics. Chapman & Hall, New York.
7. Basu, D. (1964). Recovery of ancillary information. Sankhȳa 26, 3-16.
8. Berger, J. O. and Wolpert, R. L. (1984). The Likelihood Principle. IMS Lecture Notes- Monograph Series, Volume 6. Institute of Mathematical Statistics, Hayward.
9. Bhapkar, V. P. (1989). Conditioning on ancillary statistics and loss of information in the presence of nuisance parameters. J. Statist. Plann. Inference 21, 139-160.
10. Bhapkar, V. P. (1991). Loss of information in the presence of nuisance parameters and partial sufficiency. J. Statist. Plann. Inference 28, 185-203.
11. Brazzale, A. R., Davison, A. C. and Reid, N. (2007). Applied Asymptotics: Case Studies in Small Sample Statistics. Cambridge University Press, Cambridge.
12. Brown, L. D. (1990). An ancillarity paradox which appears in multiple regression models. Ann. Statist. 18, 471-538.
13. Buehler, R. J. (1982). Some ancillary statistics and their properties. J. Amer. Statist. Assoc. 77, 581-594.
14. Cohen, A. (1966). All admissible linear estimators of the mean vector. Ann. Math. Statist. 37, 458-463.
15. Cox, D. R. (1958). Some problems connected with statistical inference. Ann. Math. Statist. 29, 357-372.
16. Cox, D. R. (1971). The choice between alternative ancillary statistics. J. Roy. Statist. Soc., B 33, 251-252.
17. Cox, D. R. (1980). Local ancillarity. Biometrika 67, 273-278.
18. Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman & Hall, New York.
19. Daniels, H. E. (1954). Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631-650.
20. Datta, G.S., Ghosh, M., Smith, D.D. and Lahiri, P. (2002). On an asymptotic theory of conditional and unconditional coverage probabilities of empirical Bayes confidence intervals. Scand. J. Stat. 29, 139-152.
21. Durbin, J. (1980). Approximations for densities of sufficient estimators Biometrika 67, 311-333.
22. Edgeworth, F. Y. (1893). Exercises in the calculation of errors. Philosophical Magazine (Fifth Series) 36, 98-111.
23. Efron, B. and Hinkley, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika 65, 457-487.
24. Ferreira, P. E. and Minder, C. E. (1981). On a concept of ancillarity. Biometrika 68, 344.
25. Fisher, R. A. (1925). Theory of statistical estimation. Proc. Camb. Phil. Soc. 22, 700-725.
26. Fisher, R. A. (1934). Two new properties of mathematical likelihood. Proc. Roy. Soc. Ser. A 144, 285-307.
27. Fisher, R. A. (1935). The logic of inductive inference. J. Roy. Statist. Soc. Ser. B 98, 39-54.
28. Fraser, D. A. S. (1968). The Structure of Inference. John Wiley and Sons, New York.
29. Fraser, D. A. S. (1979). Inference and Linear Models. McGraw-Hill, New York.
30. Fraser, D. A. S. (2004). Ancillaries and conditional inference. Statist. Sci. 19, 333-369.
31. Fraser, D. A. S. and McDunnogh, P. (1980). Some remarks on conditional and unconditional inference for location-scale models. Statist. Hefte. 21, 224-231.
32. Fraser, D. A. S. and Reid, N. (1993). Third order assymptotic models: likelihood functions leading to accurate approximation of distribution functions. Statist. Sinica 3, 67-82.
33. Fraser, D. A. S. and Reid, N. (1995). Ancillaries and third order significance. Utilitias Mathematica 47, 33-53.
34. Fraser, D. A. S.,Wong, A. C. M. andWu, J. (1999). Regression analysis, nonlinear or nonnormal simple and accurate p-values from likelihood analysis. J. Amer. Statist. Assoc. 94, 1286-1295.
35. Ghosh, J. K. (1988). Statistical Information and Likelihood: A Collection of Critical Essays by Dr. D. Basu. Springer-Verlag, New York.
36. Godambe, V. P. (1960). An optimum property of regular maximum likelihood equation. Ann. Math. Statist. 31, 1208-1211.
37. Godambe, V. P. (1976). Conditional likelihood and unconditional optimal estimating equations. Biometrika 63, 277-284.
38. Godambe, V. P. (1980). On sufficiency and ancillarity in the presence of nuisance parameters. Biometrika 67, 155-162.
39. Godambe, V. P. (1982). Ancillarity principle and a statistical paradox. J. Amer. Statist. Assoc. 77, 931-933.
40. Hill, J. R. (1990). A general framework for model based statistics. Biometrika 77, 115-126.
41. Iglesias-Gonzalez, S. (2007). Highly accurate tests for the mixed linear model. Ph.D. Dissertation, University of Toronto.
42. Kalbfleisch, J. D. (1975). Sufficiency and conditionality. Biometrika 62, 251-268.
43. Kalbfleisch, J. D. (1982). Ancillary statistics. In Encyclopedia of Statistical Sciences, V1. (Edied by S. Kotz, N. L. Johnson and C.B. Read), 77-81. Wiley, New York.
44. Lehmann, E. L. (1981). An interpretation of completeness and Basu's theorem. J. Amer. Statist. Assoc., 76 335-340.
45. Lehmann, E. L. and Scholz, F. W. (1992). Ancillarity. In Current Issues in Statistical Inference: Essays in Honor of D. Basu. (Edited by M. Ghosh and P. K. Pathak). Ims Lecture Notes and Monograph Series 17, 32-51.
46. Lugannani, R. and Rice, S. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab. 12, 475-490.
47. McCullagh, P. (1984). Local sufficiency. Biometrika 71, 233-244.
48. Pearson, K. P. (1896). Mathematical contributions to the theory of evolution, III: regression, heredity, and panmixia. Philos. Trans. R. Soc. Lond. Ser. A 187, 253-318.
49. Pena, E. A., Rohatgi, V. K. and Szekely, G. J. (1992). On the non-existence of ancillary statistics. Statist. Probab. Lett. 15, 357-360.
50. Reid, N. (2003). Asymptotics and the theory of inference. Ann. Statist. 31, 1695-1731.
51. Reid, N. and Fraser, D. A. S. (2008). Mean likelihood and higher order approximations. Submitted for publication.
52. Sandved, E. (1965). A principle for conditioning on an ancillary statistic. Skandinavisk Aktuarietidskrift, 49, 39-47.
53. Sartori, N. (2003). A note on likelihood asymptotics in normal linear regression. Ann. Inst. Math. Statist. 55, 187-195.
54. Severini, T. A. (1993). Local ancillarity in the presence of a nuisance parameter. Biometrika 81, 649-661.
55. Severini, T. A. (1995). Comment on "The roles of conditioning in inference" by N. Reid. Stat. Science 187-189.
56. Severini, T. A. (2000). Likelihood Methods in Statistics. Oxford University Press, Oxford.
57. Skovgaard, I. M. (1986). Successive improvement of the order of ancillarity. Biometrika 73, 516-519.
58. Skovgaard, I. M. (1990). On the density of minimum contrast estimators. Ann. Statist. 18, 779-789.
59. Skovgaard, I. M. (1996). An explicit large-deviation approximation to one-parameter tests. Bernoulli 2, 145-165.
60. Stigler, S. M. (2001). Ancillary history. State of the Art in Probability and Statistics 36, 555-567.
61. Sverdrup, E. (1966). The present state of decision theory and Neyman-Pearson theory. Rev. Inst. Internat. Statist. 34, 309-333.
62. Wang, S. (1993). Saddlepoint approximations in conditional inference. J. Appl. Probab. 30, 397-404.
63. Welch, B. L. (1939). On confidence limits and sufficiency with particular reference to parameters of location. Ann. Math. Statist. 10, 58-69.