Ancillaries and conditional inference (with comments and rejoinder).

*(English)*Zbl 1100.62534Summary: Sufficiency has long been regarded as the primary reduction procedure to simplify a statistical model, and the assessment of the procedure involves an implicit global repeated sampling principle. By contrast, conditional procedures are almost as old and yet appear only occasionally in the central statistical literature. Recent likelihood theory examines the form of a general large sample statistical model and finds that certain natural conditional procedures provide, in wide generality, the definitive reduction from the initial variable to a variable of the same dimension as the parameter, a variable that can be viewed as directly measuring the parameter. We begin with a discussion of two intriguing examples from the literature that compare conditional and global inference methods, and come quite extraordinarily to opposite assessments concerning the appropriateness and validity of the two approaches. We then take two simple normal examples, with and without known scaling, and progressively replace the restrictive normal location assumption by more general distributional assumptions. We find that sufficiency typically becomes inapplicable and that conditional procedures from large sample likelihood theory produce the definitive reduction for the analysis. We then examine the vector parameter case and find that the elimination of nuisance parameters requires a marginalization step, not the commonly proffered conditional calculation that is based on exponential model structure. Some general conditioning and modelling criteria are then introduced. This is followed by a survey of common ancillary examples, which are then assessed for conformity to the criteria. In turn, this leads to a discussion of the place for the global repeated sampling principle in statistical inference. It is argued that the principle in conjunction with various optimality criteria has been a primary factor in the long-standing attachment to the sufficiency approach and in the related neglect of the conditioning procedures based directly on available evidence.

##### MSC:

62B05 | Sufficient statistics and fields |

62A01 | Foundations and philosophical topics in statistics |

##### Keywords:

Ancillaries; conditional inference; inference directions; likelihood; sensitivity directions; pivotal
Full Text:
DOI

**OpenURL**

##### References:

[1] | Barndorff-Nielsen, O. E. (1986). Inference on full or partial parameters based on the standardized, signed log likelihood ratio. Biometrika 73 307–322. · Zbl 0605.62020 |

[2] | Brown, L. D. (1990). An ancillarity paradox which appears in multiple linear regression (with discussion). Ann. Statist. 18 471–538. JSTOR: · Zbl 0721.62011 |

[3] | Buehler, R. J. (1982). Some ancillary statistics and their properties (with discussion). J. Amer. Statist. Assoc. 77 581–594. · Zbl 0494.62005 |

[4] | Casella, G. and Berger, R. L. (2002). Statistical Inference , 2nd ed. Duxbury, Pacific Grove, CA. · Zbl 0699.62001 |

[5] | Cox, D. R. (1958). Some problems connected with statistical inference. Ann. Math. Statist. 29 357–372. · Zbl 0088.11702 |

[6] | Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics . Chapman and Hall, London. · Zbl 0334.62003 |

[7] | Fisher, R. A. (1925). Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22 700–725. · JFM 51.0385.01 |

[8] | Fisher, R. A. (1934). Two new properties of mathematical likelihood. Proc. Roy. Soc. London Ser. A 144 285–307. · Zbl 0009.21902 |

[9] | Fisher, R. A. (1935). The logic of inductive inference (with discussion). J. Roy. Statist. Soc. 98 39–82. · Zbl 0011.03205 |

[10] | Fisher, R. A. (1956). Statistical Methods and Scientific Inference. Oliver and Boyd, London. · Zbl 0070.36903 |

[11] | Fisher, R. A. (1961). Sampling the reference set. Sankhyā Ser. A 23 3–8. · Zbl 0095.33402 |

[12] | Fraser, D. A. S. (1972). Bayes, likelihood or structural. Ann. Math. Statist. 43 777–790. · Zbl 0281.62005 |

[13] | Fraser, D. A. S. (1979). Inference and Linear Models. McGraw–Hill, New York. · Zbl 0455.62052 |

[14] | Fraser, D. A. S. (2003). Likelihood for component parameters. Biometrika 90 327–339. · Zbl 1035.62012 |

[15] | Fraser, D. A. S. and McDunnough, P. (1980). Some remarks on conditional and unconditional inference for location–scale models. Statist. Hefte 21 224–231. · Zbl 0444.62045 |

[16] | Fraser, D. A. S., Monette, G., Ng, K. W. and Wong, A. (1994). Higher order approximations with generalized linear models. In Multivariate Analysis and Its Applications (T. W. Anderson, K. T. Fang and I. Olkin, eds.) 253–262. IMS, Hayward, CA. |

[17] | 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. · Zbl 0831.62016 |

[18] | Fraser, D. A. S. and Reid, N. (1995). Ancillaries and third order significance. Utilitas Math. 47 33–53. · Zbl 0829.62006 |

[19] | Fraser, D. A. S. and Reid, N. (2001). Ancillary information for statistical inference. Empirical Bayes and Likelihood Inference. Lecture Notes in Statist. 148 185–209. Springer, New York. |

[20] | Fraser, D. A. S. and Reid, N. (2003). Strong matching of frequentist and Bayesian parametric inference. J. Statist. Plann. Inference 103 263–285. · Zbl 1005.62005 |

[21] | Fraser, D. A. S., Reid, N., Li, R. and Wong, A. (2003). \(p\)-value formulas from likelihood asymptotics: Bridging the singularities. J. Statist. Res. 37 1–15. |

[22] | Fraser, D. A. S., Reid, N. and Wong, A. (1997). Simple and accurate inference for the mean of the gamma model. Canad. J. Statist. 25 91–99. · Zbl 0876.62014 |

[23] | 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. · Zbl 0932.62003 |

[24] | 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. · Zbl 0998.62059 |

[25] | Fraser, D. A. S. and Yi, G. Y. (2002). Location reparameterization and default priors for statistical analysis. J. Iranian Statist. Soc. 1 55–78. · Zbl 06657064 |

[26] | Godambe, V. P. (1982). Ancillarity principle and a statistical paradox. J. Amer. Statist. Assoc. 77 931–933. · Zbl 0512.62008 |

[27] | Godambe, V. P. (1985). Discussion of “Resolution of Godambe’s paradox,” by C. Genest and M. J. Schervish. Canad. J. Statist. 13 300. · Zbl 0624.62004 |

[28] | 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. · Zbl 0425.60042 |

[29] | Reid, N. (1995). The roles of conditioning in inference (with discussion). Statist. Sci. 10 138–157, 173–196. · Zbl 0955.62524 |

[30] | Welch, B. L. (1939). On confidence limits and sufficiency, with particular reference to parameters of location. Ann. Math. Statist. 10 58–69. · Zbl 0020.38202 |

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.