# Ancillary Information for Statistical Inference

• D. A. S. Fraser
• N. Reid
Chapter
Part of the Lecture Notes in Statistics book series (LNS, volume 148)

## Abstract

Inference for a scalar interest parameter in the presence of nuisance parameters is obtained in two steps: an initial ancillary reduction to a variable having the dimension of the full parameter and a subsequent maxginalization to a scalar pivotal quantity for the component parameter of interest. Recent asymptotic likelihood theory has provided highly accurate third order approximations for the second maxginalization to a pivotal quantity, but no general procedure for the initial ancillary conditioning. We develop a second order location type ancillary generalizing (1995) and a first order affine type ancillary generalizing (1980). The second order ancillary leads to third order p-values and the first order ancillary leads to second order p-values. For an n dimensional variable with p dimensional parameter, we also show that the only information needed concerning the ancillary is an array V = (v 1 ... v p ) of p linearly independent vectors tangent to the ancillary surface at the observed data. For the two types of ancillarity simple expressions for the array V are given; these are obtained from a full dimensional pivotal quantity. A brief summary describes the use of V to obtain second and third order p-values and some examples are discussed.

## Keywords

Statistical Inference Exponential Model Tangent Direction Nuisance Parameter Component Parameter
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 9 References

1. Abebe, F., D.A.S. Fraser, N. Reid, and A. Wong (1996). Nonlinear regression: third order significance. Utilitas Mathematica 7, 1–17.Google Scholar
2. Barndorff-Nielsen, O.E. (1980). Conditionality resolutions. Biometrika 67, 293–310.
3. Barndorff-Nielsen, O.E. (1986). Inference on full or partial parameters based on the standardized, signed log likelihood ratio. Biometrika 73, 307–322.
4. Barndorff-Nielsen, O.E. (1991). Modified signed log likelihood ratio. Biometrika 78, 557–563.
5. Barndorff-Nielsen, O.E. and S.R. Chamberlin (1994). Stable and invariant adjusted directed likelihoods. Biometrika 81, 485–499.
6. Brenner, D., D.A.S. Fraser, and P. McDunnough (1982). On asymptotic normality of likelihood and conditional analysis. Canad. J. Statist. 10, 163–172.
7. Cox, D.R. and E.J. Snell (1981). Applied Statistics. London: Chapman and Hall.
8. DiCiccio, T.J., C.A. Field, and D.A.S. Fraser (1990). Approximation of marginal tail probabilities and inference for scalar parameters. Biometrika 77, 77–95.
9. DiCiccio, T.J. and M.A. Martin (1993). Simple modifications for signed likelihood ratio statistics. J. Roy. Statist. Soc. Ser. B 55, 305–316.
10. Feigl, P. and M. Zelen (1965). Estimation of exponential survival probability with concomitant information. Biometrika 21, 826–838.Google Scholar
11. Fraser, D.A.S. (1964). Local conditional sufficiency. J. Roy. Statist. Soc. Ser. B 26, 52–62.
12. Fraser, D.A.S. (1988). Normed likelihood as saddlepoint approximation. J. Mult. Anal. 27, 181–193.
13. Fraser, D.A.S., P. McDunnough, and N.A. Taback (1997). Improper priors, posterior asymptotic normality, and conditional inference. In N. Johnson and N. Balakrishnan (Eds.), Advances in the Theory and Practice of Statistics, pp. 563–569. New York: Wiley.Google Scholar
14. Fraser, D.A.S., G. Monette, K.W. Ng, and A. Wong (1994). Higher order approximations with generalized linear models. In T. Anderson, K. Fang, and I. Olkin (Eds.), Multivariate Analysis and Its Applications, Volume 24 of Inst. Math. Statist. Lect. Notes and Monograph Series, Hayward, pp. 253–262.Google Scholar
15. Fraser, D.A.S. and N. Reid (1993). Simple asymptotic connections between densities and cumulant generating function leading to accurate approximations for distribution functions. Statist. Sinica 3, 67–82.
16. Fraser, D.A.S. and N. Reid (1995). Ancillaries and third order significance. Utilitas Mathematica 47, 33–53.
17. Fraser, D.A.S., N. Reid, and J. Wu (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86, 249–264.
18. Fraser, D.A.S., A. Wong, and J. Wu (1999). Regression analysis, nonlinear or nonnormal: simple and accurate p-values from likelihood analysis. J. Amer. Statist Assoc. 94, 1286–1295.
19. Lugannani, R. and S.O Rice (1980). Saddlepoint approximation for the distribution of the sums of independent random variables. Adv. Appl. Prob. 12, 475–490.
20. Pierce, D.A. and D. Peters (1992). Practical use of higher order asymptotics for multiparameter exponential families (with discussion). J. Roy. Statist. Soc. Ser. B 54, 701–738.
21. Skovgaard, I.M. (1986). Successive improvements of the order of ancillarity. Biometrika 73, 516–519.