Ancillary Information for Statistical Inference
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.
KeywordsLeukemia Covariance Eter Dinate
Unable to display preview. Download preview PDF.
- Abebe, F., D.A.S. Fraser, N. Reid, and A. Wong (1996). Nonlinear regression: third order significance. Utilitas Mathematica 7, 1–17.Google Scholar
- Feigl, P. and M. Zelen (1965). Estimation of exponential survival probability with concomitant information. Biometrika 21, 826–838.Google Scholar
- 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
- 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