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.
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Fraser, D.A.S., Reid, N. (2001). Ancillary Information for Statistical Inference. In: Ahmed, S.E., Reid, N. (eds) Empirical Bayes and Likelihood Inference. Lecture Notes in Statistics, vol 148. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0141-7_12
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DOI: https://doi.org/10.1007/978-1-4613-0141-7_12
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