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Likelihood p-values ancillaries and the vector quantile function

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Likelihood p-values and ancillaries have been around for much longer than most would acknowledge, yet continue to be clouded by lack of agreement and direction. Asymptotics from Daniels to Barndorff-Nielsen has has extended core versions to give highly accurate approximations in more general contexts, but with variable and parameter essentially of the same dimension. More recently the vector quantile function obtained by inverting coordinate distribution functions to coordinate quantile functions has clarified the role of continuity and enabled the further extension to sample spaces with dimension larger than the parameter. We survey these developments and discuss two examples, nonlinear regression and Cauchy analysis; equivocal results are resolved.

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