Asymptotic Behaviour of the Modified Likelihood Root
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We examine the normal approximation of the modified likelihood root, an inferential tool from higher-order asymptotic theory, for the linear exponential and location-scale family. We show that the r^⋆ statistic can be thought of as a location and scale adjustment to the likelihood root r up to O_p(n^-3/2), and more generally r^⋆ can be expressed as a polynomial in r. We also show the linearity of the modified likelihood root in the likelihood root for these two families.
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