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Asymptotic Behaviour of the Modified Likelihood Root

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 $r^\star$ 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 $r^\star$ 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.