Abstract: We examine the normal approximation to the distribution of the modified likelihood root, an inferential tool of higher-order asymptotic theory, for the linear exponential and location-scale families. We show that the modified likelihood root, r^⋆, can be expressed as a location and scale adjustment to the likelihood root, r, to O_p(n^–3/2), and more generally can be expressed as a polynomial in r. We discuss some extensions of these results to the high-dimensional regime.

Key words and phrases: High-dimensional statistics, higher-order asymptotics, linear exponential families, location-scale families, statistical inference.