Likelihood computations without Bartlett identities

Abstract

The signed square root statistic R is given by sgn( \hatθ- θ) ( l( \hatθ) - l( θ) )^1/2, where l is the log-likelihood and \hatθ is the maximum likelihood estimator. The pth cumulant of R is typically of the form n^-{p/2}k_p + O(n^-{p+2)/2) , where n is the number of observations. This paper shows how to symbolically compute k_p without invoking the Bartlett identities. As an application, we show how the family of alternatives influences the coverage accuracy of R.