CONSTRUCTING APPROXIMATELY STANDARD NORMAL PIVOTS FROM SIGNED ROOTS OF ADJUSTED LIKELIHOOD RATIO STATISTICS

For inference about a scalar parameter psi in the presence of nuisance parameters, several authors have suggested that the usual log profile likelihood function M(psi) be replaced by an objective function of th e form ($) over bar M(psi) = M(psi) + B(psi), where the derivatives of the adjustment function B(psi) are of order O-p(1). An adjusted likel ihood ratio statistic ($) over bar(psi)) can be defined in terms of ($ ) over bar M(psi). The distribution of ($) over bar(psi), the signed r oot of ($) over bar W(psi, is typically standard normal to error of or der O(n(-1/2)). This paper concerns the use of mean and variance corre ctions to construct approximate pivots based on ($) over bar(psi) that have the standard normal distribution to error of order O(n(-3/2)). G eneral formulae for the mean and variance corrections are provided, an d these formulae are evaluated for specific adjustment functions B(psi ) that have been proposed in the literature. Use of the corrections is illustrated in numerical examples.