Test statistics arising from quasi likelihood: Bartlett adjustment and higher-order power

With reference to the quasi-likelihood arising from an unbiased estimating function, we consider a large class of test statistics which includes the likelihood ratio, Rao's score and Wald's statistics in particular. We study Bartlett adjustability and third-order power in a possibly non-iid setting and provide explicit formulae. Since the relevant Bartlett identities may not hold while working with a quasi-likelihood, our results can differ from those based on the usual likelihood. The prospects regarding posterior Bartlett adjustability have been briefly indicated.