Information bias and adjusted profile likelihoods

The bias and information bias of the ordinary profile score statistic are both typically of order O(1). Several additive adjustments to the profile score statistic that reduce its bias to order O(n(-1)) have been proposed. In certain situations, the information bias of these adjusted profile score statistics is also reduced to order O(n(-1)); specifically, the modified profile likelihood of Bamdorff-Nielsen yields an adjusted profile score statistic having both reduced bias and reduced information bias. In general, however, a bias reducing adjustment to the profile score statistic will not automatically reduce the order of the information bias as well. In this paper, an analytical formula is obtained for the information bias of a general bias-adjusted profile score statistic. This formula is used to compare bias reducing adjustments to the profile score statistic, as well as to construct further adjustments that reduce the information bias to O(n(-1)). Several examples are presented to illustrate use of the formula for information bias. In particular, the information bias formula may be utilized in a criterion for choosing between orthogonal parameterizations for the conditional profile likelihood of Cox and Reid.