Information on parameters of interest decreases under transformations

An important property of Fisher information is that it decreases weakly under transformation of random variables. Kagan and Rao (2003) [A. Kagan, C.R. Rao, Some properties and applications of the efficient Fisher score, J. Statist. Plann. Inference 116 (2003) 343–352] showed that, in the presence of nuisance parameters, Fisher information on the interest parameters decreases similarly. We prove here a general algebraic result on partitioned positive-definite matrices, and use it to show that the decrease in Fisher information on the interest parameters is bounded below by the conditional Fisher information on the interest parameters. A consequence is that standard large-sample confidence regions for parameters of interest based on the deviance, score and Wald statistics become asymptotically ‘wider’ under transformations, both in the context of independent identically distributed random variables and for the binomial detectability models of Fewster and Jupp (2009) [R.M. Fewster, P.E. Jupp, Inference on population size in binomial detectability models, Biometrika 96 (2009) 805–820]. One implication is that models that combine different data sources for inference on the interest parameters are asymptotically more efficient than models for any of the individual data sources, despite the possible need for further nuisance parameters when combining the sources.