Sub- and superadditivity à la Carlen of matrices related to the Fisher information

Let Z=(Z(1),Z(2)) be an s-variate random vector partitioned into r- and q-variate subvectors whose distribution depends on an s-variate location parameter θ=(θ(1),θ(2)) partitioned in the same way as Z. For the s×s matrix I of Fisher information on θ contained in Z and r×r and q×q matrices I1 and I2 of Fisher information on θ(1) and θ(2) in Z(1) and Z(2), it is proved that trace(I-1)⩽trace(I1-1)+trace(I2-1). The inequality is similar to Carlen's superadditivity but has a different statistical meaning: it is a large sample version of an inequality for the covariance matrices of Pitman estimators. If the distribution of Z depends also on an m-variate nuisance parameter η (of a general nature) and I^,I^(1) and I^(2) are the efficient matrices of information on θ,θ(1),θ(2) in Z,Z(1) and Z(2), respectively, then trace(I^)⩾trace(I^(1))+trace(I^(2)).