Sub- and superadditivity a la Carlen of matrices related to the fisher information

Abstract

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(I^-1₁)+trace(I^-1₂). 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(I¹)+trace(I¹).