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 over(I, ^), over(I, ^) ⁽¹⁾ and over(I, ^) ⁽²⁾ are the efficient matrices of information on θ, θ⁽¹⁾, θ⁽²⁾ in Z, Z⁽¹⁾ and Z⁽²⁾, respectively, then trace (over(I, ^)) ≥ trace (over(I, ^) ⁽¹⁾) + trace (over(I, ^) ⁽²⁾).