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 I 1 and I 2 of Fisher information on θ ( 1 ) and θ ( 2 ) in Z ( 1 ) and Z ( 2 ) , it is proved that trace ( I - 1 ) ⩽ trace ( I 1 - 1 ) + trace ( I 2 - 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 ) ) .
Published Version
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