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For example, in a health survey, d may consist of height, weight, smoking status and other variables of interest. In principle, the value of this variable is observable for each of the N units that make up the surveyed population, which we denote by U . Let DU denote the matrix containing these N values. When the survey variable is a scalar, then DU becomes a vector of N elements, which we denote by the corresponding lower case dU . We assume that DU is generated from a distribution with density DU , which is completely known except for a parameter vector θ.

This leads −1 z s and µz = z U . to the maximum likelihood estimates ν = y s − σyz σzz Since −1 µd = ν + σdz σzz µz , it immediately follows from the invariance properties of maximum likelihood that the maximum likelihood estimate of µ is µ= = µd µz = −1 ν + σdz σzz µz µz −1 ds + σdz σzz (z U − z s ) zU . 5) is not the sample mean vector (ds , z s )T that maximizes log ds , zs , but rather this sample mean vector modified by the imbalance between the nonsample and sample z-values, as well as the strength of the relationship between d and z, as measured ILLUSTRATIVE EXAMPLES WITH COMPLETE RESPONSE 35 by the covariance terms in V.

The important concepts of sufficiency and ancillarity give a more refined view of where in the likelihood this information is and is not contained. We refer to ancillarity in this book so it is important to give a definition; however, we only use it occasionally so the following definitions can be skipped at a first reading. BIBLIOGRAPHIC NOTES 21 A statistic t(Dobs ) is a sufficient statistic for a model if the conditional distribution of Dobs given t(Dobs ) = t is the same for all distributions in the model.

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