  Consider a semiparametric model with a Euclidean parameter and an
infinite-dimensional parameter, to be called a Banach parameter. Assume: (a)
There exists an efficient estimator of the Euclidean parameter. (b) When the
value of the Euclidean parameter is known, there exists an estimator of the
Banach parameter, which depends on this value and is efficient within this
restricted model. Substituting the efficient estimator of the Euclidean
parameter for the value of this parameter in the estimator of the Banach
parameter, one obtains an efficient estimator of the Banach parameter for the
full semiparametric model with the Euclidean parameter unknown. This hereditary
property of efficiency completes estimation in semiparametric models in which
the Euclidean parameter has been estimated efficiently. Typically, estimation
of both the Euclidean and the Banach parameter is necessary in order to
describe the random phenomenon under study to a sufficient extent. Since
efficient estimators are asymptotically linear, the above substitution method
is a particular case of substituting asymptotically linear estimators of a
Euclidean parameter into estimators that are asymptotically linear themselves
and that depend on this Euclidean parameter. This more general substitution
case is studied for its own sake as well, and a hereditary property for
asymptotic linearity is proved.
