  This paper studies a class of exponential family models whose canonical
parameters are specified as linear functionals of an unknown
infinite-dimensional slope function. The optimal minimax rates of convergence
for slope function estimation are established. The estimators that achieve the
optimal rates are constructed by constrained maximum likelihood estimation with
parameters whose dimension grows with sample size. A change-of-measure
argument, inspired by Le Cam's theory of asymptotic equivalence, is used to
eliminate the bias caused by the nonlinearity of exponential family models.
