  This paper studies minimax rates of convergence for nonparametric
location-scale models, which include mean, quantile and expectile regression
settings. Under Hellinger differentiability on the error distribution and other
mild conditions, we show that the minimax rate of convergence for estimating
the regression function under the squared $L_2$ loss is determined by the
metric entropy of the nonparametric function class. Different error
distributions, including asymmetric Laplace distribution, asymmetric connected
double truncated gamma distribution, connected normal-Laplace distribution,
Cauchy distribution and asymmetric normal distribution are studied as examples.
Applications on low order interaction models and multiple index models are also
given.
