  Linear regression is a fundamental and popular statistical method. There are
various kinds of linear regression, such as mean regression and quantile
regression. In this paper, we propose a new one called distribution regression,
which allows broad-spectrum of the error distribution in the linear regression.
Our method uses nonparametric technique to estimate regression parameters. Our
studies indicate that our method provides a better alternative than mean
regression and quantile regression under many settings, particularly for
asymmetrical heavy-tailed distribution or multimodal distribution of the error
term. Under some regular conditions, our estimator is $\sqrt n$-consistent and
possesses the asymptotically normal distribution. The proof of the asymptotic
normality of our estimator is very challenging because our nonparametric
likelihood function cannot be transformed into sum of independent and
identically distributed random variables. Furthermore, penalized likelihood
estimator is proposed and enjoys the so-called oracle property with diverging
number of parameters. Numerical studies also demonstrate the effectiveness and
the flexibility of the proposed method.
