### Yale University

Department of Statistics

Seminar

#### Wednesday, January 24, 1996

Yuhong Yang

Department of Statistics

Yale University

#### "Model Selection for Density Estimation "

Information-theoretic tools are used to derive minimax risk bounds for
density estimation. A metric entropy condition alone determines the minimax
rate of convergence in each class of density functions. To achieve the
minimax rates simultaneously for multiple function classes, we consider
lists of finite-dimensional
approximating models and use model selection criteria to select adaptively
a good model based on data. The use of many candidate models, as in the
case of subset selection, provides more flexibility for adaptation, yet
significant selection bias can occur with criteria such as AIC. We
incorporate a model complexity
term in the model selection criteria to handle this selection bias. It is
shown that the risk of the estimated density is bounded by an index of
resolvability, which characterizes the best tradeoff among approximation
error, estimation error, and model complexity. As an application, we show
that the optimal rate of convergence is simultaneously achieved for density
in the Sobolev space * W_2^s(U) * without knowing the smoothness
parameter * s * and norm parameter U in advance.

#### Seminar to be held in Room 107, 24 Hillhouse Ave at 12:15 pm