MINIMUM-PENALIZED DISTANCE ESTIMATORS OF A DENSITYAND ITS DERIVATIVES
Lesław Gajek
Abstract: Let be an -times differentiable distribution function (df)
generating the data. Let be the density of Let denote the empirical
df. The paper concerns fitting an -times differentiable function to
the data by minimizing where denotes the -norm and is a sequence of smoothing parameters.
Let be an (approximate) minimizer of the above problem. We establish an
upper bound for with respect to the choice of
In particular, the choice of results in the optimal -rate of
convergence of to The estimation is also evaluated.
be an 
-times differentiable distribution function (df)
generating the data. Let 
be the density of 
Let 
denote the empirical
df. The paper concerns fitting an 
-times differentiable function 
to
the data by minimizing 
where 
denotes the 
-norm and 
is a sequence of smoothing parameters.
Let 
be an (approximate) minimizer of the above problem. We establish an
upper bound for 
with respect to the choice of
In particular, the choice of 
results in the optimal 
-rate of
convergence of 
to 
The estimation 
is also evaluated.