Seminarium i matematisk statistik
fredagen den 11 februari 13.15 i MH:227

Natalia Markovich
Institute of Control Sciences
Russian Academy of Science, Moscow

Titel:
Nonparametric estimation of the distribution density
and its functions by the regularization method

Abstract:
The restoration of the distribution density and its
functions such as hazard rate, renewal functions on a sequence of 
independent observations of some random variable is considered 
as an inverse problems of the approximate solution of the 
Fredholm's integral equations.

This means, that a stochastically ill-posed problem with an
inaccurately specified right-hand side function
and sometimes an operator is solved. The regularization method 
to the solution of this problem is applied. 
This method gives the possibility to
obtain various estimators as for the distribution density
(kernel estimators, projection estimators, histogram estimators)
as for its functions.

The main problem of regularization method is the choice 
of regularization parameter. This smoothing parameter is determined 
to obtain the most statistically reliable solutions of the operator
equation for the given number of empirical data. 
To select this parameter different versions of the discrepancy 
method on empirical sample are used as well as a priori 
parameter choice strategies, where the parameter is chosen 
as a function of sample size before 
the calculations begin. Since the main problem of the application
of the dicrepancy method is that the value of the 
discrepancy in practice is given in a very 
inaccurate manner then a selection of the discrepancy as a mode 
of distribution of Kolmogorov-Smirnov's
or Mises-Smirnov's statistics is proposed.

The convergence rates in L2 of the projection estimate 
for the probability density with a bounded variation of the 
kth derivative, obtained by the method of stochastic regularization 
is determined in the case when the regularization parameter 
is selected by the discrepancy method and as a function of
the sample size.The convergence in C for the regularized estimates 
of the hazard rate as well as a convergence rate in L2 metric 
in the case of a limited variation of the kth derivative of 
the hazard rate are considered.