Cumulative Slice Estimation Under Multivariate Mormal Mixture Distribution

In the field of statistics,it is an important issue to estimate the conditional distribution of the response given some indicators.When the relationship type between the response and the regressors is unknown,we will be faced with the non-parametric regression problem.If the number of regressors is small,we can directly fit the relationship between the response and the independent variables in non-parametric approaches.However,when the number of regressors is large,a problem called "curse of dimensionality" will invalidate the fit.In order to overcome such a problem,reducing the dimension of the independent variables is of great necessity.Most of the existing sufficient dimension reduction methods depend on some assumptions on the predictors.The first-order methods such as sliced inverse regression(SIR)require the linear design conditions.The second-order methods such as sliced average variance estimator(SAVE)and the principal Hessian direction(PHD)method require,in addition,the constant conditional variance assumption.Among these methods,as a method taking full use of sample information,cumulative slicing estimation(CUME)has a special advantage.It can get a more accurate estimation of the center subspace.Therefore,cumulative slicing estimation has received a lot of attention in the field of sufficient dimension reduction.However,linear design condition is not always satisfied in practice.Among those scenarios,mixture distributions are very typical,especially in the case of categorical data.Under mixture distributions,we cannot use the above methods to reduce the dimension of indicator vector.In view of this,we study how to improve cumulative slicing estimation to make it applicable with a typical and common mixture distribution--multivariate normal mixture distribution.We propose two methods.Using these two methods,we construct new kernel matrices,and obtain the estimators of dimension reduction space,directions and structural dimension.An algorithm is proposed.The proposedmethodologies are illustrated by simulation studies.