In this research, some robust non-parametric methods were used to estimate the semi-parametric regression model, and then  these methods were compared using the MSE comparison criterion, different sample sizes, levels of variance, pollution rates, and three different models were used. These methods are S-LLS S-Estimation -local smoothing, (M-LLS)M- Estimation -local smoothing, (S-NW) S-Estimation-NadaryaWatson Smoothing, and (M-NW) M-Estimation-Nadarya-Watson Smoothing.

The results in the first model proved that the (S-LLS) method was the best in the case of large sample sizes, and small sample sizes showed that the

As the process of  estimate for model and variable selection significant is a crucial process in the semi-parametric modeling At the beginning of the modeling process often At there are many explanatory variables to Avoid the loss of any explanatory elements may be important as a result , the selection of significant variables become necessary , so the process of variable selection is not intended to simplifying  model complexity explanation , and also predicting. In this research was to use some of the semi-parametric methods (LASSO-MAVE , MAVE and The proposal method (Adaptive LASSO-MAVE) for variable selection and estimate semi-parametric single index model (SSIM) at the same time .

Let  be a commutative ring with 1 and  be left unitary  . In this paper we introduced and studied concept of semi-small compressible module (a     is said to be semi-small compressible module if  can be embedded in every nonzero semi-small submodule of . Equivalently,  is  semi-small compressible module if there exists a monomorphism  , ,     is said to be semi-small retractable module if  , for every non-zero  semi-small sub module in . Equivalently,  is semi-small retractable if there exists a homomorphism  whenever  .     In this paper we introduce and study the concept of semi-small compressible and semi-small retractable s as a generalization of compressible  and retractable  respectively and give some of their adv

A submoduleA of amodule M is said to be strongly pure , if for each finite subset {ai} in A , (equivalently, for each a ?A) there exists ahomomorphism f : M ?A such that f(ai) = ai, ?i(f(a)=a).A module M is said to be strongly F–regular if each submodule of M is strongly pure .The main purpose of this paper is to develop the properties of strongly F–regular modules and study modules with the property that the intersection of any two strongly pure submodules is strongly pure .

The concept of -closedness, a kind of covering property for topological spaces, has already been studied with meticulous care from different angles and via different approaches. In this paper, we continue the said investigation in terms of a different concept viz. grills. The deliberations in the article include certain characterizations and a few necessary conditions for the -closedness of a space, the latter conditions are also shown to be equivalent to -closedness in a - almost regular space. All these and the associated discussions and results are done with grills as the prime supporting tool.

We define and study new ideas of fibrewise topological space on D namely fibrewise multi-topological space on D. We also submit the relevance of fibrewise closed and open topological space on D. Also fibrewise multi-locally sliceable and fibrewise multi-locally section able multi-topological space on D. Furthermore, we propose and prove a number of statements about these ideas.