The semi-parametric regression model is an intermediate case between the fully parametric and fully nonparametric model. There are some of different definitions of semi-parametric models. This dissertation concentrates on a semi-parametric model which contains a nonparametric component and a parametric component and estimating the parametric component using Bayesian theorem.
The semi-parametric model has the advantages of both parametric and nonparametric models. This type of model tries to combine the flexibility of a nonparametric model with the advantages of a parametric model. A fully nonparametric model will be more robust than semi-parametric and parametric models since the first doesn't suffer from risk of misspecification. On the other hand, nonparametric estimators have low convergence rates, which deteriorate when considering higher order derivatives and multidimensional random variables. In contrast, the parametric model carries a risk of misspecification but if it is correctly specified it will normally enjoy √n-consistency with no deterioration caused by derivatives and multivariate data.
This dissertation provides a new bayesian estimation and inference of the unknown parameters for the generalized partial linear model (GPLM) and the semi-parametric logistic regression model (SLRM) using some multivariate prior distributions under the square error loss function.
Anew algorithm for estimating the GPLM and SLRM parameters by using Bayesian theorem presented as follows:
Algorithm (1) Obtain the probability distribution of response variable▁Y.
Algorithm (2) Obtain the likelihood function of the probability distribution of response variable ▁Y.
Algorithm (3) Choosing Suitable prior distribution of β.
Algorithm (4) Obtain the posterior distribution.
Algorithm (5) Calculate bayesian estimator under the square error loss function.
Algorithm (6) Replace bayesian estimator with the initial value of.
Algorithm (7) Use Profile likelihood method, generalized speckman method and Backftting method with the new initial value of β to estimate (GPLM).
Algebraic derivations for obtaining the posterior distribution of response variable▁Y and for Calculating bayesian estimators under the square error loss function are presented in more detail.
The usefulness of the two models is illustrated by using a real and generated data for estimating the GPLM and SLRM parameters. The estimation results and statistical analysis are obtained by using a statistical package called (XploRe, 4.8).
The results of simulation study are based on 500 replications, a certain sample size (n=50, n=100, n=200, n=500, n=100, n=2000), and a certain bandwidth parameter (h=2, h=1, h=0.5, h=0.2), the real data set consists of 400 observations on business loans and 14 covariates. This dissertation consists of seven chapters as follows:-
Chapter one:
This chapter contains a general introduction to this study, describes the issue of the study, the area of research and the structure of this dissertation.
Chapter two:
The aim of this chapter is to present the parametric regression models, literature review of previous studies and method of estimation.
Chapter three:
This chapter introduces the non parametric regression models, literature review of previous studies and method of estimation.
Chapter four:
The generalized partial linear model (GPLM), Semi-parametric logistic regression model (SLRM) and other semi-parametric regression models are presented in more detail in this chapter.
Chapter five:
The aim of this chapter is to present Bayesian theorem, algebraic derivations for obtaining the posterior distribution of response variable▁Y , Calculate bayesian estimators under the square error loss function and the new algorithm for estimating the GPLM and SLRM parameters by using Bayesian theorem.
Chapter six:
This chapter introduces
1. Simulation study with two stages.
2. An application to real data.
Chapter seven:
This chapter contains all results of this dissertation and some future researches.
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