Equivalence of Maximum Likelihood and Modified Least Squares for Generalized Linear Model | Chapter 2 | Research Updates in Mathematics and Computer Science Vol. 5

In this chapter, we focus, on the one hand, on the estimation of the linear mixed model parameters and, on the other hand, on the solution of the generalized estimating equations. The most frequently used models for statistical data analysis are regression models. In linear regression, the objective is to study the relationship between a response variable (explained variable) and one or more explanatory variables, based on linear models (LM). The estimation of the considered parameters model is one of the most crucial processes in the statistical data analysis process. The greatest likelihood and least squares are the two most used estimation techniques. There is no preference for one approach over the other when the data are gaussian as the two procedures are equivalent. However, if the data are not gaussian, this equivalence is no longer valid. Also, if the normal equations are not linear, we make use of iterative methods (Newton-Raphson algorithm, Fisher, etc ...). In this work, we consider a particular case where the data are not normal and solving equations are not linear and that it leads to the equivalence between the maximum likelihood and the least squares methods, but the last is modified. In addition, we concluded by referring to the application of this modified method for solving the equations of Liang and Zeger. At the end of the work, we showed the existing relationship between the maximum likelihood for the GLM and the GEE resolution method which is the iteratively reweighted least squares ones (IRLS).

Author(s) Details:

Ahsene Lanani,
Department of Mathematics, Mentouri Brothers University of Constantine 1, Algeria.

Rahima Benchabi,
Department of Mechanical Engineering, Mentouri Brothers University of Constantine1, Algeria.

Please see the link here: https://stm.bookpi.org/RUMCS-V5/article/view/14213

The Non-Informative and Informative Prior Distributions in Estimating Variance Parameter of Normal Distribution | Chapter 7 | Research Updates in Mathematics and Computer Science Vol. 4

This chapter aims to elucidate the concepts of non-informative and informative prior distributions concerning the variance, a crucial component of the unknown parameter within a normal distribution. The variance serves as a representation of the distribution's spread or variability. The estimation variance is exhibited in the point and interval estimations based on non-informative and informative prior distributions. Point estimation entails furnishing a specific value to estimate a population parameter. In contrast, interval estimation provides a range or interval of values to estimate a population parameter, commonly called a confidence interval. While non-informative priors express a need for prior knowledge, informative priors bring valuable insight into the modeling process. The non-informative priors encompass the maximum likelihood method, the Jackknife method, and the bootstrap method. Maximum likelihood is widely recognized for approximating parameters and boasting properties such as unbiased estimation, consistency, and efficiency. The informative prior distribution employs the Bayesian and Markov Chain Monte Carlo methods. These methods involve the prior distribution, given the probability distribution and approach to the posterior distribution.

Author(s) Details:

Autcha Araveeporn,
Department of Statistics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok-10520, Thailand.

Please see the link here: https://stm.bookpi.org/RUMCS-V4/article/view/14101

The Parametric and Nonparametric Estimations of Normal Distribution for Mean Parameter | Chapter 12 | Research and Applications Towards Mathematics and Computer Science Vol. 7

 This stage's objective describes the point and pause estimations on the mean that is a part of life after death parameter of sane distribution. The mean represents the allocation's central locale or average, while the standard deviation measures the spread or instability of the data.   Point estimation includes providing a specific advantage to estimate a population limit. The interval estimation supplies a range or interval of principles to estimate a population parameter, named a confidence pause. These estimations are estimated for one parametric method by using the maximum prospect, Bayesian, and Markov Chain Monte Carlo procedures. The nonparametric method consists of the start operating system and the Jackknife arrangements. Maximum likelihood is the familiar method to approximate parameters cause there are the characteristics of the unbiased estimator, consistency, and adeptness estimator. The Bayesian method is a mathematical approach based on feasibility, prior, and posterior distribution. The Markov Chain Gambling establishment technique influences Markov chains and random sampling to estimate complex anticipation distributions and involves the Bayesian method. The start operating system and Jackknife forms are the resampling techniques by repeatedly illustration samples from the available dossier.

Author(s) Details:

Autcha Araveeporn,
Department of Statistics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok-10520, Thailand.

Somsri Banditvilai,
Department of Statistics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok-10520, Thailand.

Please see the link here: https://stm.bookpi.org/RATMCS-V7/article/view/12883

Study on Bayesian Regression Model and Applications| Chapter 8 | Novel Research Aspects in Mathematical and Computer Science Vol. 7

 We introduce a sparse vector regression model. Bayesian formulation and the Gaussian process are used to create the method. When compared to similar Bayesian vector regression models, the number of parameters in the method is minimised by applying a particular prior hyperparameter setting during the development process. The algorithm is created using an iterative computational approach. Examples of applications to the inverse scattering issue and function approximations are given.

Author(s) Details:

Yijun Yu,
Department of Mathematics, Tuskegee University, Tuskegee, AL, USA.

Please see the link here: https://stm.bookpi.org/NRAMCS-V7/article/view/7953

A Simulation Study to Investigate the Performance of Survival Mixture Model of Three Different Distributions to Anlayze Heterogeneous Data | Chapter 14 | Current Topics on Mathematics and Computer Science Vol. 1

 A simulation study of a parametric mixture model with three different distributions is considered in this paper to model heterogeneous survival data. The proposed parametric mixture of Exponential, Gamma, and Weibull has some properties that are investigated. To estimate the maximum likelihood estimators of three different postulated parametric mixture model parameters, the Expectation Maximization Algorithm (EM) is used. The simulations are carried out by simulating data sampled from a population of three component parametric mixtures of three different distributions, and they are repeated 10, 30, 50, 100, and 500 times to investigate the effects. The EM scheme's consistency and stability The developed EM Algorithm scheme can estimate mixture parameters that are very close to the parameters of the postulated model. As the number of simulation repetitions increases, the parameters get closer and closer to the postulated models, with relatively small standard errors. The results showed that the EM estimated the parameters of the three component mixture model correctly.

Author(s) Details

Yusuf Abbakar MohammedI
Deptartment of Mathematical Sciences, Faculty of Sciences, University of Maiduguri, Nigeria.

Bidin Yatim
School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, Malaysia.

Suzilah Ismail
School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, Malaysia.

View Book :-  https://stm.bookpi.org/CTMCS-V1/article/view/1395