scholarly journals Penalized Likelihood Estimation of Gamma Distributed Response Variable via Corrected Solution of Regression Coefficients

2021 ◽  
Vol 19 (1) ◽  
Author(s):  
Rasaki Olawale Olanrewaju
Keyword(s):  
Generalized Linear Models ◽  
Linear Models ◽  
Penalized Likelihood ◽  
Likelihood Estimation ◽  
Regression Coefficients ◽  
Response Variable ◽  
Penalized Likelihood Estimation ◽  
Selection Operator ◽  
Minimax Concave Penalty

A Gamma distributed response is subjected to regression penalized likelihood estimations of Least Absolute Shrinkage and Selection Operator (LASSO) and Minimax Concave Penalty via Generalized Linear Models (GLMs). The Gamma related disturbance controls the influence of skewness and spread in the corrected path solutions of the regression coefficients.

Smoothing of X-ray diffraction data andKα2elimination using penalized likelihood and the composite link model

2014 ◽  
Vol 47 (3) ◽  
pp. 852-860 ◽  
Author(s):  
Johan J. de Rooi ◽  
Niek M. van der Pers ◽  
Ruud W. A. Hendrikx ◽  
Rob Delhez ◽  
Amarante J. Böttger ◽  
...  
Keyword(s):  
Linear Models ◽  
Penalized Likelihood ◽  
Likelihood Estimation ◽  
X Ray Diffraction ◽  
X Ray ◽  
Local Distortion ◽  
Penalized Likelihood Estimation ◽  
Log Likelihood ◽  
Link Model ◽  
Expected Values

X-ray diffraction scans consist of series of counts; these numbers obey Poisson distributions with varying expected values. These scans are often smoothed and theKα2component is removed. This article proposes a framework in which both issues are treated. Penalized likelihood estimation is used to smooth the data. The penalty combines the Poisson log-likelihood and a measure for roughness based on ideas from generalized linear models. To remove theKα doublet the model is extended using the composite link model. As a result the data are decomposed into two smooth components: aKα1and aKα2part. For both smoothing andKα2removal, the weight of the applied penalty is optimized automatically. The proposed methods are applied to experimental data and compared with the Savitzky–Golay algorithm for smoothing and the Rachinger method forKα2stripping. The new method shows better results with less local distortion. Freely available software in MATLAB and R has been developed.

Generalized Linear Models and Penalized Likelihood

2020 ◽  
pp. 227-285
Author(s):  
Jianqing Fan ◽  
Runze Li ◽  
Cun-Hui Zhang ◽  
Hui Zou
Keyword(s):  
Generalized Linear Models ◽  
Linear Models ◽  
Penalized Likelihood

More generalized linear modelling.

2021 ◽  
pp. 171-186
Author(s):  
Donald Quicke ◽  
Buntika A. Butcher ◽  
Rachel Kruft Welton
Keyword(s):  
Generalized Linear Models ◽  
Count Data ◽  
Binary Data ◽  
Linear Models ◽  
Explanatory Variable ◽  
Response Variable ◽  
Explanatory Variables ◽  
Continuous Response ◽  
Generalized Linear Modelling ◽  
Linear Modelling

Abstract This chapter employs generalized linear modelling using the function glm when we know that variances are not constant with one or more explanatory variables and/or we know that the errors cannot be normally distributed, for example, they may be binary data, or count data where negative values are impossible, or proportions which are constrained between 0 and 1. A glm seeks to determine how much of the variation in the response variable can be explained by each explanatory variable, and whether such relationships are statistically significant. The data for generalized linear models take the form of a continuous response variable and a combination of continuous and discrete explanatory variables.

scholarly journals Catalytic prior distributions with application to generalized linear models

2020 ◽  
Vol 117 (22) ◽  
pp. 12004-12010
Author(s):  
Dongming Huang ◽  
Nathan Stein ◽  
Donald B. Rubin ◽  
S. C. Kou
Keyword(s):  
Generalized Linear Models ◽  
Linear Models ◽  
Synthetic Data ◽  
Interval Estimation ◽  
Likelihood Estimation ◽  
Predictive Distribution ◽  
Point Estimation ◽  
Tuning Parameter ◽  
Working Model ◽  
Coverage Accuracy

A catalytic prior distribution is designed to stabilize a high-dimensional “working model” by shrinking it toward a “simplified model.” The shrinkage is achieved by supplementing the observed data with a small amount of “synthetic data” generated from a predictive distribution under the simpler model. We apply this framework to generalized linear models, where we propose various strategies for the specification of a tuning parameter governing the degree of shrinkage and study resultant theoretical properties. In simulations, the resulting posterior estimation using such a catalytic prior outperforms maximum likelihood estimation from the working model and is generally comparable with or superior to existing competitive methods in terms of frequentist prediction accuracy of point estimation and coverage accuracy of interval estimation. The catalytic priors have simple interpretations and are easy to formulate.

Posterior contraction in sparse generalized linear models

Biometrika ◽  
2020 ◽  
Author(s):  
Seonghyun Jeong ◽  
Subhashis Ghosal
Keyword(s):  
Generalized Linear Models ◽  
Bayesian Methods ◽  
Generalized Linear Model ◽  
Linear Models ◽  
Continuous Distribution ◽  
Fractional Power ◽  
Regression Coefficients ◽  
Link Function ◽  
Convergence Properties ◽  
Set Up

Summary We study posterior contraction rates in sparse high-dimensional generalized linear models using priors incorporating sparsity. A mixture of a point mass at zero and a continuous distribution is used as the prior distribution on regression coefficients. In addition to the usual posterior, the fractional posterior, which is obtained by applying Bayes theorem with a fractional power of the likelihood, is also considered. The latter allows uniformity in posterior contraction over a larger subset of the parameter space. In our set-up, the link function of the generalized linear model need not be canonical. We show that Bayesian methods achieve convergence properties analogous to lasso-type procedures. Our results can be used to derive posterior contraction rates in many generalized linear models including logistic, Poisson regression and others.

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