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Determining effects of controllable factors on flawless operations of tanker terminals: Tupras case study
Published in Selma Ergin, C. Guedes Soares, Sustainable Development and Innovations in Marine Technologies, 2022
Generalized Linear Models (GLM), introduced by Nelder & Wedderburn (1972), are a combination of linear and non-linear regression models that allow the choice of one of the members of the very rich and flexible family, called the exponential distribution family, as a response distribution by the researcher. The Normal, Binomial, Poisson, Exponential, Gamma, Inverse Normal, and Negative Binomial distributions are all members of the exponential family. Normality and constant variance assumptions are not required with Generalized Linear Models. However, variance in all exponential family members is a function of the mean. A Generalized Linear Model consists of three components. These components are: Distribution of the responseSystematic structureLink function.
Generalized Linear Models
Published in Taylor Arnold, Michael Kane, Bryan W. Lewis, A Computational Approach to Statistical Learning, 2019
Taylor Arnold, Michael Kane, Bryan W. Lewis
An exponential family is a class of probability distributions that all share a specific form of density functions. The class is primarily defined for mathematical convenience, with general results allowing for the computation of properties such as sufficient statistics and specification of closed‐form conjugate priors. Both discrete and continuous distributions are contained in the family. Specifically, single‐parameter distributions in the exponential family are those that can be written as f(zθ)=h(z)·exp{η(θ)·T(z)-A(θ)} $$ f(\left. z \right|\theta ) = h(z) \cdot {\text{~exp~}}\{ \eta (\theta ) \cdot T(z) - A(\theta )\} $$
Generalized least squares and the analysis of heteroscedasticity
Published in Raymond J. Carroll, David Ruppert, Transfor mation and Weighting in Regression, 2017
Raymond J. Carroll, David Ruppert
These two examples indicate that generalized least-squares estimates might be maximum-likelihood estimates for a wide class of models. The Poisson and gamma distributions are examples of distributions in the exponential family (see McCullagh and Nelder, 1983). The probability density or mass function of these distributions is given as () exp{[yη−b(η)]/σ2+c(y,σ)}
A simulation-based estimation method for bias reduction
Published in IISE Transactions, 2018
The uniform convergence in probability can be easily verified for some frequently used distributions. For instance, let be the MLE of the mean of an exponential distribution with mean equal to θ. We have because − 1/n∑ni = 1log (ωi) converges to one in probability by the weak law of large numbers (Feller, 1968). Therefore, converges to θ in probability uniformly. Generally, it can also be verified that under some conditions, the MLE of parameters of an exponential family of distributions converge in probability uniformly. The details are included in the Appendix.
A penalized-likelihood approach to characterizing bridge-related crashes in New Jersey
Published in Traffic Injury Prevention, 2021
Mohammad Jalayer, Mahdi Pour-Rouholamin, Deep Patel, Subasish Das, Hooman Parvardeh
Firth introduced a penalized MLE into the binary model that can counteract the bias associated with MLE (Firth 1993). The log-likelihood can be expressed as an exponential family model as follows: where βn is the regression parameter to be estimated, t is the vector of the observed sufficient statistics, and K is the number of parameters estimated. The derivative of the log-likelihood, the score function, is employed to calculate the MLE of parameter βn as follows:
A novel approach for non-normal multi-response optimisation problems
Published in International Journal of Production Research, 2021
Jianjun Wang, Yanan Tu, Yan Ma, Linhan Ouyang, Yiliu Tu
Generalised linear models (GLMs), firstly introduced by Nelder and Wedderburn (1972), have been developed to fit regression models for response data that follows a general distribution called the exponential family. The exponential family includes a large number of probability distributions such as normal, binomial, Poisson, Gamma, exponential, and negative binomial distributions. The GLM allows the linear model to be related to the response variable via a link function and assumes the variance of the response to be a function of its predicted mean value (Myers, Montgomery, and Vinning 2002).