scholarly journals PERBANDINGAN METODE MAXIMUM LIKELIHOOD DAN METODE BAYES DALAM MENGESTIMASI PARAMETER MODEL REGRESI LINIER BERGANDA UNTUK DATA BERDISTRIBUSI NORMAL

2019 ◽  
Vol 4 (2) ◽  
pp. 100
Author(s):  
Catrin Muharisa ◽  
Ferra Yanuar ◽  
Hazmira Yozza
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Gibbs Sampler

Analisis regresi merupakan salah satu metode untuk melihat hubungan antara variabel bebas (independent) dengan variabel terikat (dependent) yang dinyatakan dalam model regresi. Beberapa metode yang bisa digunakan untuk mengestimasi parameter model regresi, diantaranya adalah metode klasik dan metode Bayes. Salah satu metode klasik adalah metode maximum likelihood. Penelitian ini membahas tentang perbandingan metode maximum likelihood dan metode Bayes dalam mengestimasi parameter model regresi linear berganda untuk data berdistribusi normal. Adapun rumus untuk mengestimasi parameter dengan metode maximum likelihood adalah βˆ=(XTX)-1XTY dan ˆσ2 = 1 n P∞ k=1 ei. Sedangkan untuk mengestimasi parameter dengan metode Bayes adalah dengan menggunakan distribusi prior dan fungsi likelihood. Distribusi prior yag dipilih pada kajian ini adalah f(β, σ2 ) = Qn i=1 f(βj |σ 2 )f(σ 2 ) dengan βj ∼ N(µβj , σ2 ) dan σ 2 ∼ IG(a, b). Distribusi prior konjugat tersebut kemudian dikalikan dengan fungsi likelihood L(β, σ2 ) sehingga membentuk distribusi posterior f(β|σ 2 ). Distribusi posterior inilah yang digunakan untuk mengestimasi parameter model melalui proses Markov Chain Monte Carlo (MCMC). Algoritma MCMC yang digunakan adalah algoritma Gibbs Sampler. Model regresi linear berganda yang diperoleh dengan metode maximum likelihood adalahyˆ = −27, 8210000 + 0, 0307430X1 + 0, 0039211X2 + 0, 0034631X3 + 0, 6537000X4dengan kecocokan modelnya adalah sebesar 95,7 %. Sedangkan model regresi linear berganda yang diperoleh dengan metode Bayes adalahyˆ = −26, 620000 + 0, 029380X1 + 0, 004204X2 + 0, 003321X3 + 0, 656200X4dengan kecocokan modelnya adalah sebesar 99,99 %. Dengan demikian dapat disimpulkan bahwa metode Bayes lebih baik dari pada metode maximum likelihood.Kata Kunci: Model Regresi Linear Berganda, metode Maximum Likelihood, dan metode Bayes

ON THE ADAN–WEISS LOSS MODEL HAVING SKILL-BASED SERVERS AND LONGEST IDLE ASSIGNMENT RULE

2015 ◽  
Vol 29 (2) ◽  
pp. 181-189 ◽  
Author(s):  
Babak Haji
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Gibbs Sampler ◽  
Service Time ◽  
Loss System ◽  
Multiplicative Constant ◽  
Assignment Rule ◽  
Poisson Arrivals

We consider a queueing loss system with heterogeneous skill based servers with arbitrary distributions. We assume Poisson arrivals, with each arrival having a vector indicating which of the servers are eligible to serve it. Arrivals can only be assigned to a server that is both idle and eligible. We assume arrivals are assigned to the idle eligible server that has been idle the longest and derive, up to a multiplicative constant, the limiting distribution for this system. We show that the limiting probabilities of the ordered list of idle servers depend on the service time distributions only through their means. Moreover, conditional on the ordered list of idle servers, the remaining service times of the busy servers are independent and have their respective equilibrium service distributions. We also provide an algorithm using Gibbs sampler Markov Chain Monte Carlo method for estimating the limiting probabilities and other desired quantities of this system.

The estimated parameter of logistic regression model by Markov Chain Monte Carlo method with multicollinearity

2020 ◽  
Vol 36 (4) ◽  
pp. 1253-1259
Author(s):  
Autcha Araveeporn ◽  
Yuwadee Klomwises
Keyword(s):  
Monte Carlo ◽  
Logistic Regression ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Regression Model ◽  
Ridge Regression ◽  
Logistic Regression Model ◽  
Classical Method ◽  
Likelihood Method

Markov Chain Monte Carlo (MCMC) method has been a popular method for getting information about probability distribution for estimating posterior distribution by Gibbs sampling. So far, the standard methods such as maximum likelihood and logistic ridge regression methods have represented to compare with MCMC. The maximum likelihood method is the classical method to estimate the parameter on the logistic regression model by differential the loglikelihood function on the estimator. The logistic ridge regression depends on the choice of ridge parameter by using crossvalidation for computing estimator on penalty function. This paper provides maximum likelihood, logistic ridge regression, and MCMC to estimate parameter on logit function and transforms into a probability. The logistic regression model predicts the probability to observe a phenomenon. The prediction accuracy evaluates in terms of the percentage with correct predictions of a binary event. A simulation study conducts a binary response variable by using 2, 4, and 6 explanatory variables, which are generated from multivariate normal distribution on the positive and negative correlation coefficient or called multicollinearity problem. The criterion of these methods is to compare by a maximum of predictive accuracy. The outcomes find that MCMC satisfies all situations.

scholarly journals A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models With Small Sample Sizes

2021 ◽  
Vol 12 ◽  
Author(s):  
Oliver Lüdtke ◽  
Esther Ulitzsch ◽  
Alexander Robitzsch
Keyword(s):  
Monte Carlo ◽  
Factor Analysis ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Confirmatory Factor Analysis ◽  
Parameter Estimates ◽  
Penalized Maximum Likelihood ◽  
Confirmatory Factor ◽  
Bayesian Estimators

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as non-convergence or parameter estimates outside the admissible parameter space. In this article, we distinguish different Bayesian estimators that can be used to stabilize the parameter estimates of a CFA: the mode of the joint posterior distribution that is obtained from penalized maximum likelihood (PML) estimation, and the mean (EAP), median (Med), or mode (MAP) of the marginal posterior distribution that are calculated by using Markov Chain Monte Carlo (MCMC) methods. In two simulation studies, we evaluated the performance of the Bayesian estimators from a frequentist point of view. The results show that the EAP produced more accurate estimates of the latent correlation in many conditions and outperformed the other Bayesian estimators in terms of root mean squared error (RMSE). We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded, and we suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

scholarly journals A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models with Small Sample Sizes

2020 ◽  
Author(s):  
Oliver Lüdtke ◽  
Esther Ulitzsch ◽  
Alexander Robitzsch
Keyword(s):  
Monte Carlo ◽  
Factor Analysis ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Posterior Distribution ◽  
Parameter Estimates ◽  
Penalized Maximum Likelihood ◽  
Confirmatory Factor ◽  
Joint Posterior Distribution

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as nonconvergence or parameter estimates that are outside the admissible parameter space. In the present article, we discuss two Bayesian estimation methods for stabilizing parameter estimates of a CFA: Penalized maximum likelihood (PML) estimation and Markov Chain Monte Carlo (MCMC) methods. We clarify that these use different Bayesian point estimates from the joint posterior distribution—the mode (PML) of the joint posterior distribution, and the mean (EAP) or mode (MAP) of the marginal posterior distribution—and discuss under which conditions the two methods produce different results. In a simulation study, we show that the MCMC method clearly outperforms PML and that these performance gains can be explained by the fact that MCMC uses the EAP as a point estimate. We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded and suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

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