Browsing by Author "Yilmaz, Asuman"

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Bayesian Inference for Geometric Process With Lindley Distribution and Its Applications
(World Scientific Publ Co Pte Ltd, 2022) Yilmaz, Asuman; Kara, Mahmut; Kara, Hasan
The geometric process (GP) plays an important role in the reliability theory and life span models. It has been used extensively as a stochastic model in many areas of application. Therefore, the parameter estimation problem is very crucial in a GP. In this study, the parameter estimation problem for GP is discussed under the assumption that X1 has a Lindley distribution with parameter theta. The maximum likelihood (ML) estimators of a,mu and sigma(2) of the GP and their asymptotic distributions are derived. A test statistic is developed based on ML estimators for testing whether a=1 or not. The same problem is also studied by using Bayesian methods. Bayes estimators of the unknown model parameters are obtained under squared error loss function (SELF) using uniform and gamma priors on the ratio a and theta parameters. It is not possible to obtain Bayes estimators in explicit forms. Therefore, Markov Chain Monte Carlo (MCMC), Lindley (LD), and Tierney-Kadane (T-K) methods are used to estimate the parameters a,mu and sigma(2) in GP. The efficiencies of the ML estimators are compared with Bayes estimators via an extensive Monte Carlo simulation study. It is seen that the Bayes estimators perform better than the ML estimators. Two real-life examples are also presented for application purposes. The first data set concerns the coal mining disaster. The second is the number of COVID-19 patients in Turkey.
Bayesian Parameter Estimation for Geometric Process With Rayleigh Distribution
(2024) Yilmaz, Asuman
The main purpose of this study is to deal with the parameter estimation problem for the geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Rayleigh. For this purpose, maximum likelihood and Bayesian parameter estimation methods are discussed. Lindley and MCMC approximation methods are used in Bayesian calculations. Additionally, a novel method called the Modified-Lindley approximation has been proposed as an alternative to the Lindley approximation. An extensive simulation study was conducted to compare the performances of the prediction methods. Finally, a real data set is analyzed for illustrative purposes.
Comparison of Different Estimation Methods for Extreme Value Distribution
(Taylor & Francis Ltd, 2021) Yilmaz, Asuman; Kara, Mahmut; Ozdemir, Onur
The extreme value distribution was developed for modeling extreme-order statistics or extreme events. In this study, we discuss the distribution of the largest extreme. The main objective of this paper is to determine the best estimators of the unknown parameters of the extreme value distribution. Thus, both classical and Bayesian methods are used. The classical estimation methods under consideration are maximum likelihood estimators, moment's estimators, least squares estimators, and weighted least squares estimators, percentile estimators, the ordinary least squares estimators, best linear unbiased estimators, L-moments estimators, trimmed L-moments estimators, and Bain and Engelhardt estimators. We also propose new estimators for the unknown parameters. Bayesian estimators of the parameters are derived by using Lindley's approximation and Markov Chain Monte Carlo methods. The asymptotic confidence intervals are considered by using maximum likelihood estimators. The Bayesian credible intervals are also obtained by using Gibbs sampling. The performances of these estimation methods are compared with respect to their biases and mean square errors through a simulation study. The maximum daily flood discharge (annual) data sets of the Meric River and Feather River are analyzed at the end of the study for a better understanding of the methods presented in this paper.
Estimation of Parameters for the Gumbel Type-I Distribution Under Type-Ii Censoring Scheme
(Coll Science Women, Univ Baghdad, 2023) Yilmaz, Asuman; Kara, Mahmut
This paper aims to decide the best parameter estimation methods for the parameters of the Gumbel type-I distribution under the type-II censorship scheme. For this purpose, classical and Bayesian parameter estimation procedures are considered. The maximum likelihood estimators are used for the classical parameter estimation procedure. The asymptotic distributions of these estimators are also derived. It is not possible to obtain explicit solutions of Bayesian estimators. Therefore, Markov Chain Monte Carlo, and Lindley techniques are taken into account to estimate the unknown parameters. In Bayesian analysis, it is very important to determine an appropriate combination of a prior distribution and a loss function. Therefore, two different prior distributions are used. Also, the Bayesian estimators concerning the parameters of interest under various loss functions are investigated. The Gibbs sampling algorithm is used to construct the Bayesian credible intervals. Then, the efficiencies of the maximum likelihood estimators are compared with Bayesian estimators via an extensive Monte Carlo simulation study. It has been shown that the Bayesian estimators are considerably more efficient than the maximum likelihood estimators. Finally, a real-life example is also presented for application purposes.
Reliability Estimation and Parameter Estimation for Inverse Weibull Distribution Under Different Loss Functions
(Academic Publication Council, 2022) Yilmaz, Asuman; Kara, Mahmut
In this paper, the classical and Bayesian estimators of the unknown parameters and the reliability function of the inverse Weibull distribution are considered. The maximum likelihood estimators (MLEs) and modified maximum likelihood estimators (MMLEs) are used in the classical parameter estimation. Bayesian estimators of the parameters are obtained by using symmetric and asymmetric loss functions under informative and non-informative priors. Bayesian computations are derived by using Lindley approximation and Markov chain Monte Carlo (MCMC) methods. The asymptotic confidence intervals are constructed based on the maximum likelihood estimators. The Bayesian credible intervals of the parameters are obtained by using the MCMC method. Furthermore, the performances of these estimation methods are compared concerning their biases and mean square errors through a simulation study. It is seen that the Bayes estimators perform better than the classical estimators. Finally, two real-life examples are given for illustrative purposes.
Statistical Analysis of Wind Speed Data with Different Distributions: Bitlis, Türkiye
(2024) Yilmaz, Asuman; Kara, Mahmut
Rüzgar hızının doğru bir şekilde modellenmesi, belirli bir bölgenin rüzgar enerjisi potansiyelinin tahmin edilmesi açısından önemlidir. İki parametreli Weibull dağılımı enerji literatüründe en yaygın kullanılan ve kabul edilen dağılımdır. Ancak doğada karşılaşılan tüm rüzgar hızı verilerini modellemez. Bu nedenle bu çalışmada rüzgâr enerjisinin modellenmesinde Gamma, log-normal, Genelleştirilmiş Rayleigh gibi farklı dağılımlar kullanılmıştır. Bu dağılımların bilinmeyen parametrelerinin tahmin edicileri, maksimum olabilirlik tahmin edicileri kullanılarak bulunur.
Statistical Inference for Geometric Process With the Generalized Rayleigh Distribution
(Univ Nis, 2020) Bicer, Cenker; Bicer, Hayrinisa D.; Kara, Mahmut; Yilmaz, Asuman
In the present paper, the statistical inference problem is considered for the geometric process (GP) by assuming the distribution of the first arrival time with generalized Rayleigh distribution with the parameters alpha and lambda. We have used the maximum likelihood method for obtaining the ratio parameter of the GP and distributional parameters of the generalized Rayleigh distribution. By a series of Monte-Carlo simulations evaluated through the different samples of sizes - small, moderate and large, we have also compared the estimation performances of the maximum likelihood estimators with the other estimators available in the literature such as modified moment, modified L-moment, and modified least squares. Furthermore, wehave presented two real-life datasets analyses to show the modeling behavior of GP with generalized Rayleigh distribution.
A Study on Comparisons of Bayesian and Classical Parameter Estimation Methods for the Two-Parameter Weibull Distribution
(Ankara Univ, Fac Sci, 2020) Yilmaz, Asuman; Kara, Mahmut; Aydogdu, Halil
The main objective of this paper is to determine the best estimators of the shape and scale parameters of the two parameter Weibull distribution. Therefore, both classical and Bayesian approximation methods are considered. For parameter estimation of classical approximation methods maximum likelihood estimators (MLEs), modified maximum likelihood estimators-I (MMLEs-I), modified maximum likelihood estimators-II (MMLEs-II), least square estimators (LSEs), weighted least square estimators (WLSEs), percentile estimators (PEs), moment estimators (MEs), L-moment estimators (LMEs) and TL-moment estimators (TLMEs) are used. Since the Bayesian estimators don't have the explicit form. There are Bayes estimators are obtained by using Lindley's and Tierney Kadane's approximation methods in this study. In Bayesian approximation, the choice of loss function and prior distribution is very important. Hence, Bayes estimators are given based on both the non-informative and informative prior distribution. Moreover, these estimators have been calculated under different symmetric and asymmetric loss functions. The performance of classical and Bayesian estimators are compared with respect to their biases and MSEs through a simulation study. Finally, a real data set taken from Turkish State Meteorological Service is analysed for better understanding of methods presented in this paper.