Browsing by Author "Senoglu, Birdal"

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Estimation of the Location and Scale Parameters of Moyal Distribution
(Turkic World Mathematical Soc, 2020) Arslan, Talha; Acitas, Sukru; Senoglu, Birdal
In this study, we estimate the parameters of the Moyal distribution by using well-known and widely-used maximum likelihood (ML) and method of moments (MoM) methodologies. The ML estimators of the location and scale parameters of the Moyal distribution cannot be obtained in closed forms therefore iterative methods should be utilized. To make the study complete, modifed ML (MML) estimators for the location and the scale parameters of the Moyal distribution are also derived. The MML estimators are in closed forms and asymptotically equivalent to the ML estimators. Efficiencies of the MML estimators are compared with their ML and MoM counterparts using Monte-Carlo (MC) simulation study. Results of the simulation study show that the ML estimators are more efficient than the MML and MoM estimators for small sample sizes. However when the sample size increases performances of the ML and MML estimators are almost same in terms of the Defficiency (Def) criterion as expected. At the end of the study, a real data set is used to show the implementation of the methodology developed in this paper.
Estimation of the Parameters of the Gamma Geometric Process
(Taylor & Francis Ltd, 2022) Kara, Mahmut; Guven, Gamze; Senoglu, Birdal; Aydogdu, Halil
There is no doubt that finding the estimators of model parameters accurately and efficiently is very important in many fields. In this study, we obtain the explicit estimators of the unknown model parameters in the gamma geometric process (GP) via the modified maximum likelihood (MML) methodology. These estimators are as efficient as maximum likelihood (ML) estimators. The marginal and joint asymptotic distributions of the MML estimators are also derived and efficiency comparisons between ML and MML estimators are made through an extensive Monte Carlo simulations. Moreover, a real data example is considered to illustrate the performances of the MML estimators together with their ML counterparts. According to simulation results, the performances of MML and ML estimators are close to each other even for small sample sizes.
Modified Minimum Distance Estimators: Definition, Properties and Applications
(Springer Heidelberg, 2022) Arslan, Talha; Acitas, Sukru; Senoglu, Birdal
Estimating the location and scale parameters of a distribution is one of themost crucial issues in Statistics. Therefore, various estimators are proposed for estimating them, such as maximum likelihood, method of moments and minimum distance (e.g. Cramervon Mises-CvM and Anderson Darling-AD), etc. However, in most of the cases, estimators of the location parameter mu and scale parameter s cannot be obtained in closed forms because of the nonlinear function(s) included in the corresponding estimating equations. Therefore, numerical methods are used to obtain the estimates of these parameters. However, they may have some drawbacks such as multiple roots, wrong convergency, and non-convergency of iterations. In this study, we adopt the idea of Tiku (Biometrika 54:155-165, 1967) into the CvM and AD methodologies with the intent of eliminating the aforementioned difficulties and obtaining closed form estimators of the parameters mu and s. Resulting estimators are called as modified CvM (MCvM) and modified AD (MAD), respectively. Proposed estimators are expressed as functions of sample observations and thus their calculations are straightforward. This property also allows us to avoid computational cost of iteration. A Monte-Carlo simulation study is conducted to compare the efficiencies of the CvM and AD estimators with their modified counterparts, i.e. the MCvM and MAD, for the normal, extreme value and Weibull distributions for an illustration. Real data sets are used to show the implementation of the proposed estimation methodologies.
Parameter Estimation for Thetwo-Parameter Maxwell Distribution Under Complete and Censored Samples
(inst Nacional Estatistica-ine, 2021) Arslan, Talha; Acitas, Sukru; Senoglu, Birdal
The Maxwell distribution is one of the basic distributions in Physics besides being popular in Statistics for modeling lifetime data. This paper considers the parameter estimation of the Maxwell distribution via modified maximum likelihood (MML) methodology for both complete and censored samples. The MML estimators for the location and scale parameters of the Maxwell distribution have explicit forms and they are robust against the plausible deviations from the assumed model. A Monte Carlo simulation study is conducted to compare the performances of the MML estimators with the corresponding maximum likelihood (ML), least squares (LS) and method of moments (MoM) estimators.
Slash Maxwell Distribution: Definition, Modified Maximum Likelihood Estimation and Applications
(Gazi Univ, 2020) Acitas, Sukru; Arslan, Talha; Senoglu, Birdal
In this study slash Maxwell (SM) distribution, defined as a ratio of a Maxwell random variate to a power of an independent uniform random variate, is introduced. Its stochastic representation and some distributional properties such as moments, skewness and kurtosis measures are provided. The maximum likelihood (ML) method is used for estimating the unknown parameters. However, closed forms of the ML estimators cannot be obtained since the likelihood equations include nonlinear functions of the unknown parameters. We therefore use Tiku's (1967,1968) modified maximum likelihood (MML) methodology which allows to obtain explicit forms of the estimators. Some asymptotic properties of the MML estimators are derived. A Monte-Carlo simulation study is also carried out to compare the performances of the ML and MML estimators. Two data sets taken from the literature are modelled using the SM distribution in application part of the study.
Statistical Inference for Α-Series Process With Gamma Distribution
(Taylor & Francis inc, 2017) Kara, Mahmut; Aydogdu, Halil; Senoglu, Birdal
The explicit estimators of the parameters alpha, mu and sigma(2) are obtained by using the methodology known as modifiedmaximum likelihood (MML) when the distribution of the first occurrence time of an event is assumed to be Weibull in series process. The efficiencies of the MML estimators are compared with the corresponding nonparametric (NP) estimators and it is shown that the proposed estimators have higher efficiencies than the NP estimators. In this study, we extend these results to the case, where the distribution of the first occurrence time is Gamma. It is anotherwidely used andwell-known distribution in reliability analysis. A real data set taken fromthe literature is analyzed at the end of the study for better understanding the methodology presented in this paper.