Halil Aydoğdu

Parameter estimation in geometric process with Weibull distribution

We consider geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Weibull. Explicit estimators of the parameters in GP are derived by using the method of modified maximum likelihood (MML) proposed by Tiku [24]. Asymptotic distributions and consistency properties of these estimators are obtained. We show that our estimators are more efficient than the widely used modified moment (MM) estimators via Monte Carlo simulation study. Further, two real life examples are given at the end of the paper.

Statistical inference for geometric process with the inverse Gaussian distribution

In this study, the statistical inference problem for the geometric process (GP) is considered when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). The parameters a, μ and of the GP are estimated by using the maximum likelihood (ML) method, where a, μ and are the ratio of the GP, the mean and the variance of the IG distribution, respectively. Asymptotic distributions and consistency properties of the ML estimators are obtained. These asymptotic distributions enable us to give a test statistic which distinguishes a renewal process from a geometric process. Monte Carlo simulations are performed to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators. It is seen from the results that the ML estimators are more efficient than the MM estimators. Further, three real-life examples are given for application purposes.

A pointwise estimator for the k-fold convolution of a distribution function

ABSTRACT This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.

Computation of the mean value and variance functions in geometric process

Geometric process (GP) is widely used as a non-stationary stochastic model in reliability analysis. In many of applications related with GP its mean value and variance functions are needed. Since there are no analytical forms of these functions in a lot of situations their computations are of importance. In this study, a numerical approximation and Monte Carlo estimation method based on the convolutions of distribution functions have been proposed for both the mean value and variance functions.

Statistical inference for α-series process with gamma distribution

ABSTRACT The explicit estimators of the parameters α, μ and σ2 are obtained by using the methodology known as modified maximum 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 another widely used and well-known distribution in reliability analysis. A real data set taken from the literature is analyzed at the end of the study for better understanding the methodology presented in this paper.

Statistical inference for α-series process with the inverse Gaussian distribution

ABSTRACT Statistical inferences for the geometric process (GP) are derived when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). An α-series process, as a possible alternative to the GP, is introduced since the GP is sometimes inappropriate to apply some reliability and scheduling problems. In this study, statistical inference problem for the α-series process is considered where the distribution of first occurrence time is IG. The estimators of the parameters α, μ, and σ2 are obtained by using the maximum likelihood (ML) method. Asymptotic distributions and consistency properties of the ML estimators are derived. In order to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators, Monte Carlo simulations are performed. The results showed that the ML estimators are more efficient than the MM estimators. Moreover, two real life datasets are given for application purposes.

Parameter estimation in α-series process with lognormal distribution

Abstract The -series process (ASP) is widely used as a monotonic stochastic model in the reliability context. So the parameter estimation problem in an ASP is of importance. In this study parameter estimation problem for the ASP is considered when the distribution of the first occurrence time of an event is assumed to be lognormal. The parameters and of the ASP are estimated via maximum likelihood (ML) method. Asymptotic distributions and consistency properties of these estimators are derived. A test statistic is conducted to distinguish the ASP from renewal process (RP). Further, modified moment (MM) estimators are proposed for the parameters and and their consistency is proved. A nonparametric (NP) novel method is presented to test whether the ASP is a suitable model for data sets. Monte Carlo simulations are performed to compare the efficiencies of the ML and MM estimators. A real life data example is also studied to illustrate the usefulness of the ASP.