hood (parametric), conditional least squares (non-parametric), and Yule-Walker (non-parametric) e... more hood (parametric), conditional least squares (non-parametric), and Yule-Walker (non-parametric) estimation methods. Several INAR(1) processes are considered to fitting the COVID-19 data sets. The goodness of fit measures consists of the Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and Hanna-Quinn (HQIC). Results and Discussion The simulation comparison is conducted concerning the mean square errors (MSE), which shows the superiority of the conditional maximum likelihood estimation method. Due to evaluating the performance of estimators in terms of MSE, 100 iterations are considered for different sample sizes. Based on the goodness of fit measures, it is concluded that the DEW-INAR(1) process is preferred among other INAR(1) processes. Conclusion The main focus of the manuscript is to introduce the discrete version of the Exponential-Weibull distribution and the modeling of its corresponding integer-valued autoregressive process. The flexibility and comprehensiveness of the discrete Exponential-Weibull distribution in fitting different types of count data is deduced by examining the statistical characteristics. Also, parameter estimation methods and Monte Carlo simulation studies are presented. According to the simulation results, the conditional maximum likelihood parametric estimation method performs better than non-parametric approaches. Using the data of COVID-19, the efficiency of the new process is confirmed in comparison with the classical INAR(1) models.
In this paper, the definition of probability, conditional probability and likelihood function are... more In this paper, the definition of probability, conditional probability and likelihood function are generalized to the intuitionistic fuzzy observations. We focus on different estimation approaches of two-parameter Weibull (TW) distribution based on the intuitionistic fuzzy lifetime data including, maximum likelihood (ML) and Bayesian estimation methodology. The ML estimation of the parameters and reliability function of TW distribution is provided using the Newton–Raphson (NR) and Expectation–Maximization (EM) algorithms. The Bayesian estimates are provided via Tierney and Kadane’s approximation. In the Bayesian estimation approach, for the shape and scale parameters, the Gamma and inverse-Gamma priors are considered, respectively. Finally, a simulated data set is analyzed for illustrative purposes to show the applicability of the proposed estimation methods. The Monte Carlo simulations are performed to find the more efficient estimator in the intuitionistic fuzzy environment. The pe...
In this paper, the two-parameter Pareto lifetime distribution is considered with vague shape and ... more In this paper, the two-parameter Pareto lifetime distribution is considered with vague shape and scale parameters, where parameters are set as generalized intuitionistic fuzzy numbers. A new L-R type intuitionistic fuzzy number is introduced, and cuts of the new fuzzy set are provided. The generalized intuitionistic fuzzy reliability characteristics such as reliability, conditional reliability, hazard rate and mean time to failure functions are defined, along with the special case of the twoparameter Pareto generalized intuitionistic fuzzy reliability analysis. Furthermore, the series and parallel system reliability are evaluated by the generalized intuitionistic fuzzy sets. Finally, for certain cases of the fuzzy shape and scale parameters and cut set values, the generalized intuitionistic fuzzy reliability characteristics are provided and compared through several illustrative plots. Keywords Generalized L-R type intuitionistic fuzzy numbers • (α 1 , α 2)-cut set • Generalized intuitionistic fuzzy reliability • Generalized intuitionistic fuzzy probability • Two-parameter Pareto distribution
In this paper, we propose an estimate of reliability in a multicomponent system. The system has $... more In this paper, we propose an estimate of reliability in a multicomponent system. The system has $$k$$ components strengths are given by independently and identically distributed random variables $${X}_{1}$$ , $${X}_{2}$$ ,…, $${X}_{k}$$ and each component is exposed to random stress $${\rm Y}$$ . The reliability of such a system is obtained when strength and stress variables are given by inverse Weibull (IW) distribution with scale parameters $${\lambda }_{1}$$ , $${\lambda }_{2}$$ and common shape parameter $$\alpha$$ . The system reliability is estimated using maximum likelihood estimation (MLE) and the best two-observational percentile estimation (BTPE) methods in samples drawn from strength and stress distributions. Also, the asymptotic confidence interval for system reliability is obtained. The reliability estimators obtained from both methods are compared using average bias, mean squares error, and confidence interval length via Monte Carlo simulation. In the end, using two re...
Journal of Statistical Computation and Simulation, 2021
In this paper, we focussed on the scale parameter and reliability estimations of the inverse gene... more In this paper, we focussed on the scale parameter and reliability estimations of the inverse generalized Weibull distribution. Both classical and Bayesian approaches are considered with various loss functions as general entropy, squared log error and weight squared error. For the Bayesian method, both informative and non-informative priors are applied for the reliability and scale parameter estimation. Furthermore, we introduce a new loss function that exhibits some attractive performances. The reliability function and scale parameter of the inverse generalized Weibull distribution are estimated based on the new loss function. By the Monte Carlo simulation procedure, we demonstrate the efficiency of the new proposed loss function among some competitors in estimating the reliability function . Finally, the analysis of two real data set has also been represented for illustration purposes. Some goodness of fit measures affirmed the adequacy of the inverse generalized Weibull distribution in modelling real data sets.
Journal of Statistical Computation and Simulation, 2016
ABSTRACT In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the para... more ABSTRACT In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the parameter derived from Pareto distribution under different loss functions. The definition of the E-Bayesian estimation of the parameter is provided. Moreover, for Pareto distribution, under the condition of the scale parameter is known, based on the different loss functions, formulas of the E-Bayesian estimation and hierarchical Bayesian estimations for the shape parameter are given, respectively, properties of the E-Bayesian estimation – (i) the relationship between of E-Bayesian estimations under different loss functions are provided, (ii) the relationship between of E-Bayesian and hierarchical Bayesian estimations under the same loss function are also provided, and using the Monte Carlo method simulation example is given. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.
In this paper, we consider the estimation of the PDF and the CDF of the Frechet distribution. In ... more In this paper, we consider the estimation of the PDF and the CDF of the Frechet distribution. In this regard, following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the bias and the mean squared error. As the result of simulation studies and real data applications indicate, the ML estimator performs better than the others.
Journal of Statistical Computation and Simulation, 2017
ABSTRACT In this paper, we consider the estimation of the probability density function and the cu... more ABSTRACT In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.
Communications in Statistics - Simulation and Computation, 2016
The Weibull extension model is a useful extension of the Weibull distribution, allowing for batht... more The Weibull extension model is a useful extension of the Weibull distribution, allowing for bathtub shaped hazard rates among other things. Here, we consider estimation of the PDF and the CDF of the Weibull extension model. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least squares (LS) estimator, and weighted least squares (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.
hood (parametric), conditional least squares (non-parametric), and Yule-Walker (non-parametric) e... more hood (parametric), conditional least squares (non-parametric), and Yule-Walker (non-parametric) estimation methods. Several INAR(1) processes are considered to fitting the COVID-19 data sets. The goodness of fit measures consists of the Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and Hanna-Quinn (HQIC). Results and Discussion The simulation comparison is conducted concerning the mean square errors (MSE), which shows the superiority of the conditional maximum likelihood estimation method. Due to evaluating the performance of estimators in terms of MSE, 100 iterations are considered for different sample sizes. Based on the goodness of fit measures, it is concluded that the DEW-INAR(1) process is preferred among other INAR(1) processes. Conclusion The main focus of the manuscript is to introduce the discrete version of the Exponential-Weibull distribution and the modeling of its corresponding integer-valued autoregressive process. The flexibility and comprehensiveness of the discrete Exponential-Weibull distribution in fitting different types of count data is deduced by examining the statistical characteristics. Also, parameter estimation methods and Monte Carlo simulation studies are presented. According to the simulation results, the conditional maximum likelihood parametric estimation method performs better than non-parametric approaches. Using the data of COVID-19, the efficiency of the new process is confirmed in comparison with the classical INAR(1) models.
In this paper, the definition of probability, conditional probability and likelihood function are... more In this paper, the definition of probability, conditional probability and likelihood function are generalized to the intuitionistic fuzzy observations. We focus on different estimation approaches of two-parameter Weibull (TW) distribution based on the intuitionistic fuzzy lifetime data including, maximum likelihood (ML) and Bayesian estimation methodology. The ML estimation of the parameters and reliability function of TW distribution is provided using the Newton–Raphson (NR) and Expectation–Maximization (EM) algorithms. The Bayesian estimates are provided via Tierney and Kadane’s approximation. In the Bayesian estimation approach, for the shape and scale parameters, the Gamma and inverse-Gamma priors are considered, respectively. Finally, a simulated data set is analyzed for illustrative purposes to show the applicability of the proposed estimation methods. The Monte Carlo simulations are performed to find the more efficient estimator in the intuitionistic fuzzy environment. The pe...
In this paper, the two-parameter Pareto lifetime distribution is considered with vague shape and ... more In this paper, the two-parameter Pareto lifetime distribution is considered with vague shape and scale parameters, where parameters are set as generalized intuitionistic fuzzy numbers. A new L-R type intuitionistic fuzzy number is introduced, and cuts of the new fuzzy set are provided. The generalized intuitionistic fuzzy reliability characteristics such as reliability, conditional reliability, hazard rate and mean time to failure functions are defined, along with the special case of the twoparameter Pareto generalized intuitionistic fuzzy reliability analysis. Furthermore, the series and parallel system reliability are evaluated by the generalized intuitionistic fuzzy sets. Finally, for certain cases of the fuzzy shape and scale parameters and cut set values, the generalized intuitionistic fuzzy reliability characteristics are provided and compared through several illustrative plots. Keywords Generalized L-R type intuitionistic fuzzy numbers • (α 1 , α 2)-cut set • Generalized intuitionistic fuzzy reliability • Generalized intuitionistic fuzzy probability • Two-parameter Pareto distribution
In this paper, we propose an estimate of reliability in a multicomponent system. The system has $... more In this paper, we propose an estimate of reliability in a multicomponent system. The system has $$k$$ components strengths are given by independently and identically distributed random variables $${X}_{1}$$ , $${X}_{2}$$ ,…, $${X}_{k}$$ and each component is exposed to random stress $${\rm Y}$$ . The reliability of such a system is obtained when strength and stress variables are given by inverse Weibull (IW) distribution with scale parameters $${\lambda }_{1}$$ , $${\lambda }_{2}$$ and common shape parameter $$\alpha$$ . The system reliability is estimated using maximum likelihood estimation (MLE) and the best two-observational percentile estimation (BTPE) methods in samples drawn from strength and stress distributions. Also, the asymptotic confidence interval for system reliability is obtained. The reliability estimators obtained from both methods are compared using average bias, mean squares error, and confidence interval length via Monte Carlo simulation. In the end, using two re...
Journal of Statistical Computation and Simulation, 2021
In this paper, we focussed on the scale parameter and reliability estimations of the inverse gene... more In this paper, we focussed on the scale parameter and reliability estimations of the inverse generalized Weibull distribution. Both classical and Bayesian approaches are considered with various loss functions as general entropy, squared log error and weight squared error. For the Bayesian method, both informative and non-informative priors are applied for the reliability and scale parameter estimation. Furthermore, we introduce a new loss function that exhibits some attractive performances. The reliability function and scale parameter of the inverse generalized Weibull distribution are estimated based on the new loss function. By the Monte Carlo simulation procedure, we demonstrate the efficiency of the new proposed loss function among some competitors in estimating the reliability function . Finally, the analysis of two real data set has also been represented for illustration purposes. Some goodness of fit measures affirmed the adequacy of the inverse generalized Weibull distribution in modelling real data sets.
Journal of Statistical Computation and Simulation, 2016
ABSTRACT In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the para... more ABSTRACT In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the parameter derived from Pareto distribution under different loss functions. The definition of the E-Bayesian estimation of the parameter is provided. Moreover, for Pareto distribution, under the condition of the scale parameter is known, based on the different loss functions, formulas of the E-Bayesian estimation and hierarchical Bayesian estimations for the shape parameter are given, respectively, properties of the E-Bayesian estimation – (i) the relationship between of E-Bayesian estimations under different loss functions are provided, (ii) the relationship between of E-Bayesian and hierarchical Bayesian estimations under the same loss function are also provided, and using the Monte Carlo method simulation example is given. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.
In this paper, we consider the estimation of the PDF and the CDF of the Frechet distribution. In ... more In this paper, we consider the estimation of the PDF and the CDF of the Frechet distribution. In this regard, following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the bias and the mean squared error. As the result of simulation studies and real data applications indicate, the ML estimator performs better than the others.
Journal of Statistical Computation and Simulation, 2017
ABSTRACT In this paper, we consider the estimation of the probability density function and the cu... more ABSTRACT In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.
Communications in Statistics - Simulation and Computation, 2016
The Weibull extension model is a useful extension of the Weibull distribution, allowing for batht... more The Weibull extension model is a useful extension of the Weibull distribution, allowing for bathtub shaped hazard rates among other things. Here, we consider estimation of the PDF and the CDF of the Weibull extension model. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least squares (LS) estimator, and weighted least squares (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.
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Papers by Einolah deiri