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Zawar Hussain - One of the best experts on this subject based on the ideXlab platform.
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bayesian analysis of heterogeneous doubly censored lifetime data using the 3 component mixture of rayleigh Distributions a monte carlo simulation study
Scientia Iranica, 2018Co-Authors: Madeeha Tahir, Muhammad Aslam, Zawar Hussain, Muhammad A, Haider S BhattiAbstract:This article is about Bayesian estimation of parameters of a heterogeneous 3-component mixture of Rayleigh Distributions (3-CMRD) generating a mixture data. Being the most popular and reasonable sampling scheme in reliability and survival analyses, the doubly censored sampling scheme is considered. The Bayes estimators and their Posterior risks are derived under various situations. In addition, elicitation of hyperparameters is presented. Algebraic expressions for Posterior Predictive Distribution and Bayesian Predictive intervals are derived. Assuming the informative and the non-informative priors, a comprehensive Monte Carlo simulation is conducted to examine the performance of the Bayes estimators under symmetric and asymmetric loss functions. Finally, to highlight the practical importance, the proposed 3-compnent mixture model is applied to a doubly censored lifetime data from a real life situation. It is observed that the analysis of doubly censored data in Bayesian framework, the SRIGP paired with SELF (DLF) is suitable choice for estimating mixing proportion (component) parameters.
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Bayesian Analysis of a 3-Component Mixture of Rayleigh Distributions under Type-I Right Censoring Scheme
Atlantis Press, 2017Co-Authors: Muhammad Tahir, Muhammad Aslam, Zawar HussainAbstract:Since the last few decades, constructing flexible parametric classes of probability Distributions has been the most popular approach in the Bayesian analysis. As compared to simple probability models, a mixture model of some suitable lifetime Distributions may be more capable of capturing the heterogeneity of the nature. In this study, a 3- component mixture of Rayleigh Distributions is investigated by considering type-I right censoring scheme to obtain data from a heterogeneous population. The closed form expressions for the Bayes estimators and Posterior risks assuming the non-informative (uniform and Jeffreysr) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. The Posterior Predictive Distribution for a future observation and the Bayesian Predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and Posterior risks are derived. Simulated data sets are used for the different comparisons and the model is finally illustrated using the real data
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bayesian estimation of finite3 component mixture of burr type xii Distributions assuming type i right censoring scheme
alexandria engineering journal, 2016Co-Authors: Muhammad Tahir, Muhammad Aslam, Zawar HussainAbstract:Abstract As compared to simple models, the mixture models of underlying lifetime Distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII Distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII Distributions are discussed. These mathematical properties include Bayes estimators and Posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the Posterior Predictive Distribution for a future observation and the Bayesian Predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and Posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data.
Muhammad Tahir - One of the best experts on this subject based on the ideXlab platform.
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Bayesian Analysis of a 3-Component Mixture of Rayleigh Distributions under Type-I Right Censoring Scheme
Atlantis Press, 2017Co-Authors: Muhammad Tahir, Muhammad Aslam, Zawar HussainAbstract:Since the last few decades, constructing flexible parametric classes of probability Distributions has been the most popular approach in the Bayesian analysis. As compared to simple probability models, a mixture model of some suitable lifetime Distributions may be more capable of capturing the heterogeneity of the nature. In this study, a 3- component mixture of Rayleigh Distributions is investigated by considering type-I right censoring scheme to obtain data from a heterogeneous population. The closed form expressions for the Bayes estimators and Posterior risks assuming the non-informative (uniform and Jeffreysr) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. The Posterior Predictive Distribution for a future observation and the Bayesian Predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and Posterior risks are derived. Simulated data sets are used for the different comparisons and the model is finally illustrated using the real data
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bayesian estimation of finite3 component mixture of burr type xii Distributions assuming type i right censoring scheme
alexandria engineering journal, 2016Co-Authors: Muhammad Tahir, Muhammad Aslam, Zawar HussainAbstract:Abstract As compared to simple models, the mixture models of underlying lifetime Distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII Distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII Distributions are discussed. These mathematical properties include Bayes estimators and Posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the Posterior Predictive Distribution for a future observation and the Bayesian Predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and Posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data.
Muhammad Aslam - One of the best experts on this subject based on the ideXlab platform.
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bayesian analysis of heterogeneous doubly censored lifetime data using the 3 component mixture of rayleigh Distributions a monte carlo simulation study
Scientia Iranica, 2018Co-Authors: Madeeha Tahir, Muhammad Aslam, Zawar Hussain, Muhammad A, Haider S BhattiAbstract:This article is about Bayesian estimation of parameters of a heterogeneous 3-component mixture of Rayleigh Distributions (3-CMRD) generating a mixture data. Being the most popular and reasonable sampling scheme in reliability and survival analyses, the doubly censored sampling scheme is considered. The Bayes estimators and their Posterior risks are derived under various situations. In addition, elicitation of hyperparameters is presented. Algebraic expressions for Posterior Predictive Distribution and Bayesian Predictive intervals are derived. Assuming the informative and the non-informative priors, a comprehensive Monte Carlo simulation is conducted to examine the performance of the Bayes estimators under symmetric and asymmetric loss functions. Finally, to highlight the practical importance, the proposed 3-compnent mixture model is applied to a doubly censored lifetime data from a real life situation. It is observed that the analysis of doubly censored data in Bayesian framework, the SRIGP paired with SELF (DLF) is suitable choice for estimating mixing proportion (component) parameters.
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Bayesian Analysis of a 3-Component Mixture of Rayleigh Distributions under Type-I Right Censoring Scheme
Atlantis Press, 2017Co-Authors: Muhammad Tahir, Muhammad Aslam, Zawar HussainAbstract:Since the last few decades, constructing flexible parametric classes of probability Distributions has been the most popular approach in the Bayesian analysis. As compared to simple probability models, a mixture model of some suitable lifetime Distributions may be more capable of capturing the heterogeneity of the nature. In this study, a 3- component mixture of Rayleigh Distributions is investigated by considering type-I right censoring scheme to obtain data from a heterogeneous population. The closed form expressions for the Bayes estimators and Posterior risks assuming the non-informative (uniform and Jeffreysr) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. The Posterior Predictive Distribution for a future observation and the Bayesian Predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and Posterior risks are derived. Simulated data sets are used for the different comparisons and the model is finally illustrated using the real data
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bayesian estimation of finite3 component mixture of burr type xii Distributions assuming type i right censoring scheme
alexandria engineering journal, 2016Co-Authors: Muhammad Tahir, Muhammad Aslam, Zawar HussainAbstract:Abstract As compared to simple models, the mixture models of underlying lifetime Distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII Distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII Distributions are discussed. These mathematical properties include Bayes estimators and Posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the Posterior Predictive Distribution for a future observation and the Bayesian Predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and Posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data.
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on the bayesian analysis of the mixture of power function Distribution using the complete and the censored sample
Journal of Applied Statistics, 2010Co-Authors: M Saleem, Muhammad Aslam, Polychronis EconomouAbstract:The power function Distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function Distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The Posterior Predictive Distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the Predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.
Jacob R Gardner - One of the best experts on this subject based on the ideXlab platform.
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parametric gaussian process regressors
arXiv: Machine Learning, 2019Co-Authors: Martin Jankowiak, Geoff Pleiss, Jacob R GardnerAbstract:The combination of inducing point methods with stochastic variational inference has enabled approximate Gaussian Process (GP) inference on large datasets. Unfortunately, the resulting Predictive Distributions often exhibit substantially underestimated uncertainties. Notably, in the regression case the Predictive variance is typically dominated by observation noise, yielding uncertainty estimates that make little use of the input-dependent function uncertainty that makes GP priors attractive. In this work we propose two simple methods for scalable GP regression that address this issue and thus yield substantially improved Predictive uncertainties. The first applies variational inference to FITC (Fully Independent Training Conditional; Snelson et.~al.~2006). The second bypasses Posterior approximations and instead directly targets the Posterior Predictive Distribution. In an extensive empirical comparison with a number of alternative methods for scalable GP regression, we find that the resulting Predictive Distributions exhibit significantly better calibrated uncertainties and higher log likelihoods--often by as much as half a nat per datapoint.
Gruen D. - One of the best experts on this subject based on the ideXlab platform.
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Dark energy survey internal consistency tests of the joint cosmological probes analysis with Posterior Predictive Distributions
'Oxford University Press (OUP)', 2021Co-Authors: Doux C., Baxter E., Lemos P., Chang C., Alarcon A., Amon A., Campos A., Choi A., Gatti M., Gruen D.Abstract:International audienceBeyond ΛCDM, physics or systematic errors may cause subsets of a cosmological data set to appear inconsistent when analysed assuming ΛCDM. We present an application of internal consistency tests to measurements from the Dark Energy Survey Year 1 (DES Y1) joint probes analysis. Our analysis relies on computing the Posterior Predictive Distribution (PPD) for these data under the assumption of ΛCDM. We find that the DES Y1 data have an acceptable goodness of fit to ΛCDM, with a probability of finding a worse fit by random chance of p = 0.046. Using numerical PPD tests, supplemented by graphical checks, we show that most of the data vector appears completely consistent with expectations, although we observe a small tension between large- and small-scale measurements. A small part (roughly 1.5 per cent) of the data vector shows an unusually large departure from expectations; excluding this part of the data has negligible impact on cosmological constraints, but does significantly improve the p-value to 0.10. The methodology developed here will be applied to test the consistency of DES Year 3 joint probes data sets
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Dark energy survey internal consistency tests of the joint cosmological probes analysis with Posterior Predictive Distributions
'Oxford University Press (OUP)', 2021Co-Authors: Doux Cyrille, Baxter E., Alarcon A., Amon A., Campos A., Choi A., Gatti M., Lemos Pablo, Chang, Chihway L., Gruen D.Abstract:ABSTRACT Beyond ΛCDM, physics or systematic errors may cause subsets of a cosmological data set to appear inconsistent when analysed assuming ΛCDM. We present an application of internal consistency tests to measurements from the Dark Energy Survey Year 1 (DES Y1) joint probes analysis. Our analysis relies on computing the Posterior Predictive Distribution (PPD) for these data under the assumption of ΛCDM. We find that the DES Y1 data have an acceptable goodness of fit to ΛCDM, with a probability of finding a worse fit by random chance of p = 0.046. Using numerical PPD tests, supplemented by graphical checks, we show that most of the data vector appears completely consistent with expectations, although we observe a small tension between large- and small-scale measurements. A small part (roughly 1.5 per cent) of the data vector shows an unusually large departure from expectations; excluding this part of the data has negligible impact on cosmological constraints, but does significantly improve the p-value to 0.10. The methodology developed here will be applied to test the consistency of DES Year 3 joint probes data sets
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Dark Energy Survey internal consistency tests of the joint cosmological probes analysis with Posterior Predictive Distributions
'Oxford University Press (OUP)', 2021Co-Authors: Doux C., Baxter E., Lemos P., Chang C., Alarcon A., Amon A., Campos A., Choi A., Gatti M., Gruen D.Abstract:Beyond-$\Lambda$CDM physics or systematic errors may cause subsets of a cosmological data set to appear inconsistent when analyzed assuming $\Lambda$CDM. We present an application of internal consistency tests to measurements from the Dark Energy Survey Year 1 (DES Y1) joint probes analysis. Our analysis relies on computing the Posterior Predictive Distribution (PPD) for these data under the assumption of $\Lambda$CDM. We find that the DES Y1 data have an acceptable goodness of fit to $\Lambda$CDM, with a probability of finding a worse fit by random chance of ${p = 0.046}$. Using numerical PPD tests, supplemented by graphical checks, we show that most of the data vector appears completely consistent with expectations, although we observe a small tension between large- and small-scale measurements. A small part (roughly 1.5%) of the data vector shows an unusually large departure from expectations; excluding this part of the data has negligible impact on cosmological constraints, but does significantly improve the $p$-value to 0.10. The methodology developed here will be applied to test the consistency of DES Year 3 joint probes data sets.Comment: v2: Minor modification in calibration methodology; conclusions unchanged. v3: Matches version accepted in MNRAS; generalises calibration of p-values, while leaving qualitative conclusions unchanged. Comments are welcom
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Dark Energy Survey internal consistency tests of the joint cosmological probes analysis with Posterior Predictive Distributions
'Oxford University Press (OUP)', 2020Co-Authors: Doux C., Baxter E., Lemos P., Chang C., Alarcon A., Amon A., Campos A., Choi A., Gatti M., Gruen D.Abstract:Beyond-$Lambda$CDM physics or systematic errors may cause subsets of a cosmological data set to appear inconsistent when analyzed assuming $Lambda$CDM. We present an application of internal consistency tests to measurements from the Dark Energy Survey Year 1 (DES Y1) joint probes analysis. Our analysis relies on computing the Posterior Predictive Distribution (PPD) for these data under the assumption of $Lambda$CDM. We find that the DES Y1 data have an acceptable goodness of fit to $Lambda$CDM, with a probability of finding a worse fit by random chance of ${p = 0.046}$. Using numerical PPD tests, supplemented by graphical checks, we show that most of the data vector appears completely consistent with expectations, although we observe a small tension between large- and small-scale measurements. A small part (roughly 1.5%) of the data vector shows an unusually large departure from expectations; excluding this part of the data has negligible impact on cosmological constraints, but does significantly improve the $p$-value to 0.10. The methodology developed here will be applied to test the consistency of DES Year 3 joint probes data sets...