Artykuły w czasopismach: „Bayesian estimation”

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Eldemery, E. M., A. M. Abd-Elfattah, K. M. Mahfouz, and Mohammed M. El Genidy. "Bayesian and E-Bayesian Estimation for the Generalized Rayleigh Distribution under Different Forms of Loss Functions with Real Data Application." Journal of Mathematics 2023 (August 31, 2023): 1–25. http://dx.doi.org/10.1155/2023/5454851.

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This paper investigates the estimation of an unknown shape parameter of the generalized Rayleigh distribution using Bayesian and expected Bayesian estimation techniques based on type-II censoring data. Subsequently, these estimators are obtained using four different loss functions: the linear exponential loss function, the weighted linear exponential loss function, the compound linear exponential loss function, and the weighted compound linear exponential loss function. The weighted compound linear exponential loss function is a novel suggested loss function generated by combining weights with the compound linear exponential loss function. We use the gamma distribution as a prior distribution. In addition, the expected Bayesian estimator is obtained through three different prior distributions of the hyperparameters. Moreover, depending on the four distinct forms of loss functions, Bayesian and expected Bayesian estimation techniques are performed using Monte Carlo simulations to verify the effectiveness of the suggested loss function and to compare Bayesian and expected Bayesian estimation methods. Furthermore, the simulation results indicate that, depending on the minimum mean squared error, the Bayesian and expected Bayesian estimations corresponding to the weighted compound linear exponential loss function suggested in this paper have significantly better performance compared to other loss functions, and the expected Bayesian estimator also performs better than the Bayesian estimator. Finally, the proposed techniques are demonstrated using a set of real data from the medical field to clarify the applicability of the suggested estimators to real phenomena and to show that the discussed weighted compound linear exponential loss function is efficient and can be applied in a real-life scenario.

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admin, admin. "On The Bayesian Estimation of Parameters of SQDM." Neutrosophic and Information Fusion 3, no. 1 (2024): 34–41. http://dx.doi.org/10.54216/nif.030105.

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This work is concerned with the problem of estimating parameters of spatial quadratic models by Bayesian technique (SQDM). This technique involves the prior information of the first and second moment of the parameters, where its estimation model is called the Bayesian quadratic unbiased estimator. The results of the estimation are taken in compared with the estimates of minimum norm quadratic unbiased estimators.

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Al-Bossly, Afrah. "E-Bayesian and Bayesian Estimation for the Lomax Distribution under Weighted Composite LINEX Loss Function." Computational Intelligence and Neuroscience 2021 (December 11, 2021): 1–10. http://dx.doi.org/10.1155/2021/2101972.

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The main contribution of this work is the development of a compound LINEX loss function (CLLF) to estimate the shape parameter of the Lomax distribution (LD). The weights are merged into the CLLF to generate a new loss function called the weighted compound LINEX loss function (WCLLF). Then, the WCLLF is used to estimate the LD shape parameter through Bayesian and expected Bayesian (E-Bayesian) estimation. Subsequently, we discuss six different types of loss functions, including square error loss function (SELF), LINEX loss function (LLF), asymmetric loss function (ASLF), entropy loss function (ENLF), CLLF, and WCLLF. In addition, in order to check the performance of the proposed loss function, the Bayesian estimator of WCLLF and the E-Bayesian estimator of WCLLF are used, by performing Monte Carlo simulations. The Bayesian and expected Bayesian by using the proposed loss function is compared with other methods, including maximum likelihood estimation (MLE) and Bayesian and E-Bayesian estimators under different loss functions. The simulation results show that the Bayes estimator according to WCLLF and the E-Bayesian estimator according to WCLLF proposed in this work have the best performance in estimating the shape parameters based on the least mean averaged squared error.

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Mohammed Alomari, Huda. "Bayes Estimations for Parameter of the Poisson distribution with Progressive Schemes." Academic Journal of Applied Mathematical Sciences, no. 102 (October 9, 2024): 14–23. https://doi.org/10.32861/ajams.10.2.14.23.

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This study introduces maximum likelihood and Bayesian approaches to Poisson parameter estimation using posterior distribution. I discuss three types of loss functions: the asymmetric linear exponential loss function, non-linear exponential loss function, and squared error loss function. Their performance is compared with the maximum likelihood estimator using mean squared error (MSE) as the test criterion. The proposed method with the classical estimator (maximum likelihood estimator) is better than that with the non-classical estimators for point estimation with different sample sizes. Maximum likelihood estimation provides the optimal performance in estimating the Poisson distribution, as evidenced by the asymptotically smallest MSE values. For small true parameter values the results reveal that the Bayesian approaches have good estimation performance.

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Xiang, Ning, and Christopher Landschoot. "Bayesian Inference for Acoustic Direction of Arrival Analysis Using Spherical Harmonics." Entropy 21, no. 6 (June 10, 2019): 579. http://dx.doi.org/10.3390/e21060579.

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This work applies two levels of inference within a Bayesian framework to accomplish estimation of the directions of arrivals (DoAs) of sound sources. The sensing modality is a spherical microphone array based on spherical harmonics beamforming. When estimating the DoA, the acoustic signals may potentially contain one or multiple simultaneous sources. Using two levels of Bayesian inference, this work begins by estimating the correct number of sources via the higher level of inference, Bayesian model selection. It is followed by estimating the directional information of each source via the lower level of inference, Bayesian parameter estimation. This work formulates signal models using spherical harmonic beamforming that encodes the prior information on the sensor arrays in the form of analytical models with an unknown number of sound sources, and their locations. Available information on differences between the model and the sound signals as well as prior information on directions of arrivals are incorporated based on the principle of the maximum entropy. Two and three simultaneous sound sources have been experimentally tested without prior information on the number of sources. Bayesian inference provides unambiguous estimation on correct numbers of sources followed by the DoA estimations for each individual sound sources. This paper presents the Bayesian formulation, and analysis results to demonstrate the potential usefulness of the model-based Bayesian inference for complex acoustic environments with potentially multiple simultaneous sources.

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Guure, Chris Bambey, Noor Akma Ibrahim, and Al Omari Mohammed Ahmed. "Bayesian Estimation of Two-Parameter Weibull Distribution Using Extension of Jeffreys' Prior Information with Three Loss Functions." Mathematical Problems in Engineering 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/589640.

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The Weibull distribution has been observed as one of the most useful distribution, for modelling and analysing lifetime data in engineering, biology, and others. Studies have been done vigorously in the literature to determine the best method in estimating its parameters. Recently, much attention has been given to the Bayesian estimation approach for parameters estimation which is in contention with other estimation methods. In this paper, we examine the performance of maximum likelihood estimator and Bayesian estimator using extension of Jeffreys prior information with three loss functions, namely, the linear exponential loss, general entropy loss, and the square error loss function for estimating the two-parameter Weibull failure time distribution. These methods are compared using mean square error through simulation study with varying sample sizes. The results show that Bayesian estimator using extension of Jeffreys' prior under linear exponential loss function in most cases gives the smallest mean square error and absolute bias for both the scale parameterαand the shape parameterβfor the given values of extension of Jeffreys' prior.

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Shadmehr, Reza, and David Z. D'Argenio. "A Neural Network for Nonlinear Bayesian Estimation in Drug Therapy." Neural Computation 2, no. 2 (June 1990): 216–25. http://dx.doi.org/10.1162/neco.1990.2.2.216.

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The feasibility of developing a neural network to perform nonlinear Bayesian estimation from sparse data is explored using an example from clinical pharmacology. The problem involves estimating parameters of a dynamic model describing the pharmacokinetics of the bronchodilator theophylline from limited plasma concentration measurements of the drug obtained in a patient. The estimation performance of a backpropagation trained network is compared to that of the maximum likelihood estimator as well as the maximum a posteriori probability estimator. In the example considered, the estimator prediction errors (model parameters and outputs) obtained from the trained neural network were similar to those obtained using the nonlinear Bayesian estimator.

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Wang, Wei, and Wenhao Gui. "Estimation and Bayesian Prediction for the Generalized Exponential Distribution Under Type-II Censoring." Symmetry 17, no. 2 (February 2, 2025): 222. https://doi.org/10.3390/sym17020222.

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This research focuses on the prediction and estimation problems for the generalized exponential distribution under Type-II censoring. Firstly, maximum likelihood estimations for the parameters of the generalized exponential distribution are computed using the EM algorithm. Additionally, confidence intervals derived from the Fisher information matrix are developed and analyzed alongside two bootstrap confidence intervals for comparison. Compared to classical maximum likelihood estimation, Bayesian inference proves to be highly effective in handling censored data. This study explores Bayesian inference for estimating the unknown parameters, considering both symmetrical and asymmetrical loss functions. Utilizing Gibbs sampling to produce Markov Chain Monte Carlo samples, we employ an importance sampling approach to obtain Bayesian estimates and compute the corresponding highest posterior density (HPD) intervals. Furthermore, for one-sample prediction and, separately, for the two-sample case, we provide the corresponding posterior distributions, along with methods for computing point predictions and predictive intervals. Through Monte Carlo simulations, we evaluate the performance of Bayesian estimation in contrast to maximum likelihood estimation. Finally, we conduct an analysis of a real dataset derived from deep groove ball bearings, calculating Bayesian point predictions and predictive intervals for future samples.

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Itagaki, Hiroshi, Hiroo Asada, and Seiichi Itoh. "Bayesian Estimation." Journal of the Society of Naval Architects of Japan 1985, no. 157 (1985): 285–94. http://dx.doi.org/10.2534/jjasnaoe1968.1985.285.

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Liu, Kaiwei, and Yuxuan Zhang. "The E-Bayesian Estimation for Lomax Distribution Based on Generalized Type-I Hybrid Censoring Scheme." Mathematical Problems in Engineering 2021 (May 19, 2021): 1–19. http://dx.doi.org/10.1155/2021/5570320.

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This article studies the E-Bayesian estimation of the unknown parameter of Lomax distribution based on generalized Type-I hybrid censoring. Under square error loss and LINEX loss functions, we get the E-Bayesian estimation and compare its effectiveness with Bayesian estimation. To measure the error of E-Bayesian estimation, the expectation of mean square error (E-MSE) is introduced. With Markov chain Monte Carlo technology, E-Bayesian estimations are computed. Metropolis–Hastings algorithm is applied within the process. Similarly, the credible interval for the parameter is calculated. Then, we can compare the MSE and E-MSE to evaluate whose result is more effective. For the purpose of illustration in real datasets, cases of generalized Type-I hybrid censored samples are presented. In order to judge whether the sample data can be directly fitted by the Lomax distribution, we adopt the Kolmogorov–Smirnov tests for evaluation. Finally, we can get the conclusion after comparing the results of E-Bayesian and Bayesian estimation.

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Ren, Haiping, Qin Gong, and Xue Hu. "Estimation of Entropy for Generalized Rayleigh Distribution under Progressively Type-II Censored Samples." Axioms 12, no. 8 (August 10, 2023): 776. http://dx.doi.org/10.3390/axioms12080776.

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This paper investigates the problem of entropy estimation for the generalized Rayleigh distribution under progressively type-II censored samples. Based on progressively type-II censored samples, we first discuss the maximum likelihood estimation and interval estimation of Shannon entropy for the generalized Rayleigh distribution. Then, we explore the Bayesian estimation problem of entropy under three types of loss functions: K-loss function, weighted squared error loss function, and precautionary loss function. Due to the complexity of Bayesian estimation computation, we use the Lindley approximation and MCMC method for calculating Bayesian estimates. Finally, using a Monte Carlo statistical simulation, we compare the mean square errors to examine the superiority of maximum likelihood estimation and Bayesian estimation under different loss functions. An actual example is provided to verify the feasibility and practicality of various estimations.

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Mudasir, Sofi, Ajaz Bhat, Sheikh Ahmad, Aasimeh Rehman, Taghreed Jawa, Neveen Sayed-Ahmed, and Ahlam Tolba. "A dual approach to parameter estimation classical vs. Bayesian methods in power Rayleigh modelling." Thermal Science 28, no. 6 Part B (2024): 4877–94. https://doi.org/10.2298/tsci2406877m.

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In this article, we investigated the problem of estimating the parameters of power Rayleigh distribution using a range of classical and Bayesian estimate strategies. For applied statisticians and reliability engineers, parameter estimation provides a guide for choosing the best method of estimating the model parameters. Six frequentist estimation methods, including maximum likelihood estimation, Cramer-von Mises estimation, Anderson-Darling estimation, least square estimation, weighted least square estimation, and maximum product of spacing estimation, were taken into consideration when estimating the parameters of the power Rayleigh model. The expressions for Bayes estimators of the scale parameter are derived under squared error and precautionary loss functions and utilizing extensions of Jeffrey?s prior and natural conjugate priors. To investigate the finite sample properties of the parameter estimations, Monte Carlo simulations are also performed. Finally, two applications to real data are used to highlight the versatility of the suggested model and the comparison is made with the Rayleigh and some of its well-known extensions such as exponentiated Rayleigh and weighted Rayleigh distributions.

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Gao, Huiqing, Zhanshou Chen, and Fuxiao Li. "Linear Bayesian Estimation of Misrecorded Poisson Distribution." Entropy 26, no. 1 (January 11, 2024): 62. http://dx.doi.org/10.3390/e26010062.

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Parameter estimation is an important component of statistical inference, and how to improve the accuracy of parameter estimation is a key issue in research. This paper proposes a linear Bayesian estimation for estimating parameters in a misrecorded Poisson distribution. The linear Bayesian estimation method not only adopts prior information but also avoids the cumbersome calculation of posterior expectations. On the premise of ensuring the accuracy and stability of computational results, we derived the explicit solution of the linear Bayesian estimation. Its superiority was verified through numerical simulations and illustrative examples.

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Hu, Xue, and Haiping Ren. "Statistical inference of the stress-strength reliability for inverse Weibull distribution under an adaptive progressive type-Ⅱ censored sample." AIMS Mathematics 8, no. 12 (2023): 28465–87. http://dx.doi.org/10.3934/math.20231457.

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<abstract><p>In this paper, we investigate classical and Bayesian estimation of stress-strength reliability $\delta = P(X &gt; Y)$ under an adaptive progressive type-Ⅱ censored sample. Assume that $X$ and $Y$ are independent random variables that follow inverse Weibull distribution with the same shape but different scale parameters. In classical estimation, the maximum likelihood estimator and asymptotic confidence interval are deduced. An approximate maximum likelihood estimator approach is used to obtain the explicit form. In Bayesian estimation, the Bayesian estimators are derived based on symmetric entropy loss function and LINEX loss function. Due to the complexity of integrals, we proposed Lindley's approximation to get the approximate Bayesian estimates. To compare the different estimators, we performed Monte Carlo simulations. Under gamma prior, the approximate maximum likelihood estimator performs better than Bayesian estimators. Under non-informative prior, the approximate maximum likelihood estimator has the same behavior as Bayesian estimators. In the end, two data sets are used to prove the effectiveness of the proposed methods.</p></abstract>

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Yanuar, Ferra, Sintya Wulandari, Yudiantri Asdi, Aidinil Zetra, and Haripamyu. "Modeling of Human Development Index Using Bayesian Spatial Autoregressive Approach." Science and Technology Indonesia 10, no. 1 (January 1, 2025): 72–79. https://doi.org/10.26554/sti.2025.10.1.72-79.

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Spatial regression analysis is a technique employed to examine the relationship between independent and dependent variables in datasets that exhibit regional neighborhood influences or spatial effects. When a spatial effect exists for the independent variable, the Spatial Autoregressive (SAR) regression can be utilized. The Maximum Likelihood Estimation (MLE) is a commonly used parameter estimator for SAR. However, due to the limitations of MLE, the Bayesian method provides an alternative approach for parameter estimation. This study compares the results of SAR estimations using both MLE and Bayesian methods to determine the most accurate estimation model. Both methods were implemented in this research to model the factors affecting the Human Development Index (HDI) in East Java Province for the year 2022. The findings indicate that the Bayesian SAR offers a superior proposed model compared to the MLE SAR. The factors influencing the HDI in East Java Province in 2022 include poverty, per capita expenditure, and the presence of an upper middle-class manufacturing industry.

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Liu, Zhi-Qiang. "Bayesian Paradigms in Image Processing." International Journal of Pattern Recognition and Artificial Intelligence 11, no. 01 (February 1997): 3–33. http://dx.doi.org/10.1142/s0218001497000020.

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A large number of image and spatial information processing problems involves the estimation of the intrinsic image information from observed images, for instance, image restoration, image registration, image partition, depth estimation, shape reconstruction and motion estimation. These are inverse problems and generally ill-posed. Such estimation problems can be readily formulated by Bayesian models which infer the desired image information from the measured data. Bayesian paradigms have played a very important role in spatial data analysis for over three decades and have found many successful applications. In this paper, we discuss several aspects of Bayesian paradigms: uncertainty present in the observed image, prior distribution modeling, Bayesian-based estimation techniques in image processing, particularly, the maximum a posteriori estimator and the Kalman filtering theory, robustness, and Markov random fields and applications.

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Verma, Vivek, Dilip C. Nath, and S. N. Dwivedi. "Bayesian estimation of fertility rates under imperfect age reporting." Statistics in Transition new series 24, no. 2 (March 15, 2023): 39–57. http://dx.doi.org/10.59170/stattrans-2023-019.

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This article outlines the application of the Bayesian method of parameter estimation to situations where the probability of age misreporting is high, leading to transfers of an individual from one age group to another. An essential requirement for Bayesian estimation is prior distribution, derived for both perfect and imperfect age reporting. As an alternative to the Bayesian methodology, a classical estimator based on the maximum likelihood principle has also been discussed. Here, the age misreporting probability matrix has been constructed using a performance indicator, which incorporates the relative performance of estimators based on age when reported correctly instead of misreporting. The initial guess of performance indicators can either be empirically or theoretically derived. The method has been illustrated by using data on Empowered Action Group (EAG) states of India from National Family Health Survey-3 (2005–2006) to estimate the total marital fertility rates. The present study reveals through both a simulation and real-life set-up that the Bayesian estimation method has been more promising and reliable in estimating fertility rates, even in situations where age misreporting is higher than in case of classical maximum likelihood estimates.

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Gustafson, Steven C., Christopher S. Costello, Eric C. Like, Scott J. Pierce, and Kiran N. Shenoy. "Bayesian Threshold Estimation." IEEE Transactions on Education 52, no. 3 (August 2009): 400–403. http://dx.doi.org/10.1109/te.2008.930092.

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Adams, W. J., and P. Mamassian. "Bayesian slant estimation." Journal of Vision 1, no. 3 (March 14, 2010): 175. http://dx.doi.org/10.1167/1.3.175.

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Abdel-Mottaleb, Mohamed, and Azriel Rosenfeld. "Inexact Bayesian estimation." Pattern Recognition 25, no. 6 (June 1992): 641–46. http://dx.doi.org/10.1016/0031-3203(92)90080-3.

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Liechty, J. C. "Bayesian correlation estimation." Biometrika 91, no. 1 (March 1, 2004): 1–14. http://dx.doi.org/10.1093/biomet/91.1.1.

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Christensen, Ronald. "Inconsistent Bayesian estimation." Bayesian Analysis 4, no. 4 (December 2009): 759–62. http://dx.doi.org/10.1214/09-ba428.

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Kramer, S. C., and H. W. Sorenson. "Bayesian parameter estimation." IEEE Transactions on Automatic Control 33, no. 2 (1988): 217–22. http://dx.doi.org/10.1109/9.395.

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Yuanxi, Yang. "Robust bayesian estimation." Bulletin Géodésique 65, no. 3 (September 1991): 145–50. http://dx.doi.org/10.1007/bf00806343.

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Han, Ming. "E-Bayesian Estimation and Hierarchical Bayesian Estimation of Failure Probability." Communications in Statistics - Theory and Methods 40, no. 18 (September 15, 2011): 3303–14. http://dx.doi.org/10.1080/03610926.2010.498643.

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Han, Ming. "E-Bayesian estimation and hierarchical Bayesian estimation of failure rate." Applied Mathematical Modelling 33, no. 4 (April 2009): 1915–22. http://dx.doi.org/10.1016/j.apm.2008.03.019.

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Jia, Tianyi, Hongwei Liu, Penghui Wang, and Chang Gao. "Bayesian Direction of Arrival Estimation with Prior Knowledge from Target Tracker." Remote Sensing 15, no. 13 (June 24, 2023): 3255. http://dx.doi.org/10.3390/rs15133255.

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The performance of traditional direction of arrival (DOA) estimation methods always deteriorates at a low signal-to-noise ratio (SNR) or without sufficient observations. This paper investigates the Bayesian DOA estimation problem aided by the prior knowledge from the target tracker. The Bayesian Cramér–Rao lower bounds (CRLB) and the expected CRLB are first derived to evaluate the theoretical performance of Bayesian DOA estimation. Based on the maximum a posterior (MAP) estimator in the Bayesian framework, two methods are proposed. One is a two-step grid search method for a single target DOA case. The other is a gradient-based iterative solution for multiple targets DOA case, which extends the traditional Newton method by incorporating the prior knowledge. We also propose a minimum mean square error (MMSE) estimator using a Monte Carlo method, which requires trading off accuracy against computational complexity. By comparing with the maximum likelihood (ML) estimators and the MUSIC algorithm, the proposed three Bayesian estimators improve the DOA estimation performance in low SNR or with limited snapshots. Moreover, the performance is not affected by the correlation between sources.

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Wijayanti, Rina. "PENAKSIRAN PARAMETER ANALISIS REGRESI COX DAN ANALISIS SURVIVAL BAYESIAN." PRISMATIKA: Jurnal Pendidikan dan Riset Matematika 1, no. 2 (June 1, 2019): 16–26. http://dx.doi.org/10.33503/prismatika.v1i2.427.

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In the theory of estimation, there are two approaches, namely the classical statistical approach and global statistical approach (Bayesian). Classical statistics are statistics in which the procedure is the decision based only on the data samples taken from the population. While Bayesian statistics in making decisions based on new information from the observed data (sample) and prior knowledge. At this writing Cox Regression Analysis will be taken as an example of parameter estimation by the classical statistical approach Survival Analysis and Bayesian statistical approach as an example of global (Bayesian). Survival Bayesial parameter estimation using MCMC algorithms for model complex / complicated and difficult to resolve while the Cox regression models using the method of partial likelihood. Results of the parameter estimates do not close form that needs to be done by the method of Newton-Raphson iteration.

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Asteris, Georgios, and Sahotra Sarkar. "Bayesian Procedures for the Estimation of Mutation Rates from Fluctuation Experiments." Genetics 142, no. 1 (January 1, 1996): 313–26. http://dx.doi.org/10.1093/genetics/142.1.313.

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Bayesian procedures are developed for estimating mutation rates from fluctuation experiments. Three Bayesian point estimators are compared with four traditional ones using the results of 10,000 simulated experiments. The Bayesian estimators were found to be at least as efficient as the best of the previously known estimators. The best Bayesian estimator is one that uses (1/m 2) as the prior probability density function and a quadratic loss function. The advantage of using these estimators is most pronounced when the number of fluctuation test tubes is small. Bayesian estimation allows the incorporation of prior knowledge about the estimated parameter, in which case the resulting estimators are the most efficient. It enables the straightfonvard construction of confidence intervals for the estimated parameter. The increase of efficiency with prior information and the narrowing of the confidence intervals with additional experimental results are investigated. The results of the simulations show that any potential inaccuracy of estimation arising from lumping together all cultures with more than n mutants (the jackpots) almost disappears at n = 70 (provided that the number of mutations in a culture is low). These methods are applied to a set of experimental data to illustrate their use.

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Alduais, Fuad. "Comparison of classical and Bayesian estimators to estimate the parameters in Weibull distribution under weighted general entropy loss function." International Journal of ADVANCED AND APPLIED SCIENCES 8, no. 3 (March 2021): 57–62. http://dx.doi.org/10.21833/ijaas.2021.03.008.

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In this work, we have developed a General Entropy loss function (GE) to estimate parameters of Weibull distribution (WD) based on complete data when both shape and scale parameters are unknown. The development is done by merging weight into GE to produce a new loss function called the weighted General Entropy loss function (WGE). Then, we utilized WGE to derive the parameters of the WD. After, we compared the performance of the developed estimation in this work with the Bayesian estimator using the GE loss function. Bayesian estimator using square error (SE) loss function, Ordinary Least Squares Method (OLS), Weighted Least Squared Method (WLS), and maximum likelihood estimation (MLE). Based on the Monte Carlo simulation method, those estimators are compared depending on the mean squared errors (MSE’s). The results show that the performance of the Bayes estimator under developed method (WGE) loss function is the best for estimating shape parameters in all cases and has good performance for estimating scale parameter.

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Adepoju, Adedayo A., Oluwayemisi O, Alaba, and P. Ogundunmadetayo. "Bayesian estimation of simultaneous equation model with lagged endogenous variables and first order serially correlated errors." Global Journal of Pure and Applied Sciences 24, no. 2 (December 18, 2018): 235–44. http://dx.doi.org/10.4314/gjpas.v24i2.14.

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Most simultaneous equation models involve the inclusion of lagged endogenous and/or exogenous variables and sometimes it may be misleading to assume that the errors are normally distributed when in reality they exhibit functional formsthat are not normal especially in practical situations. The classical methods of estimating parameters of simultaneous equation models are usually affected by the presence of autocorrelation among the error terms. Unfortunately, in practice the form of correlation between the pairs of the random deviates is unknown.In this paper classical and Bayesian methods for the estimation of simultaneous equation model withlagged endogenous variables and first order serially correlated errors are considered. The smallsample properties of the methods at different levels of correlation for ρ = 0.2, 0.5 and 0.8are compared.Better parameter estimates were produced by the Bayesian estimator with smaller standard errors compared to the classical method. The standard deviations of the Bayesian estimator are consistently better than those of the OLS estimator for the sample sizes considered. For example, the standard deviations of the Bayesian for b14 (the coefficient of the lagged endogenous variable,y 1t-1) when ρ = 0.2 at N = 10, 15, 20 and 25 were 0.07712781, 0.05433923, 0.03230012 and 0.03177252 respectively while those of OLS were 0.0784732, 0.4718914, 0.05701936 and 1.31422868. However, when ρ = 0.8, the standard deviations were 0.0548055, 0.03860254, 0.02572899 and 0.02126175 for Bayesian and 0.0562190, 0.03882345, 0.053676 and 0.0315632 for OLS. Interestingly, notice that even at high correlation level, the estimates produced by the Bayesian method are closer to the parameter values and the standard deviations decrease as the sample size increases. Hence, the Bayesian estimation method might be a better choice when lagged endogenous variables are included in a simultaneous equation model with auto-correlated disturbances since it appeared to give better results compared to the classical approach.Keywords: Bayesian estimation, Lagged endogenous variables, Simultaneous equations, Monte-Carlo Simulation, First-order autoregressive process.

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Acquah, Henry De-Graft. "Bayesian Logistic Regression Modelling via Markov Chain Monte Carlo Algorithm." Journal of Social and Development Sciences 4, no. 4 (April 30, 2013): 193–97. http://dx.doi.org/10.22610/jsds.v4i4.751.

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This paper introduces Bayesian analysis and demonstrates its application to parameter estimation of the logistic regression via Markov Chain Monte Carlo (MCMC) algorithm. The Bayesian logistic regression estimation is compared with the classical logistic regression. Both the classical logistic regression and the Bayesian logistic regression suggest that higher per capita income is associated with free trade of countries. The results also show a reduction of standard errors associated with the coefficients obtained from the Bayesian analysis, thus bringing greater stability to the coefficients. It is concluded that Bayesian Markov Chain Monte Carlo algorithm offers an alternative framework for estimating the logistic regression model.

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Ozguven, Eren Erman, and Kaan Ozbay. "Nonparametric Bayesian Estimation of Freeway Capacity Distribution from Censored Observations." Transportation Research Record: Journal of the Transportation Research Board 2061, no. 1 (January 2008): 20–29. http://dx.doi.org/10.3141/2061-03.

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Previous studies have been made of the usefulness and effectiveness of survival analysis in transportation and traffic engineering studies with incomplete data in which the Kaplan–Meier estimate is proposed for determining traffic capacity distribution. However, well-known estimators like Kaplan–Meier and Nelson–Aalen have several disadvantages that make it difficult to obtain the traffic capacity distribution. First, neither estimator is defined for all values of traffic flows possible. That is, the maximum flow followed by a breakdown defines the final point of the estimated distribution curve. Therefore, parametric fitting tools have to be applied to obtain the remaining portion of the curve. Moreover, the discontinuity and nonsmoothness of the Kaplan–Meier and Nelson–Aalen estimates make it difficult to ensure the robustness of the estimation. In this paper the Kaplan–Meier and Nelson–Aalen nonparametric estimators are used to obtain the traffic capacity function of four freeway sections. Then a Bayesian nonparametric estimator, which is shown to be a Bayesian extension of the Kaplan–Meier estimator, is introduced for estimating the capacity distribution. This estimator assumes a Dirichlet process prior for the survival function under the minimization of a squared-error loss function. The results indicate that the curves obtained by using the Bayesian estimation method are smoother than those obtained with the other estimator. This smoothness also ensures the continuity in the vicinity of censored observations. Furthermore, the Bayesian estimates can be obtained for any traffic flow value regardless of the availability of data only for certain ranges of observations (including censored data).

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Han, Ming. "Estimation of Failure Rate and its Applications in the Case of Zero-Failure Data." Advanced Materials Research 945-949 (June 2014): 1046–49. http://dx.doi.org/10.4028/www.scientific.net/amr.945-949.1046.

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This paper introduces a new method, named E-Bayesian estimation method, to estimate failure rate in zero-failure data. The definition of E-Bayesian estimation of the failure rate is given, based on the definition, the formulas of E-Bayesian estimation and hierarchical Bayesian estimation of the failure rate were provided, and properties of the E-Bayesian estimation, i. e. relations between E-Bayesian estimation and hierarchical Bayesian estimation, was discussed. Calculations were performed on practical problems, showing that the proposed new method is feasible and easy to operate.

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Journal, Baghdad Science. "Bayes and Non-Bayes Estimation Methods for the Parameter of Maxwell-Boltzmann Distribution." Baghdad Science Journal 14, no. 4 (December 3, 2017): 808–12. http://dx.doi.org/10.21123/bsj.14.4.808-812.

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In this paper, point estimation for parameter ? of Maxwell-Boltzmann distribution has been investigated by using simulation technique, to estimate the parameter by two sections methods; the first section includes Non-Bayesian estimation methods, such as (Maximum Likelihood estimator method, and Moment estimator method), while the second section includes standard Bayesian estimation method, using two different priors (Inverse Chi-Square and Jeffrey) such as (standard Bayes estimator, and Bayes estimator based on Jeffrey's prior). Comparisons among these methods were made by employing mean square error measure. Simulation technique for different sample sizes has been used to compare between these methods.

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Stenling, Andreas, Andreas Ivarsson, Urban Johnson, and Magnus Lindwall. "Bayesian Structural Equation Modeling in Sport and Exercise Psychology." Journal of Sport and Exercise Psychology 37, no. 4 (August 2015): 410–20. http://dx.doi.org/10.1123/jsep.2014-0330.

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Bayesian statistics is on the rise in mainstream psychology, but applications in sport and exercise psychology research are scarce. In this article, the foundations of Bayesian analysis are introduced, and we will illustrate how to apply Bayesian structural equation modeling in a sport and exercise psychology setting. More specifically, we contrasted a confirmatory factor analysis on the Sport Motivation Scale II estimated with the most commonly used estimator, maximum likelihood, and a Bayesian approach with weakly informative priors for cross-loadings and correlated residuals. The results indicated that the model with Bayesian estimation and weakly informative priors provided a good fit to the data, whereas the model estimated with a maximum likelihood estimator did not produce a well-fitting model. The reasons for this discrepancy between maximum likelihood and Bayesian estimation are discussed as well as potential advantages and caveats with the Bayesian approach.

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Wickramasinghe, Lahiru, Alexandre Leblanc, and Saman Muthukumarana. "Bayesian inference on sparse multinomial data using smoothed Dirichlet distribution with an application to COVID-19 data." Model Assisted Statistics and Applications 18, no. 3 (September 28, 2023): 207–26. http://dx.doi.org/10.3233/mas-221411.

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We develop a Bayesian approach for estimating multinomial cell probabilities using a smoothed Dirichlet prior. The most important feature of the smoothed Dirichlet prior is that it forces the probabilities of neighboring cells to be closer to each other than under the standard Dirichlet prior. We propose a shrinkage-type estimator using this Bayesian approach to estimate multinomial cell probabilities. The proposed estimator allows us to borrow information across other multinomial populations and cell categories simultaneously to improve the estimation of cell probabilities, especially in a context of sparsity with ordered categories. We demonstrate the proposed approach using COVID-19 data and estimate the distribution of positive COVID-19 cases across age groups for Canadian health regions. Our approach allows improved estimation in smaller health regions where few cases have been observed.

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Li, Mingyao, and Juanping Zhu. "Bayesian Adaptive Estimation with Theoretical Bound: An Exploration-Exploitation Approach." Computational Intelligence and Neuroscience 2022 (December 12, 2022): 1–9. http://dx.doi.org/10.1155/2022/1143056.

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This paper investigates the theoretical bound to reduce the parameter uncertainty in Bayesian adaptive estimation for psychometric functions and proposes an exploration-exploitation (E-E) approach to improve the computation efficiency for parameter estimations. When the experimental trial goes on, the uncertainty of the parameters decreases dramatically and the space between the maximal mutual information and the theoretical bound gets narrower, so the advantage of classical Bayesian adaptive estimation algorithm diminishes. This approach tries to trade off the exploration (parameter posterior uncertainty) and the exploitation (parameter mean estimation). The experimental results show that the proposed E-E approach estimates parameters for psychometric functions with same convergence and reduces the computation time by more than 34.27%, compared with the classical Bayesian adaptive estimation.

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Hashimoto, Noriaki, and Koji Konbune. "DIRECTIONAL SPECTRUM ESTIMATION FROM A BAYESIAN APPROACH." Coastal Engineering Proceedings 1, no. 21 (January 29, 1988): 4. http://dx.doi.org/10.9753/icce.v21.4.

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A new directional spectral estimation method using a Bayesian approach is proposed. The proposed method is examined for numerical simulation data, and the validity of the method is discussed. Some examples of the directional spectra estimated from field observation data attained at an offshore oil rig utilizing seven wave probes are also shown in this report. The major conclusions of the report are : (1) The proposed method can be applied for more than four arbitrarily mixed instrument array measurements. (2) It has a higher resolution power than other existing methods for estimating directional spectrum. (3) It is a better method for estimating directional spectra from the cross-power spectra contaminated with estimation errors. (4.) It is more adaptable to reformulation of the estimation equations as the study of structures of directional spectrum progesses.

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Han, Ming. "E-Bayesian Estimation Method and its Applications in Reliability Engineering." Advanced Materials Research 199-200 (February 2011): 308–12. http://dx.doi.org/10.4028/www.scientific.net/amr.199-200.308.

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Evaluation method of reliability of industrial products needs to be improved effectively with the advance of science and technology. This paper introduces a new method, named E-Bayesian estimation method, to estimate failure probability in reliability engineering. The definition of E-Bayesian estimation of the failure probability is provided, moreover, the formulas of E-Bayesian estimation and hierarchical Bayesian estimation of the failure probability were provided, and properties of the E-Bayesian estimation, i.e. relations between E-Bayesian estimation and hierarchical Bayesian estimation, are also provided. Finally, calculation on practical problems shows that the provided method is feasible and easy to perform.

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Han, Ming. "Estimation of Failure Probability for Bearing and its Applications in the Case of Zero-Failure Data." Advanced Materials Research 915-916 (April 2014): 318–22. http://dx.doi.org/10.4028/www.scientific.net/amr.915-916.318.

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This paper introduces a new method, named E-Bayesian estimation method, to estimate failure probability. In the case of zero-failure data, the definition of E-Bayesian estimation of failure probability is provided; moreover, the formulas of E-Bayesian estimation and hierarchical Bayesian estimation and the property of E-Bayesian estimation of the failure probability are also provided. For the estimate failure probability, in the following sections we will see simple the E-Bayesian estimation method is method than hierarchical Bayesian estimation method. Finally, the calculated results of bearing show that the proposed method is feasible and convenient in engineering application.

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Tang, Yuanyuan, Philip G. Jones, Liangrui Sun, Suzanne V. Arnold, and John A. Spertus. "Constraint approaches to the estimation of relative risk." Statistical Methods in Medical Research 27, no. 11 (April 13, 2017): 3436–46. http://dx.doi.org/10.1177/0962280217702934.

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In medical and epidemiologic studies, relative risk is usually the parameter of interest. However, calculating relative risk using standard log-Binomial regression approach often encounters non-convergence. A modified Poisson regression, which uses robust variance, was proposed by Zou in 2004. Although the modified Poisson regression with sandwich variance estimator is valid for the estimation of relative risk, the predicted probability of the outcome may be greater than the natural boundary 1 for the unobserved but plausible covariate combinations. Moreover, the lower and upper bounds of confidence intervals for predicted probabilities could fall out of (0, 1). Chu and Cole, in 2010, proposed a Bayesian approach to overcome this issue. Posterior median was used to get the parameter estimation. However, the Bayesian approach may provide biased estimation, especially when the probability of outcome is high. In this article, we propose an alternative constraint optimization approach for estimating relative risk. Our approach can reach similar or better performance than Bayesian approach in terms of bias, root mean square error, coverage rate, and predictive probabilities. Simulation studies are conducted to demonstrate the usefulness of this approach. Our method is also illustrated by Prospective Registry Evaluating Myocardial Infarction: Event and Recovery data.

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Liu, Qing, David Pitt, Xibin Zhang, and Xueyuan Wu. "A Bayesian Approach to Parameter Estimation for Kernel Density Estimation via Transformations." Annals of Actuarial Science 5, no. 2 (April 18, 2011): 181–93. http://dx.doi.org/10.1017/s1748499511000030.

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AbstractIn this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibits a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel. In the current literature, there have been some developments in the area of estimating densities based on transformed data, where bandwidth selection usually depends on pre-determined transformation parameters. Moreover, in the bivariate situation, the transformation parameters were estimated for each dimension individually. We use a Bayesian sampling algorithm and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters simultaneously within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is better captured through the bivariate density estimator based on transformed data.

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Körding, Konrad P., Shih-pi Ku, and Daniel M. Wolpert. "Bayesian Integration in Force Estimation." Journal of Neurophysiology 92, no. 5 (November 2004): 3161–65. http://dx.doi.org/10.1152/jn.00275.2004.

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When we interact with objects in the world, the forces we exert are finely tuned to the dynamics of the situation. As our sensors do not provide perfect knowledge about the environment, a key problem is how to estimate the appropriate forces. Two sources of information can be used to generate such an estimate: sensory inputs about the object and knowledge about previously experienced objects, termed prior information. Bayesian integration defines the way in which these two sources of information should be combined to produce an optimal estimate. To investigate whether subjects use such a strategy in force estimation, we designed a novel sensorimotor estimation task. We controlled the distribution of forces experienced over the course of an experiment thereby defining the prior. We show that subjects integrate sensory information with their prior experience to generate an estimate. Moreover, subjects could learn different prior distributions. These results suggest that the CNS uses Bayesian models when estimating force requirements.

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Kim, Su-Young, David Huh, Zhengyang Zhou, and Eun-Young Mun. "A comparison of Bayesian to maximum likelihood estimation for latent growth models in the presence of a binary outcome." International Journal of Behavioral Development 44, no. 5 (January 10, 2020): 447–57. http://dx.doi.org/10.1177/0165025419894730.

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Latent growth models (LGMs) are an application of structural equation modeling and frequently used in developmental and clinical research to analyze change over time in longitudinal outcomes. Maximum likelihood (ML), the most common approach for estimating LGMs, can fail to converge or may produce biased estimates in complex LGMs especially in studies with modest samples. Bayesian estimation is a logical alternative to ML for LGMs, but there is a lack of research providing guidance on when Bayesian estimation may be preferable to ML or vice versa. This study compared the performance of Bayesian versus ML estimators for LGMs by evaluating their accuracy via Monte Carlo (MC) simulations. For the MC study, longitudinal data sets were generated and estimated using LGM via both ML and Bayesian estimation with three different priors, and parameter recovery across the two estimators was evaluated to determine their relative performance. The findings suggest that ML estimation is a reasonable choice for most LGMs, unless it fails to converge, which can occur with limiting data situations (i.e., just a few time points, no covariate or outcome, modest sample sizes). When models do not converge using ML, we recommend Bayesian estimation with one caveat that the influence of the priors on estimation may have to be carefully examined, per recent recommendations on Bayesian modeling for applied researchers.

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Gao, Xiaoguang, Yu Yang, and Zhigao Gao. "Learning Bayesian networks by constrained Bayesian estimation." Journal of Systems Engineering and Electronics 30, no. 03 (June 25, 2019): 511–24. http://dx.doi.org/10.21629/jsee.2019.03.09.

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Nassar, Mazen, Refah Alotaibi, Hassan Okasha, and Liang Wang. "Bayesian Estimation Using Expected LINEX Loss Function: A Novel Approach with Applications." Mathematics 10, no. 3 (January 29, 2022): 436. http://dx.doi.org/10.3390/math10030436.

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The loss function plays an important role in Bayesian analysis and decision theory. In this paper, a new Bayesian approach is introduced for parameter estimation under the asymmetric linear-exponential (LINEX) loss function. In order to provide a robust estimation and avoid making subjective choices, the proposed method assumes that the parameter of the LINEX loss function has a probability distribution. The Bayesian estimator is then obtained by taking the expectation of the common LINEX-based Bayesian estimator over the probability distribution. This alternative proposed method is applied to estimate the exponential parameter by considering three different distributions of the LINEX parameter, and the associated Bayes risks are also obtained in consequence. Extensive simulation studies are conducted in order to compare the performance of the proposed new estimators. In addition, three real data sets are analyzed to investigate the applicability of the proposed results. The results of the simulation and real data analysis show that the proposed estimation works satisfactorily and performs better than the conventional standard Bayesian approach in terms of minimum mean square error and Bayes risk.

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Neath, Andrew A., and Natalie Langenfeld. "A Note on the Comparison of the Bayesian and Frequentist Approaches to Estimation." Advances in Decision Sciences 2012 (October 22, 2012): 1–12. http://dx.doi.org/10.1155/2012/764254.

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Samaniego and Reneau presented a landmark study on the comparison of Bayesian and frequentist point estimators. Their findings indicate that Bayesian point estimators work well in more situations than were previously suspected. In particular, their comparison reveals how a Bayesian point estimator can improve upon a frequentist point estimator even in situations where sharp prior knowledge is not necessarily available. In the current paper, we show that similar results hold when comparing Bayesian and frequentist interval estimators. Furthermore, the development of an appropriate interval estimator comparison offers some further insight into the estimation problem.

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Shim, Heejung, and Bret Larget. "BayesCAT: Bayesian co-estimation of alignment and tree." Biometrics 74, no. 1 (January 18, 2017): 270–79. http://dx.doi.org/10.1111/biom.12640.

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Rasheed, Huda Abdullah, and Maryam N. Abd. "Bayesian Estimation for Two Parameters of Exponential Distribution under Different Loss Functions." Ibn AL-Haitham Journal For Pure and Applied Sciences 36, no. 2 (April 20, 2023): 289–300. http://dx.doi.org/10.30526/36.2.2946.

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In this paper, two parameters for the Exponential distribution were estimated using theBayesian estimation method under three different loss functions: the Squared error loss function,the Precautionary loss function, and the Entropy loss function. The Exponential distribution priorand Gamma distribution have been assumed as the priors of the scale γ and location δ parametersrespectively. In Bayesian estimation, Maximum likelihood estimators have been used as the initialestimators, and the Tierney-Kadane approximation has been used effectively. Based on the MonteCarlosimulation method, those estimators were compared depending on the mean squared errors (MSEs).The results showed that the Bayesian estimation under the Entropy loss function,assuming Exponential distribution and Gamma distribution priors for the scale and locationparameters, respectively, is the best estimator for the scale parameter. The best estimation methodfor location is the Bayesian estimation under the Entropy loss function in case of a small value ofthe scale γ (say γ < 1). Bayesian estimation under the Precautionary loss function is the best incase of a relatively large value of the scale γ (say γ > 1).

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