The interplay between the Bayesian and frequentist approaches: a general nesting spatial panel data model

Gross domestic product (GDP) serves as a vital indicator of a country’s economic health, influenced by factors such as unemployment rates, export-import dynamics, and inflation rates. Understanding the intricate relationship between GDP and unemployment is crucial for navigating the dynamic business environment. This study investigates this relationship in Somalia using both frequentist and Bayesian regression approaches. The frequentist approach, particularly ordinary least squares, is chosen for its widespread use and simplicity in estimating parameters by minimizing the sum of squared deviations. Meanwhile, the Bayesian approach is adopted for its flexibility in integrating prior information and updating estimates as new data emerge, providing a probabilistic framework that accommodates parameter uncertainty. Secondary data from the World Bank spanning 1991–2020 were analyzed. The results reveal a significant negative association between unemployment and GDP in both models. Comparative analysis employing mean absolute error, root mean square error, and mean square error indicates comparable predictive accuracy between the two approaches, with the Bayesian model demonstrating slight advantages. Bayesian convergence diagnostics affirm satisfactory model stability. These findings offer valuable insights for policymakers, aiding in the formulation of strategies to address unemployment challenges and stimulate economic growth in Somalia.The study investigates the relationship between unemployment and GDP in Somalia, employing both frequentist and Bayesian statistical methods. The frequentist approach provides point estimates and confidence intervals based on observed data, while the Bayesian approach incorporates prior knowledge and uncertainty into the analysis, yielding posterior distributions. By comparing these methods, the research aims to provide a comprehensive understanding of how GDP influences unemployment rates and vice versa. The results highlight the strengths and limitations of each approach, offering valuable insights for economic policymakers and contributing to the broader field of economic analysis in developing countries.

Until recently, uncertainty quantification in low energy nuclear theory was typically performed using frequentist approaches. However in the last few years, the field has shifted toward Bayesian statistics for evaluating confidence intervals. Although there are statistical arguments to prefer the Bayesian approach, no direct comparison is available. In this work, we compare, directly and systematically, the frequentist and Bayesian approaches to quantifying uncertainties in direct nuclear reactions. Starting from identical initial assumptions, we determine confidence intervals associated with the elastic and the transfer process for both methods, which are evaluated against data via a comparison of the empirical coverage probabilities. Expectedly, the frequentist approach is not as flexible as the Bayesian approach in exploring parameter space and often ends up in a different minimum. We also show that the two methods produce significantly different correlations. In the end, the frequentist approach produces significantly narrower uncertainties on the considered observables than the Bayesian. Our study demonstrates that the uncertainties on the reaction observables considered here within the Bayesian approach represent reality more accurately than the much narrower uncertainties obtained using the standard frequentist approach.

In this study we compared the efficiency of frequentist approach with Bayesian approach by carrying out extreme value analysis of Annual Maximum Daily Rainfall (AMDR). For frequentist frequency analysis of AMDR, we used the data of one station i.e. Lahore in Punjab province, Pakistan while for Bayesian analysis we used the data of three other neighboring stations as prior information. During frequentist approach, Generalized Extreme Value (GEV) was found to be a best-fit distribution on the data. In frequentist method, the parameters of GEV distribution were estimated using Maximum Likelihood Estimation (MLE), while in the Bayesian framework the Markov Chain Monte Carlo (MCMC) simulation technique along with Metropolis-Hasting algorithm and Gibbs sampler were implemented. Findings of this study indicate that despite the asymptotic requirements of the MLE, its performance can be enhanced by adopting the Bayesian MCMC paradigm. In order to acquire the posterior densities of GEV parameters, non-informative and informative priors based on the historical data of surrounding weather stations were employed. The result of Bayesian MCMC might be affected by the choice of priors. In addition, the performance of the parameters estimation methods was verified by employing several robustness measures. Robustness measures results proved that the Bayesian MCMC method performed better than MLE in estimating GEV parameters and future return levels. Therefore, the findings of these analyses could be helpful in adopting proper flood protection measures and designing infrastructures of culverts, buildings, bridges and dams in the region.

In the frailty Cox model, frequentist approaches often present problems of numerical resolution, convergence, and variance calculation. The Bayesian approach offers an alternative. The goal of this study was to compare, using real (calf gastroenteritis) and simulated data, the results obtained with the MCMC method used in the Bayesian approach versus two frequentist approaches: the Newton–Raphson algorithm to solve a penalized likelihood and the EM algorithm. The results obtained showed that when the number of groups in the population decreases, the Bayesian approach gives a less biased estimation of the frailty variance and of the group fixed effect than the frequentist approaches.

The most commonly used method of small area estimation (SAE) is the empirical best linear unbiased prediction method based on a linear mixed model. However, it is not appropriate in the case of the zero-inflated target variable with a mixture of zeros and continuously distributed positive values. Therefore, various model-based SAE methods for zero-inflated data are developed, such as the Frequentist approach and the Bayesian approach. Both approaches are compared with the survey regression (SR) method which ignores the presence of zero-inflation in the data. The results show that the two SAE approaches for zero-inflated data are capable to yield more accurate area mean estimates than the SR method.

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

The aim of the study was evaluate and compare the efficiency of Bayesian and frequentist approach to describe the rumen degradation of NDF. Simulated data was composed by four scenarios: regular restriction in the number of incubation times, random loss of incubation times, loss of specific parts of degradation curves, variation in the precision of the incubations procedures. Two real datasets was used, these real data encompassed the evaluation of NDF degradation of a tropical grass (Brachiaria decumbes). The model was fitted according their characteristics approach and compared by plots and assessors. The Bayesian and frequentist approach presented reliable estimates of degradation parameters for the majority of the data tested. Therefore, in specific cases with short random records number, the Bayesian approach showed greater bias of the estimates of incubation residue and estimates of degradation rate without a biological coherence of the parameters, compared to frequentist inference. In another words, the Bayesian approach fitted with prior diffuse, presented less flexible. Nevertheless, it is emphasized the importance of the background information before the modeling, mainly for the Bayesian approach, in order to define proper prior distributions. Future thorough studies about the influence of non-informative prior for the parameters are necessary.

A crucial decision in designing a spatial sample for soil survey is the number of sampling locations required to answer, with sufficient accuracy and precision, the questions posed by decision makers at different levels of geographic aggregation. In the Indian Soil Health Card (SHC) scheme, many thousands of locations are sampled per district. In this paper the SHC data are used to estimate the mean of a soil property within a defined study area, e.g., a district, or the areal fraction of the study area where some condition is satisfied, e.g., exceedence of a critical level. The central question is whether this large sample size is needed for this aim. The sample size required for a given maximum length of a confidence interval can be computed with formulas from classical sampling theory, using a prior estimate of the variance of the property of interest within the study area. Similarly, for the areal fraction a prior estimate of this fraction is required. In practice we are uncertain about these prior estimates, and our uncertainty is not accounted for in classical sample size determination (SSD). This deficiency can be overcome with a Bayesian approach, in which the prior estimate of the variance or areal fraction is replaced by a prior distribution. Once new data from the sample are available, this prior distribution is updated to a posterior distribution using Bayes’ rule. The apparent problem with a Bayesian approach prior to a sampling campaign is that the data are not yet available. This dilemma can be solved by computing, for a given sample size, the predictive distribution of the data, given a prior distribution on the population and design parameter. Thus we do not have a single vector with data values, but a finite or infinite set of possible data vectors. As a consequence, we have as many posterior distribution functions as we have data vectors. This leads to a probability distribution of lengths or coverages of Bayesian credible intervals, from which various criteria for SSD can be derived. Besides the fully Bayesian approach, a mixed Bayesian-likelihood approach for SSD is available. This is of interest when, after the data have been collected, we prefer to estimate the mean from these data only, using the frequentist approach, ignoring the prior distribution. The fully Bayesian and mixed Bayesian-likelihood approach are illustrated for estimating the mean of log-transformed Zn and the areal fraction with Zn-deficiency, defined as Zn concentration <0.9 mg kg −1, in the thirteen districts of Andhra Pradesh state. The SHC data from 2015–2017 are used to derive prior distributions. For all districts the Bayesian and mixed Bayesian-likelihood sample sizes are much smaller than the current sample sizes. The hyperparameters of the prior distributions have a strong effect on the sample sizes. We discuss methods to deal with this. Even at the mandal (sub-district) level the sample size can almost always be reduced substantially. Clearly SHC over-sampled, and here we show how to reduce the effort while still providing information required for decision-making. R scripts for SSD are provided as supplementary material.

Introduction: Recurrent event data are common in many longitudinal studies. Often, a terminating event such as death can be correlated with the recurrent event process. A shared frailty model applied to account for the association between recurrent and terminal events. In some situations, a fraction of subjects experience neither recurrent events nor death; these subjects are cured.&#x0D; Methods: In this paper, we discussed the Bayesian approach of a joint frailty model for recurrent and terminal events in the presence of cure fraction. We compared estimates of parameters in the Frequentist and Bayesian approaches via simulation studies in various sample sizes; we applied the joint frailty model in the presence of cure fraction with Frequentist and Bayesian approaches for breast cancer.&#x0D; Results: In small sample size Bayesian approach compared to Frequentist approach had a smaller standard error and mean square error, and the coverage probabilities close to nominal level of 95%. Also, in Bayesian approach, the sampling means of the estimated standard errors were close to the empirical standard error.&#x0D; Conclusion: The simulation results suggested that when sample size was small, the use of Bayesian joint frailty model in the presence of cure fraction led to more efficiency in parameter estimation and statistical inference.

This study focuses on modeling and predicting extreme rainfall based on data from the Southern Highlands region, the critical for rain-fed agriculture in Tanzania. Analyzing 31 years of annual maximum rainfall data spanning from 1990 to 2020, the Generalized Extreme Value (GEV) model proved to be the best for modeling extreme rainfall in all stations. Three estimation methods–L-moments, maximum likelihood estimation (MLE), and Bayesian Markov chain Monte Carlo (MCMC)–were employed to estimate GEV parameters and future return levels. The Bayesian MCMC approach demonstrated superior performance by incorporating noninformative priors to ensure that the prior information had minimal influence on the analysis, allowing the observed data to play a dominant role in shaping the posterior distribution. Furthermore, return levels for various future periods were estimated, providing guidance for flood protection measures and infrastructure design. Trend analysis using p value, Kendall’s tau, and Sen’s slope indicated no statistically significant trends in rainfall patterns, although a weak positive trend in extreme rainfall events was observed, suggesting a gradual and modest increase over time. Overall, the study contributes valuable insights into extreme rainfall patterns and underscores the importance of L-moments in identifying the best fit distribution and Bayesian MCMC methodology for accurate parameter estimation and prediction, enabling effective measures and infrastructure planning in the region.

Bayesian approaches have been extensively used in animal breeding sciences, but similar approaches in the context of evolutionary quantitative genetics have been rare. We compared the performance of Bayesian and frequentist approaches in estimation of quantitative genetic parameters (viz. matrices of additive and dominance variances) in datasets typical of evolutionary studies and traits differing in their genetic architecture. Our results illustrate that it is difficult to disentangle the relative roles of different genetic components from small datasets, and that ignoring, e.g. dominance is likely to lead to biased estimates of additive variance. We suggest that a natural summary statistic for G-matrix comparisons can be obtained by examining how different the underlying multinormal probability distributions are, and illustrate our approach with data on the common frog (Rana temporaria). Furthermore, we derive a simple Monte Carlo method for computation of fraternity coefficients needed for the estimation of dominance variance, and use the pedigree of a natural Siberian jay (Perisoreus infaustus) population to illustrate that the commonly used approximate values can be substantially biased.

Since the GUM has been published, measurement uncertainty has been defined in terms of the standard deviation of the probability distribution of the values that can be reasonably attributed to the measurand, and it has been evaluated using statistical or probabilistic methods. A debate has always been alive, among the metrologists, on whether a frequentist approach or a Bayesian approach should be followed to evaluate uncertainty. The Bayesian approach, based on some available a-priori knowledge about the measurand seems to prevail, nowadays. This paper starts from the consideration that the Bayesian approach is based on the well-known Bayes theorem that, as all mathematical theorems, is valid only to the extent the assumptions made to prove it are valid. The main question, when following the Bayesian approach, is hence whether these assumptions are satisfied in the practical cases, especially when the a-priori information is combined with the information coming from the measurement data to refine uncertainty evaluation. This paper will take into account some case studies to analyze when the Bayesian approach can be usefully and reliably employed by discussing the amount and pertinence of the available a-priori knowledge.

This study presents a comprehensive analysis of Bayesian estimation techniques for the parameters of the power Rayleigh (PR) distribution under a unified hybrid censoring scheme (UHCS). The research employs both Bayesian and Frequentist approaches, utilizing maximum likelihood estimation (MLE) alongside Bayesian estimates derived through Markov Chain Monte Carlo (MCMC) methods. The study incorporates symmetric and asymmetric loss functions, specifically general entropy (GE), linear expoential (LINEX), and squared error (SE), to evaluate the performance of the estimators. A Monte Carlo simulation study is conducted to assess the effectiveness of the proposed methods, revealing that Bayesian estimators generally outperform Frequentist estimators in terms of mean squared error (MSE). Additionally, the paper includes a real-world application involving ball bearing lifetimes, demonstrating the practical utility of the proposed estimation techniques. The findings indicate that both point and interval estimates exhibit strong properties for parameter estimation, with Bayesian estimates being particularly favored for their accuracy and reliability.

&lt;p&gt;In this study, we consider a multicomponent system which has k independent and identical strength components X1,...,Xk and each component is exposed to a common random stress Y when the underlying distributions are Weibull. The system is regarded as operating only if at least s out of k (1 ≤ s ≤ k) strength variables exceeds the random stress. We estimate the reliability of the system by using frequentist and Bayesian approaches. The Bayes estimate of the reliability has been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The asymptotic confidence interval and the highest probability density credible interval are constructed for the reliability. The comparison of the reliability estimators is made in terms of the estimated risks by the Monte Carlo simulations.&lt;/p&gt;

Wilcoxon Signed-Rank Test (WSRT) is a rank-based statistical nonparametric method to test two paired samples. Researchers often use a frequentist approach in testing by utilizing test statistics or p-values. This approach has limitations in providing information about the rejection of the alternative or null hypothesis. These limitations have spurred interest in Bayesian-based testing, known as the Bayes Factor. The advantage of the Bayesian approach is it can measure how much the data support one hypothesis over another. However, there is a problem of using Bayesian approach in WSRT, since there is no distribution of the rank implies no likelihood can be formed from the data rank. Van Doorn proposed a Bayesian approach for this test by using a latent normal approach, by modeling the data rank being come from latent variables that are normally distributed. The objective of this study is to test whether there is difference between farmer exchange rate 2021 and 2022 in Indonesia. We used Wilcoxon Signed Rank Test with Bayesian approach as given by Van Doorn. The test employs Bayes factor to make conclusion, by transforming the rank of data using latent variable that assuming follow normal distribution. The analysis was conducted by constructing a posterior population of difference ( ) of 475,000 using the Gibbs Sampling algorithm. It is obtained the values of Bayes Factor is of 3076.07 and concluded that there is difference of farmer exchange rate in Indonesia between 2021 and 2022. This Bayes Factor indicates extreme evidence of a significant difference in the farmer exchange rate in Indonesia between 2021 and 2022.