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In this paper, based on the mixed Gibbs sampling algorithm, a Bayesian estimation procedure is proposed for a new Pareto-type distribution in the case of complete and type II censored samples. Simulation studies show that the proposed method is consistently superior to the maximize likelihood estimation in the context of small samples. Also, an analysis of some real data is provided to test the Bayesian estimation.

In this paper, we propose a Bayesian approach to estimate the curve of a function f(·) that models the solar power generated at k moments per day for n days and to forecast the curve for the (n+1)th day by using the history of recorded values. We assume that f(·) is an unknown function and adopt a Bayesian model with a Gaussian-process prior on the vector of values f(t)=f(1),…, f(k). An advantage of this approach is that we may estimate the curves of f(·) and fn+1(·) as “smooth functions” obtained by interpolating between the points generated from a k-variate normal distribution with appropriate mean vector and covariance matrix. Since the joint posterior distribution for the parameters of interest does not have a known mathematical form, we describe how to implement a Gibbs sampling algorithm to obtain estimates for the parameters. The good performance of the proposed approach is illustrated using two simulation studies and an application to a real dataset. As performance measures, we calculate the absolute percentage error, the mean absolute percentage error (MAPE), and the root-mean-square error (RMSE). In all simulated cases and in the application to real-world data, the MAPE and RMSE values were all near 0, indicating the very good performance of the proposed approach.

Since the advent of composite quantile regression (CQR), its inherent robustness has established it as a pivotal methodology for high-dimensional data analysis. High-dimensional outlier contamination refers to data scenarios where the number of observed dimensions (p) is much greater than the sample size (n) and there are extreme outliers in the response variables or covariates (e.g., p/n > 0.1). Traditional penalized regression techniques, however, exhibit notable vulnerability to data outliers during high-dimensional variable selection, often leading to biased parameter estimates and compromised resilience. To address this critical limitation, we propose a novel empirical likelihood (EL)-based variable selection framework that integrates a Bayesian LASSO penalty within the composite quantile regression framework. By constructing a hybrid sampling mechanism that incorporates the Expectation–Maximization (EM) algorithm and Metropolis–Hastings (M-H) algorithm within the Gibbs sampling scheme, this approach effectively tackles variable selection in high-dimensional settings with outlier contamination. This innovative design enables simultaneous optimization of regression coefficients and penalty parameters, circumventing the need for ad hoc selection of optimal penalty parameters—a long-standing challenge in conventional LASSO estimation. Moreover, the proposed method imposes no restrictive assumptions on the distribution of random errors in the model. Through Monte Carlo simulations under outlier interference and empirical analysis of two U.S. house price datasets, we demonstrate that the new approach significantly enhances variable selection accuracy, reduces estimation bias for key regression coefficients, and exhibits robust resistance to data outlier contamination.

The increasing prominence of the problem of censored data in various fields has made studying how to perform parameter estimation and variable selection in censored mixed-effects models one of the hotspots of current research. In this paper, considering the situation that the response variable is restricted by the bilateral limit, a double-penalty Bayesian Tobit quantile regression model was constructed to carry out parameter estimation and variable selection in the interval-censored mixed-effects model, and at the same time, the fixed-effects and random effects coefficients are compressed in Tobit’s mixed-effects model, so as to reduce the estimation error of the model at the same time as the variable selection of the model is carried out. The posterior distribution of each unknown parameter was derived using the conditional Laplace prior and the mixed truncated normal distribution of interval-censored data, and then the Gibbs sampling algorithm for unknown parameter estimation was constructed. Through Monte Carlo simulation, it was found that the new method is more advantageous than the classical method in terms of variable selection and parameter estimation accuracy in various situations, such as different model sparsity, different data censoring ratios and different random error distributions, and the model is able to realize automatic variable selection. Finally, the new method is used to analyze the correlation between the crime rate and various economic indicators in China.

We develop a Markov-switching GARCH model (MS-GARCH) wherein the conditional mean and variance switch in time from one GARCH process to another. The switching is governed by a hidden Markov chain. We provide sufficient conditions for geometric ergodicity and existence of moments of the process. Because of path dependence, maximum likelihood estimation is not feasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible. We illustrate the model on S&P500 daily returns.

This paper puts forward a Bayesian method for multiple gross errors location and estimation, and studies the masking and swamping problem in multiple gross errors detection from a new point of view, further proposes the corresponding feasible solution. First, the Bayesian method for gross error location is established based on the posterior probabilities of classification variables, each of which is used to determine whether each observation contains gross error or not. When some interactions exist among observations with multiple gross errors, the above-mentioned method may lead to the failure of detection due to masking and swamping. For that, on the basis of analyzing the character of masking and swamping, starting from the eigen structure of the sample correlation coefficient matrix of the classification vector, we give the Bayesian unmasking method to locate multiple gross errors, and design the corresponding algorithm, namely the adaptive Gibbs sampling algorithm. Finally, applying the mean shift model, we raise a Bayesian approach to estimate gross errors. Significant applications of the approach show the promising results on overcoming masking and swamping.

For establishing the suitable monetary policy it is essential to know if there is a relevant relationship in practice between gross domestic product (G.D.P.) variations and monetary variables. The purpose of this study is to analyse the causality between output variation and money aggregate in Romania for quarterly data in the period 2000:Q1-2015:Q2. Moreover the impact on G.D.P. growth of other variables connected with money demand is assessed using Bayesian techniques. The results indicated a bidirectional relationship between G.D.P. variations and rate of real money demand in the mentioned period. The Granger causality test combined with stochastic search variable selection indicated that active interest rate and discount rata mostly explained G.D.P. variations. According to results based on Bayesian regime-switching models, contrary to expectations, the interest rate increases continued to generate higher output variations, the consumption being the engine of economic growth in Romania. In periods of economic recession, the lower interest rate stimulated the recovery of the economy.

The aim of this study was to determine genetic parameters for straightness of the trunk of Eucalyptus cladocalyx, with a view to the selection of straight trees, while keeping the impact on growth minimal. The tests were conducted at two locations in the semi-arid region of Chile, using a randomized block design, with 30 replications and 49 half-sib families. The parameters were estimated by a bi-character model of individual trees, using Bayesian inference by Gibbs algorithm. The heritability for stem straightness was shown to be moderate, with h²=0.40 [0.29-0.57]. Heritabilities for diameter and height were moderate: 0.30 [0.24-0.38] and 0.30 [0.22-0.44]. Genetic correlations between straightness and growth were statistically not different from zero. The genotype-environment interaction was not significant (p>0.05) for the traits. The moderate degree of genetic control allows significant genetic gains in environments under water stress.

We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and Metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution π. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the Metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given; these show that the negative-definiteness property is not redundant.

This chapter contains sections titled: Introduction The galaxy data The normal mixture model Bayesian analyses Posterior distributions for K (for flat prior) Conclusions from the Bayesian analyses Posterior distributions of the model deviances Asymptotic distributions Posterior deviances for the galaxy data Conclusions References