scholarly journals A Discrete Density Approach to Bayesian Quantile and Expectile Regression with Discrete Responses

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Xi Liu ◽  
Xueping Hu ◽  
Keming Yu
Keyword(s):  
Quantile Regression ◽  
Regression Models ◽  
Likelihood Function ◽  
Real Data ◽  
Model Parameters ◽  
Expectile Regression ◽  
Improper Priors ◽  
Continuous Responses ◽  
Discrete Responses ◽  
Probability Mass

AbstractFor decades, regression models beyond the mean for continuous responses have attracted great attention in the literature. These models typically include quantile regression and expectile regression. But there is little research on these regression models for discrete responses, particularly from a Bayesian perspective. By forming the likelihood function based on suitable discrete probability mass functions, this paper introduces a discrete density approach for Bayesian inference of these regression models with discrete responses. Bayesian quantile regression for discrete responses is first developed, and then this method is extended to Bayesian expectile regression for discrete responses. The posterior distribution under this approach is shown not only coherent irrespective of the true distribution of the response, but also proper with regarding to improper priors for the unknown model parameters. The performance of the method is evaluated via extensive Monte Carlo simulation studies and one real data analysis.

scholarly journals Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1000 ◽  
Author(s):  
Luis Sánchez ◽  
Víctor Leiva ◽  
Manuel Galea ◽  
Helton Saulo
Keyword(s):  
Quantile Regression ◽  
Spatial Data ◽  
Spatial Model ◽  
Regression Models ◽  
Maximum Likelihood Method ◽  
Marginal Distribution ◽  
Likelihood Method ◽  
Model Parameters ◽  
Data Set ◽  
Quantile Regression Model

In the present paper, a novel spatial quantile regression model based on the Birnbaum–Saunders distribution is formulated. This distribution has been widely studied and applied in many fields. To formulate such a spatial model, a parameterization of the multivariate Birnbaum–Saunders distribution, where one of its parameters is associated with the quantile of the respective marginal distribution, is established. The model parameters are estimated by the maximum likelihood method. Finally, a data set is applied for illustrating the formulated model.

The Odd Log-Logistic Geometric Normal Regression Model with Applications

2019 ◽  
Vol 11 (01n02) ◽  
pp. 1950003
Author(s):  
Fábio Prataviera ◽  
Gauss M. Cordeiro ◽  
Edwin M. M. Ortega ◽  
Adriano K. Suzuki
Keyword(s):  
Regression Model ◽  
Normal Distribution ◽  
Regression Models ◽  
Empirical Distribution ◽  
Real Data ◽  
Likelihood Method ◽  
Standard Normal Distribution ◽  
Model Parameters ◽  
Diagnostic Measures ◽  
Standard Normal

In several applications, the distribution of the data is frequently unimodal, asymmetric or bimodal. The regression models commonly used for applications to data with real support are the normal, skew normal, beta normal and gamma normal, among others. We define a new regression model based on the odd log-logistic geometric normal distribution for modeling asymmetric or bimodal data with support in [Formula: see text], which generalizes some known regression models including the widely known heteroscedastic linear regression. We adopt the maximum likelihood method for estimating the model parameters and define diagnostic measures to detect influential observations. For some parameter settings, sample sizes and different systematic structures, various simulations are performed to verify the adequacy of the estimators of the model parameters. The empirical distribution of the quantile residuals is investigated and compared with the standard normal distribution. We prove empirically the usefulness of the proposed models by means of three applications to real data.

scholarly journals New Flexible Regression Models Generated by Gamma Random Variables with Censored Data

2016 ◽  
Vol 5 (3) ◽  
pp. 9 ◽  
Author(s):  
Elizabeth M. Hashimoto ◽  
Gauss M. Cordeiro ◽  
Edwin M.M. Ortega ◽  
G.G. Hamedani
Keyword(s):  
Regression Model ◽  
Censored Data ◽  
Survival Data ◽  
Regression Models ◽  
Empirical Distribution ◽  
Real Data ◽  
Maximum Likelihood Estimates ◽  
Standard Normal Distribution ◽  
Model Parameters ◽  
Linear Regression Models

We propose and study a new log-gamma Weibull regression model. We obtain explicit expressions for the raw and incomplete moments, quantile and generating functions and mean deviations of the log-gamma Weibull distribution. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models which includes as sub-models several widely-known regression models and therefore can be used more effectively in the analysis of survival data. We obtain the maximum likelihood estimates of the model parameters by considering censored data and evaluate local influence on the estimates of the parameters by taking different perturbation schemes. Some global-influence measurements are also investigated. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We demonstrate that our extended regression model is very useful to the analysis of real data and may give more realistic fits than other special regression models. 

scholarly journals Power Prior Elicitation in Bayesian Quantile Regression

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Rahim Alhamzawi ◽  
Keming Yu
Keyword(s):  
Quantile Regression ◽  
Prior Distribution ◽  
Likelihood Function ◽  
Real Data ◽  
Laplace Distribution ◽  
Scale Mixture ◽  
Prior Elicitation ◽  
Bayesian Quantile Regression ◽  
Critical Issues ◽  
Power Prior

We address a quantile dependent prior for Bayesian quantile regression. We extend the idea of the power prior distribution in Bayesian quantile regression by employing the likelihood function that is based on a location-scale mixture representation of the asymmetric Laplace distribution. The propriety of the power prior is one of the critical issues in Bayesian analysis. Thus, we discuss the propriety of the power prior in Bayesian quantile regression. The methods are illustrated with both simulation and real data.

scholarly journals The McBurr XII and Log-McBurr XII Models with Applications to Lifetime Data

2015 ◽  
Vol 5 (1) ◽  
pp. 1 ◽  
Author(s):  
Gauss M. Cordeiro ◽  
Edwin M. M. Ortega ◽  
G. G. Hamedani ◽  
Diogo A. Garcia
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Regression Models ◽  
Real Data ◽  
Model Parameters ◽  
Lifetime Data ◽  
Fatigue Lifetime ◽  
Burr Xii Distribution ◽  
New Models ◽  
Lifetime Model

A new six-parameter extended fatigue lifetime model named the McDonald-Burr XII distribution is introduced, which generalizes the Burr XII, beta Burr XII (Parana\'iba {\it et al.}, 2011) and Kumaraswamy-Burr XII (Parana\'iba {\it et al.}, 2012) distributions. The proposed distribution is characterized in terms of truncated moments. We obtain the ordinary and incomplete moments and quantile and ge\-ne\-ra\-ting functions. We estimate the model parameters by maximum likelihood. A Monte Carlo simulation is performed to study the asymptotic normality of the estimates. We also propose an extended regression model based on the logarithm of the McDonald-Burr XII distribution that can be more realistic in the analysis of real data than other special regression models. Two applications to real data validate the importance of the new models.

scholarly journals Reduced-bias estimation of spatial autoregressive models with incompletely geocoded data

2021 ◽  
Author(s):  
Flavio Santi ◽  
Maria Michela Dickson ◽  
Diego Giuliani ◽  
Giuseppe Arbia ◽  
Giuseppe Espa
Keyword(s):  
Point Process ◽  
Likelihood Function ◽  
Marked Point ◽  
Spatial Models ◽  
Real Data ◽  
Marked Point Process ◽  
Model Parameters ◽  
Bias Estimation ◽  
Monte Carlo Simulation Study ◽  
Spatial Autoregressive

AbstractThe application of spatial Cliff–Ord models requires information about spatial coordinates of statistical units to be reliable, which is usually the case in the context of areal data. With micro-geographic point-level data, however, such information is inevitably affected by locational errors, that can be generated intentionally by the data producer for privacy protection or can be due to inaccuracy of the geocoding procedures. This unfortunate circumstance can potentially limit the use of the spatial autoregressive modelling framework for the analysis of micro data, as the presence of locational errors may have a non-negligible impact on the estimates of model parameters. This contribution aims at developing a strategy to reduce the bias and produce more reliable inference for spatial models with location errors. The proposed estimation strategy models both the spatial stochastic process and the coarsening mechanism by means of a marked point process. The model is fitted through the maximisation of a doubly-marginalised likelihood function of the marked point process, which cleans out the effects of coarsening. The validity of the proposed approach is assessed by means of a Monte Carlo simulation study under different real-case scenarios, whereas it is applied to real data on house prices.

scholarly journals The Novel Bivariate Distribution: Statistical Properties and Real Data Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
B. I. Mohammed ◽  
Abdulaziz S. Alghamdi ◽  
Hassan M. Aljohani ◽  
Md. Moyazzem Hossain
Keyword(s):  
Negative Binomial ◽  
Real Data ◽  
Information Criterion ◽  
Mass Function ◽  
Bivariate Distribution ◽  
Statistical Properties ◽  
Model Parameters ◽  
Probability Mass Function ◽  
New Class ◽  
Probability Mass

This article proposes a novel class of bivariate distributions that are completely defined by stating their conditionals as Poisson exponential distributions. Numerous statistical properties of this distribution are also examined here, including the conditional probability mass function (PMF) and moments of the new class. The techniques of maximum likelihood and pseudolikelihood are used to estimate the model parameters. Additionally, the effectiveness of the bivariate Poisson exponential conditional (BPEC) distribution is compared to that of the bivariate Poisson conditional (BPC), the bivariate Poisson (BP), the bivariate Poisson–Lindley (BPL), and the bivariate negative binomial (BNB) distributions using a real-world dataset. The findings of Akaike information criterion (AIC) and Bayesian information criterion (BIC) reveal that the BPEC distribution performs better than the other distributions considered in this study. As a result, the authors claim that this distribution may be used to fit dependent and overspread count data.

scholarly journals On the Maintenance Modeling of a Hybrid Model with Exponential Repair Efficiency

2020 ◽  
Vol 6 (1) ◽  
pp. 254-267
Author(s):  
Wassila Nissas ◽  
Soufiane Gasmi
Keyword(s):  
Hybrid Model ◽  
Likelihood Function ◽  
Bootstrap Method ◽  
Real Data ◽  
Random Variable ◽  
Model Parameters ◽  
Data Sets ◽  
Imperfect Maintenance ◽  
Repairable Systems ◽  
Repair Efficiency

In the reliability literature, maintenance efficiency is usually dealt with as a fixed value. Since repairable systems are subject to different degrees and types of repair, it is more convenient to regard a random variable for maintenance efficiency. This paper is devoted to the statistical study of a general hybrid model for repairable systems working under imperfect maintenance. For both failure improvement and virtual age reduction of the system, maintenance efficiency is assumed to be random, with an exponential distribution as a probability density function. The likelihood function of this model is provided, and the estimation of the model parameters is computed by considering the maximization likelihood procedure. Obtained results were tested and applied to simulated and real data sets. To construct confidence intervals, the bias-corrected accelerated bootstrap method has been used.

scholarly journals Entropy-Randomized Forecasting of Stochastic Dynamic Regression Models

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1119 ◽  
Author(s):  
Yuri S. Popkov ◽  
Alexey Yu. Popkov ◽  
Yuri A. Dubnov ◽  
Dimitri Solomatine
Keyword(s):  
Regression Models ◽  
Real Data ◽  
Maximization Problem ◽  
Model Parameters ◽  
Stochastic Dynamic ◽  
Density Functions ◽  
Electrical Load ◽  
Learning Stage ◽  
Measurement Noises ◽  
Dynamic Regression Model

We propose a new forecasting procedure that includes randomized hierarchical dynamic regression models with random parameters, measurement noises and random input. We developed the technology of entropy-randomized machine learning, which includes the estimation of characteristics of a dynamic regression model and its testing by generating ensembles of predicted trajectories through the sampling of the entropy-optimal probability density functions of the model parameters and measurement noises. The density functions are determined at the learning stage by solving the constrained maximization problem of an information entropy functional subject to the empirical balances with real data. The proposed procedure is applied to the randomized forecasting of the daily electrical load in a regional power system. We construct a two-layer dynamic model of the daily electrical load. One of the layers describes the dependence of electrical load on ambient temperature while the other simulates the stochastic quasi-fluctuating temperature dynamics.

ESTIMATION OF HIGH CONDITIONAL TAIL RISK BASED ON EXPECTILE REGRESSION

2021 ◽  
pp. 1-32
Author(s):  
Jie Hu ◽  
Yu Chen ◽  
Keqi Tan
Keyword(s):  
Quantile Regression ◽  
Value At Risk ◽  
Risk Measures ◽  
Asymptotic Properties ◽  
Estimation Method ◽  
Real Data ◽  
Tail Index ◽  
Tail Risk ◽  
Expectile Regression ◽  
Quantile Regression Method

Abstract Assessing conditional tail risk at very high or low levels is of great interest in numerous applications. Due to data sparsity in high tails, the widely used quantile regression method can suffer from high variability at the tails, especially for heavy-tailed distributions. As an alternative to quantile regression, expectile regression, which relies on the minimization of the asymmetric l2-norm and is more sensitive to the magnitudes of extreme losses than quantile regression, is considered. In this article, we develop a new estimation method for high conditional tail risk by first estimating the intermediate conditional expectiles in regression framework, and then estimating the underlying tail index via weighted combinations of the top order conditional expectiles. The resulting conditional tail index estimators are then used as the basis for extrapolating these intermediate conditional expectiles to high tails based on reasonable assumptions on tail behaviors. Finally, we use these high conditional tail expectiles to estimate alternative risk measures such as the Value at Risk (VaR) and Expected Shortfall (ES), both in high tails. The asymptotic properties of the proposed estimators are investigated. Simulation studies and real data analysis show that the proposed method outperforms alternative approaches.

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