Asymmetric Laplace Distribution Research Articles

A stochastic algorithm for quantile regression models with fixed effects

In this paper, we present a stochastic algorithm for parameter estimation based on panel quantile regression model. We propose an easy-to-implement estimator based on the proposed algorithm. We profile the quantile-specific fixed effects as functions of the parameters of interest based on the Gaussian mixture representation of the asymmetric Laplace (AL) likelihood and eliminate the fixed effects through a data transformation. Parameters of interest can be estimated via quantile regression. Under a set of sufficient conditions, the proposed estimator is consistent and asymptotically normal when n and T both go to infinity. The proposed estimator is illustrated via both simulations and real data examples.

Interval-valued linear regression model with an asymmetric Laplace distribution

Interval-valued linear regression model with an asymmetric Laplace distribution

Asymmetric Laplace scale mixtures for the distribution of cryptocurrency returns

AbstractRecent studies about cryptocurrency returns show that their distribution can be highly-peaked, skewed, and heavy-tailed, with a large excess kurtosis. To accommodate all these peculiarities, we propose the asymmetric Laplace scale mixture (ALSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional asymmetric Laplace (AL) distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector $$\varvec{\theta }$$ θ providing greater flexibility to the resulting ALSM. Advantageously concerning the AL distribution, our family members allow for a wider range of values for skewness and kurtosis. For illustrative purposes, we consider different mixing distributions; they give rise to ALSMs having a closed-form probability density function where the AL distribution is obtained as a special case under a convenient choice of $$\varvec{\theta }$$ θ . We examine some properties of our ALSMs such as hierarchical and stochastic representations and moments of practical interest. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered ALSMs. We fit these models to the returns of two cryptocurrencies, considering several classical distributions for comparison. The analysis shows how our models represent a valid alternative to the considered competitors in terms of AIC, BIC, and likelihood-ratio tests.

The spike-and-slab quantile LASSO for robust variable selection in cancer genomics studies.

Data irregularity in cancer genomics studies has been widely observed in the form of outliers and heavy-tailed distributions in the complex traits. In the past decade, robust variable selection methods have emerged as powerful alternatives to the nonrobust ones to identify important genes associated with heterogeneous disease traits and build superior predictive models. In this study, to keep the remarkable features of the quantile LASSO and fully Bayesian regularized quantile regression while overcoming their disadvantage in the analysis of high-dimensional genomics data, we propose the spike-and-slab quantile LASSO through a fully Bayesian spike-and-slab formulation under the robust likelihood by adopting the asymmetric Laplace distribution (ALD). The proposed robust method has inherited the prominent properties of selective shrinkage and self-adaptivity to the sparsity pattern from the spike-and-slab LASSO (Roc̆ková and George, J Am Stat Associat, 2018, 113(521): 431-444). Furthermore, the spike-and-slab quantile LASSO has a computational advantage to locate the posterior modes via soft-thresholding rule guided Expectation-Maximization (EM) steps in the coordinate descent framework, a phenomenon rarely observed for robust regularization with nondifferentiable loss functions. We have conducted comprehensive simulation studies with a variety of heavy-tailed errors in both homogeneous and heterogeneous model settings to demonstrate the superiority of the spike-and-slab quantile LASSO over its competing methods. The advantage of the proposed method has been further demonstrated in case studies of the lung adenocarcinomas (LUAD) and skin cutaneous melanoma (SKCM) data from The Cancer Genome Atlas (TCGA).

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

A quantile-regression approach to bivariate longitudinal joint modeling

Joint modeling of longitudinal outcomes and time-to-event data has become a major re- search interest in the last thirty years. A joint model is useful since it helps (i) to under- stand the evolution of the outcome(s) of interest over time, (ii) to understand the effects of the outcomes on the time of occurrence of some event(s) of interest (e.g. death/relapse), and (iii) to study the effects of the time-varying and time-invariant predictors on the longitudinal and time-to-event process. Traditional linear mixed models are routinely used for modeling the longitudinal process. However, for non-Gaussian/skewed outcomes it is more appealing to use quantile-regression models since such models do not assume any specific probability distribution for the outcomes. In this article, we present a bivariate quantile-regression approach for jointly modeling longitudinal process and time-to-event. In a Bayesian setting we consider an Asymmetric Laplace Distribution (ALD) for modeling different quantiles of the outcomes, and a semi-parametric Proportional Hazards (PH) model for time-to-event. Model parameters are estimated using Markov chain Monte Carlo (MCMC) algorithm, and we discuss the computational complexities through several simulation studies. Our numerical studies illustrate the usefulness of our model over the other traditional models. Journal of Statistical Research 2024, Vol. 58, No. 1, pp. 111-130.

A novel Bayesian computational approach for bridge-randomized quantile regression in high dimensional models

A bridge-randomized penalization that employs a prior for the shrinkage parameter, as opposed to the conventional bridge penalization with a fixed penalty, often delivers more superior performance compared to many other traditional shrinkage methods. In this paper, we develop an efficient Bayesian computational algorithm via the two-block Markov Chain Monte Carlo method for the bridge-randomized penalization in quantile regression to perform inference in the high-dimensional ‘large-p’ and ‘large-p-small-n’ settings. To construct a fully Bayesian formulation, we utilize the asymmetric Laplace distribution as an auxiliary error distribution and the generalized Gaussian distribution prior for the regression coefficients. Simulation studies encompassing a wide range of scenarios indicate that the proposed method performs at least as well as, and often better than, other existing procedures in terms of both parameter estimation and variable selection. Finally, a real-data application is provided for illustrative purposes.

Bayesian Estimation of Simultaneous Regression Quantiles Using Hamiltonian Monte Carlo

The simultaneous estimation of multiple quantiles is a crucial statistical task that enables a thorough understanding of data distribution for robust analysis and decision-making. In this study, we adopt a Bayesian approach to tackle this critical task, employing the asymmetric Laplace distribution (ALD) as a flexible framework for quantile modeling. Our methodology implementation involves the Hamiltonian Monte Carlo (HMC) algorithm, building on the foundation laid in priorwork, where the error term is assumed to follow an ALD. Capitalizing on the interplay between two distinct quantiles of this distribution, we endorse a straightforward and fully Bayesian method that adheres to the non-crossing property of quantiles. Illustrated through simulated scenarios, we showcase the effectiveness of our approach in quantile estimation, enhancing precision via the HMC algorithm. The proposed method proves versatile, finding application in finance, environmental science, healthcare, and manufacturing, and contributing to sustainable development goals by fostering innovation and enhancing decision-making in diverse fields.

A Nonlinear Quantile Regression for Accelerated Destructive Degradation Testing Data

Traditional regression approaches to accelerated destructive degradation test (ADDT) data have modeled the mean curve as representative. However, maximum likelihood estimates (MLEs) of the mean model are likely to be biased when the data are non-Gaussian or highly skewed. The median model can be an alternative for skewed degradation data. In this work, we introduce a nonlinear quantile regression (QR) approach for estimating quantile curves of ADDT data. We propose an iterative QR algorithm that uses the generalized expectation-maximization (GEM) framework to estimate the parameters of the nonlinear QR ADDT model, based on the asymmetric Laplace distribution to accommodate non-Gaussian and skewed errors. Using the asymptotic properties of the QR parameter estimates, we estimate variance-covariance matrix for the τ th QR parameters using order statistics and bootstrap methods. We propose a new prediction method of the quantile of the failure-time distribution in the normal use condition. Confidence intervals for the quantiles of the failure-time distribution are constructed using the parametric bootstrap method. The proposed model is illustrated using an industrial application and compared with the existing model. Various quantile curve estimates derived using the QR ADDT model provide a more flexible modeling framework than the traditional mean ADDT modeling approach.

Bayesian analysis of joint quantile regression for multi-response longitudinal data with application to primary biliary cirrhosis sequential cohort study.

This article proposes a Bayesian approach for jointly estimating marginal conditional quantiles of multi-response longitudinal data with multivariate mixed effects model. The multivariate asymmetric Laplace distribution is employed to construct the working likelihood of the considered model. Penalization priors on regression parameters are incorporated into the working likelihood to conduct Bayesian high-dimensional inference. Markov chain Monte Carlo algorithm is used to obtain the fully conditional posterior distributions of all parameters and latent variables. Monte Carlo simulations are conducted to evaluate the sample performance of the proposed joint quantile regression approach. Finally, we analyze a longitudinal medical dataset of the primary biliary cirrhosis sequential cohort study to illustrate the real application of the proposed modeling method.

The Tail Characteristics of Several Asymmetric Distributions and Their Applications in the Field of Financial Data Modelling and Machine Learning

Asymmetric distributions play a pivotal role in various fields, including finance, machine learning, and artificial intelligence. In finance, asymmetric distributions, characterized by negative skewness, peakedness, heavy tails, and asymmetric fluctuations, provide a more accurate representation of financial data. These distributions are particularly useful in capturing the complex behavior of assets and risk management. In the field of machine learning, data disturbance, noise, outliers and other disturbances will have a negative impact on the performance and reliability of the model, asymmetric distributions offer a valuable tool for describing the tail behavior of noise and interference. This is crucial in designing more robust machine learning models that can handle outliers and noise effectively. Additionally, when dealing with unbalanced data classification, asymmetric distributions can be leveraged by adjusting the decision threshold of classifiers or employing specialized loss functions. In this paper, the asymptotic tail ratio behavior of probability density function(pdf) and cumulative distribution function (cdf) of three asymmetric distributions (asymmetric Laplace, generalized Logistic and asymmetric normal distribution) and Student-t distribution were considered respectively under some regular conditions. The detailed characteristics of these asymmetric tails will have a profound impact on the fields of financial data analysis, artificial intelligence, and unbalanced classification.

Single-Index Mixed-Effects Model for Asymmetric Bivariate Clustered Data

Studies/trials assessing status and progression of periodontal disease (PD) usually focus on quantifying the relationship between the clustered (tooth within subjects) bivariate endpoints, such as probed pocket depth (PPD), and clinical attachment level (CAL) with the covariates. Although assumptions of multivariate normality can be invoked for the random terms (random effects and errors) under a linear mixed model (LMM) framework, violations of those assumptions may lead to imprecise inference. Furthermore, the response-covariate relationship may not be linear, as assumed under a LMM fit, and the regression estimates obtained therein do not provide an overall summary of the risk of PD, as obtained from the covariates. Motivated by a PD study on Gullah-speaking African-American Type-2 diabetics, we cast the asymmetric clustered bivariate (PPD and CAL) responses into a non-linear mixed model framework, where both random terms follow the multivariate asymmetric Laplace distribution (ALD). In order to provide a one-number risk summary, the possible non-linearity in the relationship is modeled via a single-index model, powered by polynomial spline approximations for index functions, and the normal mixture expression for ALD. To proceed with a maximum-likelihood inferential setup, we devise an elegant EM-type algorithm. Moreover, the large sample theoretical properties are established under some mild conditions. Simulation studies using synthetic data generated under a variety of scenarios were used to study the finite-sample properties of our estimators, and demonstrate that our proposed model and estimation algorithm can efficiently handle asymmetric, heavy-tailed data, with outliers. Finally, we illustrate our proposed methodology via application to the motivating PD study.

A Bayesian quantile joint modeling of multivariate longitudinal and time-to-event data.

Linear mixed models are traditionally used for jointly modeling (multivariate) longitudinal outcomes and event-time(s). However, when the outcomes are non-Gaussian a quantile regression model is more appropriate. In addition, in the presence of some time-varying covariates, it might be of interest to see how the effects of different covariates vary from one quantile level (of outcomes) to the other, and consequently how the event-time changes across different quantiles. For such analyses linear quantile mixed models can be used, and an efficient computational algorithm can be developed. We analyze a dataset from the Acute Lymphocytic Leukemia (ALL) maintenance study conducted by Tata Medical Center, Kolkata. In this study, the patients suffering from ALL were treated with two standard drugs (6MP and MTx) for the first two years, and three biomarkers (e.g. lymphocyte count, neutrophil count and platelet count) were longitudinally measured. After treatment the patients were followed nearly for the next three years, and the relapse-time (if any) for each patient was recorded. For this dataset we develop a Bayesian quantile joint model for the three longitudinal biomarkers and time-to-relapse. We consider an Asymmetric Laplace Distribution (ALD) for each outcome, and exploit the mixture representation of the ALD for developing a Gibbs sampler algorithm to estimate the regression coefficients. Our proposed model allows different quantile levels for different biomarkers, but still simultaneously estimates the regression coefficients corresponding to a particular quantile combination. We infer that a higher lymphocyte count accelerates the chance of a relapse while a higher neutrophil count and a higher platelet count (jointly) reduce it. Also, we infer that across (almost) all quantiles 6MP reduces the lymphocyte count, while MTx increases the neutrophil count. Simulation studies are performed to assess the effectiveness of the proposed approach.

Copula-based pairwise estimator for quantile regression with hierarchical missing data

Quantile regression can be a helpful technique for analysing clustered (such as longitudinal) data. It can characterize the change in response over time without making distributional assumptions and is robust to outliers in the response. A quantile regression model using a copula-based multivariate asymmetric Laplace distribution for addressing correlation due to clustering is introduced. Furthermore, we propose a pairwise estimator for the parameters of the model. Since it is based on pseudo-likelihood, it needs to be modified to avoid bias in presence of missingness. Therefore, we enhance the model with inverse probability weighting. In this way, our proposal is unbiased under the missing at random assumption. Based on simulations, the estimator is efficient and computationally fast. Finally, the methodology is illustrated using a study in ophthalmology.

Tail risk forecasting with semiparametric regression models by incorporating overnight information

AbstractThis research incorporates realized volatility and overnight information into risk models, wherein the overnight return often contributes significantly to the total return volatility. Extending a semiparametric regression model based on asymmetric Laplace distribution, we propose a family of RES‐CAViaR‐oc models by adding overnight return and realized measures as a nowcasting technique for simultaneously forecasting Value‐at‐Risk (VaR) and expected shortfall (ES). We utilize Bayesian methods to estimate unknown parameters and forecast VaR and ES jointly for the proposed model family. We also conduct extensive backtests based on joint elicitability of the pair of VaR and ES during the out‐of‐sample period. Our empirical study on four international stock indices confirms that overnight return and realized volatility are vital in tail risk forecasting.

A study of China’s financial market risks in the context of Covid-19, based on a rolling generalized autoregressive score model using the asymmetric Laplace distribution

A study of China’s financial market risks in the context of Covid-19, based on a rolling generalized autoregressive score model using the asymmetric Laplace distribution

Distribution of Money on Connected Graphs with Multiple Banks

This paper studies an interacting particle system of interest in econophysics inspired from a model introduced in the physics literature. The original model consists of the customers of a single bank characterized by their capital, and the dynamics consists of monetary transactions in which a random individual x gives one coin to another random individual y, the transaction being canceled when x is in debt and there are no more coins in the bank. Using a combination of numerical simulations and heuristic arguments, physicists conjectured that the distribution of money (the random number of coins owned by a given individual) at equilibrium converges to an asymmetric Laplace distribution in the large population limit when the money temperature is large. We prove and extend this conjecture to a more general model including multiple banks and interactions among customers across banks. More importantly, we assume that customers are located on a general undirected connected graph (as opposed to the complete graph in the original model) where neighbors are interpreted as business partners, and transactions occur along the edges, thus modeling the flow of money across a social network. We first derive an exact expression for the distribution of money for all population sizes and money temperatures, then prove its convergence to an asymmetric Laplace distribution in the large population limit.

A flexible Bayesian hierarchical quantile spatial model for areal data

This article introduces a new class of nested models that extends the literature standard combination of spatial autoregressive model for areal data with parametric quantile regression by considering an asymmetric Laplace distribution for the random errors. In addition to being more flexible, the new proposed model can incorporate a hierarchical structure, allowing it to deal with clustered data. Such an approach produces a robust statistical method for modeling the quantiles of areal data distributed in a geographically hierarchical setting. The proposed non-hierarchical model is evaluated using a wellknown house pricing dataset and a simulation study. In addition, its hierarchical version is applied to a real dataset of math scores related to public high schools within the metropolitan area of Rio de Janeiro, Brazil.

A Laplace-based model with flexible tail behavior

The proposed multiple scaled contaminated asymmetric Laplace (MSCAL) distribution is an extension of the multivariate asymmetric Laplace distribution to allow for a different excess kurtosis on each dimension and for more flexible shapes of the hyper-contours. These peculiarities are obtained by working on the principal component (PC) space. The structure of the MSCAL distribution has the further advantage of allowing for automatic PC-wise outlier detection – i.e., detection of outliers separately on each PC – when convenient constraints on the parameters are imposed. The MSCAL is fitted using a Monte Carlo expectation-maximization (MCEM) algorithm that uses a Monte Carlo method to estimate the orthogonal matrix of eigenvectors. A simulation study is used to assess the proposed MCEM in terms of computational efficiency and parameter recovery. In a real data application, the MSCAL is fitted to a real data set containing the anthropometric measurements of monozygotic/dizygotic twins. Both a skewed bivariate subset of the full data, perturbed by some outlying points, and the full data are considered.

Bayesian variable selection and estimation in binary quantile regression using global-local shrinkage priors

In this paper, we construct a Bayesian hierarchical model with global-local shrinkage priors for the regression coefficients, which includes the horseshoe prior and normal-gamma prior. This model is used for high-dimensional quantile regression models with dichotomous response data. We have developed an efficient sampling algorithm to generate posterior samplings for making posterior inference. We use a location-scale mixture representation of the asymmetric Laplace distribution. We assess the performance of the proposed methods through Monte Carlo simulations and two real-data applications in terms of parameter estimation and variable selection. Numerical results demonstrate that the proposed methods perform comparably with existing Bayesian methods under a variety of scenarios.