Asymmetric Laplace Distribution Research Articles

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.

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

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.

Controlling the Flexibility of Non-Gaussian Processes Through Shrinkage Priors

The normal inverse Gaussian (NIG) and generalized asymmetric Laplace (GAL) distributions can be seen as skewed and semi-heavy-tailed extensions of the Gaussian distribution. Models driven by these more flexible noise distributions are then regarded as flexible extensions of simpler Gaussian models. Inferential procedures tend to overestimate the degree of non-Gaussianity in the data and therefore we propose controlling the flexibility of these non-Gaussian models by adding sensible priors in the inferential framework that contract the model towards Gaussianity. In our venture to derive sensible priors, we also propose a new intuitive parameterization of the non-Gaussian models and discuss how to implement them efficiently in Stan. The methods are derived for a generic class of non-Gaussian models that include spatial Matérn fields, autoregressive models for time series, and simultaneous autoregressive models for aerial data. The results are illustrated with a simulation study and geostatistics application, where priors that penalize model complexity were shown to lead to more robust estimation and give preference to the Gaussian model, while at the same time allowing for non-Gaussianity if there is sufficient evidence in the data.

Locally adaptive spatial quantile smoothing: Application to monitoring crime density in Tokyo

Spatial trend estimation under potential heterogeneity is an important problem to extract spatial characteristics and hazards such as criminal activity. By focusing on quantiles, which provide substantial information on distributions compared with commonly used summary statistics such as means, it is often useful to estimate not only the average trend but also the high (low) risk trend additionally. In this paper, we propose a Bayesian quantile trend filtering method to estimate the non-stationary trend of quantiles on graphs and apply it to crime data in Tokyo between 2013 and 2017. By modeling multiple observation cases, we can estimate the potential heterogeneity of spatial crime trends over multiple years in the application. To induce locally adaptive Bayesian inference on trends, we introduce general shrinkage priors for graph differences. Introducing so-called shadow priors with multivariate distribution for local scale parameters and mixture representation of the asymmetric Laplace distribution, we provide a simple Gibbs sampling algorithm to generate posterior samples. The numerical performance of the proposed method is demonstrated through simulation studies.

A note on computing maximum likelihood estimates for the three-parameter asymmetric Laplace distribution

A finite-step procedure is proposed for maximum likelihood estimation of the three-parameter asymmetric Laplace distribution. Its performance is compared with the iterative method most commonly used for this distribution. The new procedure is much faster and reliably identifies samples for which maximum likelihood estimates lie on the boundary of the parameter space, making it a good choice for simulation studies and simulation-based methodologies.

Bayesian mixed-frequency quantile vector autoregression: Eliciting tail risks of monthly US GDP

Timely characterizations of risks in economic and financial systems play an essential role in both economic policy and private sector decisions. However, the informational content of low-frequency variables and the results from conditional mean models provide only limited evidence to investigate this problem. We propose a novel mixed-frequency quantile vector autoregression (MF-QVAR) model to address this issue. Inspired by the univariate Bayesian quantile regression literature, the multivariate asymmetric Laplace distribution is exploited under the Bayesian framework to form the likelihood. A data augmentation approach coupled with a precision sampler efficiently estimates the missing low-frequency variables at higher frequencies under the state-space representation.The proposed methods allow us to analyse conditional quantiles for multiple variables of interest and to derive quantile-related risk measures at high frequency, thus enabling timely policy interventions. The main application of the model is to detect the vulnerability in the US economy and then to nowcast conditional quantiles of the US GDP, which is strictly related to the quantification of Value-at-Risk, the Expected Shortfall and distance among percentiles of real GDP nowcasts.

Matrix variate generalized asymmetric Laplace distributions

The generalized asymmetric Laplace (GAL) distributions, also known as the variance/mean-gamma models, constitute a popular flexible class of distributions that can account for peakedness, skewness, and heavier-than-normal tails, often observed in financial or other empirical data. We consider extensions of the GAL distribution to the matrix variate case, which arise as covariance mixtures of matrix variate normal distributions. Two different mixing mechanisms connected with the nature of the random scaling matrix are considered, leading to what we term matrix variate GAL distributions of Type I and II. While Type I matrix variate GAL distribution has been studied before, there is no comprehensive account of Type II in the literature, except for their rather brief treatment as a special case of matrix variate generalized hyperbolic distributions. With this work we fill this gap, and present an account for basic distributional properties of Type II matrix variate GAL distributions. In particular, we derive their probability density function and the characteristic function, as well as provide stochastic representations related to matrix variate gamma distribution. We also show that this distribution is closed under linear transformations, and study the relevant marginal distributions. In addition, we also briefly account for Type I and discuss the intriguing connections with Type II. We hope that this work will be useful in the areas where matrix variate distributions provide an appropriate probabilistic tool for three-way or, more generally, panel data sets, which can arise across different applications.

The skewness of mean–variance normal mixtures

Mean–variance mixtures of normal distributions are very flexible: they model many nonnormal features, such as skewness, kurtosis and multimodality. Special cases include generalized asymmetric Laplace distributions, mixtures of two normal distributions with proportional covariance matrices, scale mixtures of normal distributions and normal distributions. This paper investigates the skewness of multivariate mean–variance normal mixtures. The special case of mixtures of two normal distributions with proportional covariance matrices is treated in greater detail. The paper derives the analytical forms of prominent measures of multivariate skewness and applies them to model-based clustering, normalizing linear transformations, projection pursuit and normality testing. The practical relevance of the theoretical results is assessed with both real and simulated data.

A capital asset pricing model based on the value at risk under asymmetric Laplace distribution

PurposeFinancial asset return series usually exhibit nonnormal characteristics such as high peaks, heavy tails and asymmetry. Traditional risk measures like standard deviation or variance are inadequate for nonnormal distributions. Value at Risk (VaR) is consistent with people's psychological perception of risk. The asymmetric Laplace distribution (ALD) captures the heavy-tailed and biased features of the distribution. VaR is therefore used as a risk measure to explore the problem of VaR-based asset pricing. Assuming returns obey ALD, the study explores the impact of high peaks, heavy tails and asymmetric features of financial asset return data on asset pricing.Design/methodology/approachA VaR-based capital asset pricing model (CAPM) was constructed under the ALD that follows the logic of the classical CAPM and derive the corresponding VaR-β coefficients under ALD.FindingsALD-based VaR exhibits a minor tail risk than VaR under normal distribution as the mean increases. The theoretical derivation yields a more complex capital asset pricing formula involving β coefficients compared to the traditional CAPM.The empirical analysis shows that the CAPM under ALD can reflect the β-return relationship, and the results are robust. Finally, comparing the two CAPMs reveals that the β coefficients derived in this paper are smaller than those in the traditional CAPM in 69–80% of cases.Originality/valueThe paper uses VaR as a risk measure for financial time series data following ALD to explore asset pricing problems. The findings complement existing literature on the effects of high peaks, heavy tails and asymmetry on asset pricing, providing valuable insights for investors, policymakers and regulators.

Quantile forward regression for high-dimensional survival data.

Despite the urgent need for an effective prediction model tailored to individual interests, existing models have mainly been developed for the mean outcome, targeting average people. Additionally, the direction and magnitude of covariates' effects on the mean outcome may not hold across different quantiles of the outcome distribution. To accommodate the heterogeneous characteristics of covariates and provide a flexible risk model, we propose a quantile forward regression model for high-dimensional survival data. Our method selects variables by maximizing the likelihood of the asymmetric Laplace distribution (ALD) and derives the final model based on the extended Bayesian Information Criterion (EBIC). We demonstrate that the proposed method enjoys a sure screening property and selection consistency. We apply it to the national health survey dataset to show the advantages of a quantile-specific prediction model. Finally, we discuss potential extensions of our approach, including the nonlinear model and the globally concerned quantile regression coefficients model.

Quantile regression based on the skewed exponential power distribution

Bayesian quantile regression generally relies on the asymmetric Laplace distribution (ALD) as the error distribution. We consider methods for Lp -quantile regression based on the skewed exponential power distribution (SEPD). Both Bayesian and frequentist estimation procedures are outlined and compared with previous work based on the SEPD. We find that our proposed methods greatly outperform a previous method in terms of quantile estimation. Further, compared with standard quantile regression, we find that our proposed methods generally perform better in terms of root mean square error (RMSE). Empirical evidence of the statistical properties of the proposed models is provided through a simulation study. Further, a real data application illustrates their performance.

Bayesian composite quantile regression for the single-index model.

By using a Gaussian process prior and a location-scale mixture representation of the asymmetric Laplace distribution, we develop a Bayesian analysis for the composite quantile single-index regression model. The posterior distributions for the unknown parameters are derived, and the Markov chain Monte Carlo sampling algorithms are also given. The proposed method is illustrated by three simulation examples and a real dataset.

Flexible Bayesian quantile regression for nonlinear mixed effects models based on the generalized asymmetric Laplace distribution

We propose flexible Bayesian quantile regression for a class of parametric nonlinear mixed effects models for longitudinal data based on the generalized asymmetric Laplace distribution, which exhibits more flexibility in skewness, mode and tail behaviour than the frequently used asymmetric Laplace distribution in quantile regression. An efficient Markov chain Monte Carlo procedure based on the adaptive random walk Metropolis-within-Gibbs sampling algorithm is derived for posterior inference. We demonstrate through simulation studies and empirical analysis that the proposed method could provide more accurate parameter estimation and better model fit than the existing methods.