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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). [ABSTRACT FROM AUTHOR]

In this paper, Bayesian estimates are derived for the location and scale parameters of the Laplace distribution based on complete, Type-I, and Type-II censored samples under different prior settings. Subsequently, Bayesian point and interval estimates, as well as the associated statistical inference, are discussed in detail. The developed methods are then applied to two real data sets for illustrative purposes. Moreover, a detailed Monte Carlo simulation study is carried out for evaluating the performance of the inferential methods developed here. Finally, we provide a brief discussion of the established results to demonstrate their practical utility and present some associated problems of further interest. Overall, this study fills an existing gap in the development of Bayesian inferential techniques for the parameters of the two-parameter Laplace distribution, making this research innovative and offering more investigative implications. It showcases the potential for broader methodological applications of Bayesian inference for complex real-world data sets, especially in scenarios involving different forms of censoring. This research provides a critical tool for statistical analysis in different fields such as engineering and finance, where the Laplace distribution is frequently adopted as a fundamental model. [ABSTRACT FROM AUTHOR]

The interest in considering the relation among random variables in quantiles instead of the mean has emerged in various fields, and data collected from complex survey designs are of fundamental importance to different areas. Despite the extensive literature on survey data analysis and quantile regression models, research papers exploring quantile regression estimation accounting for an informative design have primarily been restricted to a frequentist framework. In this paper, we introduce different Bayesian methods relying on the survey-weighted estimator and the estimating equations. A model-based simulation study evaluates the proposed methods compared to alternative approaches and a naïve model fitting ignoring the informative sampling design under different scenarios. In addition, we illustrate and conduct a prior sensitivity analysis in a design-based simulation study that uses data from Prova Brasil 2011. [ABSTRACT FROM AUTHOR]

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p-Laplace equation -- Δp u = μ with p ∈ (1,2). The cases when p ∈ (2 - 1/n, 2) and p ∈ (3n-2/2n-1, 2 - 1/n] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379-418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p ∈ (1, 3n-2/2n-1]. Interior and global modulus of continuity estimates of the gradients of solutions are also established. [ABSTRACT FROM AUTHOR]

The molecular movement in single particle tracking (SPT) experiments shows a crucial role of diffusion in many biological processes such as signaling, cellular organization, transport mechanisms, and more. The SPT analysis detects not only classical Brownian motion but diffusion with other features. These include directed diffusion and confined motion. The behavior remains a challenging problem for several reasons. Due to the action of many physical processes, random trajectories of cellular molecules are segmented in different diffusive modes. Often their study requires sophisticated algorithms for the analysis of statistical properties. In this paper we consider the segment analysis for trajectories of G proteins and coupled-receptors in living cells. Their movement is often transient and switches among free diffusion, confined diffusion, directed diffusion, and immobility. Moreover, the confined segments can have both Gaussian and non-Gaussian statistics. The types of alternation of diffusive modes along the trajectories of G proteins and coupled-receptors are analyzed. [ABSTRACT FROM AUTHOR]

Renewable integration in utility grid is crucial in the current energy scenario. Optimized utilization of renewable energy can minimize the energy consumption from the grid. This demands accurate forecasting of renewable contribution and planning. Most of the researches aim to find a suitable forecasting model in terms of accuracy and error metrics. However, the uncertainty and variability in these forecasts are also significant. This work combines point forecast with interval forecast to provide comprehensive information about the forecast uncertainty. In this work, solar irradiance forecasting is carried out using artificial intelligence (AI) techniques. Forecasting is done using seasonal auto-regressive moving average with exogenous factors (SARIMAX), support vector regression (SVR), long short term memory (LSTM) techniques and performance is evaluated. SVR model exhibited the best performance with R 2 values of 0.97 and 0.96 for winter and summer respectively and 0.85 for monsoon and post-monsoon seasons. This is followed by forecast error distribution studies and uncertainty analysis. For this, SVR forecast error data is fitted using laplace distribution. Uncertainty study is carried out using confidence intervals and coverage rates. Excellent coverage rates are obtained for various confidence levels for all seasons, indicating the appropriate fitting of error distribution. For the narrow 85% confidence band, coverage rates of 89%, 95%, 90%, and 88% are obtained for winter, summer, monsoon and post-monsoon respectively. The work emphasizes the need for error-distribution studies, modeling of forecast errors and their application in providing reliable forecast intervals with the perspective of enhancing system reliability. [ABSTRACT FROM AUTHOR]

Bayesian regularized composite quantile regression approaches with bridge and reciprocal bridge penalties are adopted to conduct variable selection in composite quantile regression. Two MCMC algorithms were developed for posterior inference using the normal-exponential mixture representation of the asymmetric Laplace distribution. Gamma prior is placed on the regularization parameter. The parameters of the Gamma prior are treated as unknowns and estimate them along with the other parameters. The performance of the proposed composite quantile regression methods over the existing methods is then illustrated via simulated studies and a real data set. [ABSTRACT FROM AUTHOR]

In the area of information retrieval, in order to improve search accuracy and reduce communication overhead, there is an increasing tendency to adopt ranked search in engines. Ranked search allows cloud servers to search for the top k most relevant documents based on the relevance score between the query keywords and the documents. Recently, with the increasing popularity of encrypted search technologies, ranked searchable encryption is proposed accordingly which focuses on solving ranked search problem over encrypted databases. However, recent studies show that some privacy protection methods commonly used in ranked searchable encryption, like order-preserving encryption (OPE), have some security problems. These problems may lead to the leakage of the relevant ranking privacy information. Meanwhile, most of the existing ranked searchable encryption schemes do not consider the problem of payment for outsourced services. In this paper, we propose a scheme called ranked searchable encryption based on differential privacy and blockchain (DPB-RSE). Specifically, we first add noise drawn from a Laplace distribution into the relevance score to disturb its value. Then we design a smart contract to verify the correctness of the results returned by the cloud server and realize payment fairness. The experiment results demonstrate that the accuracy of search results in this scheme can reach 94% in a small privacy budge. [ABSTRACT FROM AUTHOR]

Recommendation systems powered by artificial intelligence (AI) are widely used to improve user experience. However, AI inevitably raises privacy leakage and other security issues due to the utilization of extensive user data. Addressing these challenges can protect users' personal information, benefit service providers, and foster service ecosystems. Presently, numerous techniques based on differential privacy have been proposed to solve this problem. However, existing solutions encounter issues such as inadequate data utilization and a tenuous trade-off between privacy protection and recommendation effectiveness. To enhance recommendation accuracy and protect users' private data, we propose ID-SR, a novel privacy-preserving social recommendation scheme for trustworthy AI based on the infinite divisibility of Laplace distribution. We first introduce a novel recommendation method adopted in ID-SR, which is established based on matrix factorization with a newly designed social regularization term for improving recommendation effectiveness. We then propose a differential privacy-preserving scheme tailored to the above method that leverages the Laplace distribution's characteristics to safeguard user data. Theoretical analysis and experimentation evaluation on two publicly available datasets demonstrate that our scheme achieves a superior balance between privacy protection and recommendation effectiveness, ultimately delivering an enhanced user experience. [ABSTRACT FROM AUTHOR]

This 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. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we propose the multivariate range Value-at-Risk (MRVaR) and the multivariate range covariance (MRCov) as two risk measures and explore their desirable properties in risk management. To facilitate analytical analyses, we derive explicit expressions of the MRVaR and the MRCov in the context of the multivariate (log-)elliptical distribution family. Frequently-used cases in industry, such as normal, student-t, logistic, Laplace, and Pearson type VII distributions, are presented with numerical examples. As an application, we propose a range-based mean-variance framework for optimal portfolio selection. [ABSTRACT FROM AUTHOR]

This manuscript presents a new two-parameter unit stochastic distribution, obtained by transforming the Laplace distribution, using a generalized logistic map, into a unit interval. The distribution thus obtained is named the Laplace-logistic unit (abbreviated LLU) distribution, and its basic stochastic properties are examined in detail. Also, the procedure for estimating parameters based on quantiles is provided, along with the asymptotic properties of the obtained estimates and the appropriate numerical simulation study. Finally, the application of the LLU distribution in dynamic and regression analysis of real-world data with accentuated "peaks" and "fat" tails is also discussed. [ABSTRACT FROM AUTHOR]

Two new non convex penalty functions – Laplace and arctan – were recently introduced in the literature to obtain sparse models for high-dimensional statistical problems. In this article, we study the theoretical properties of Laplace and arctan penalized ordinary least squares linear regression models. We first illustrate the near-unbiasedness of the non zero regression weights obtained by the new penalty functions, in the orthonormal design case. In the general design case, we present theoretical results in two asymptotic settings: (a) the number of features, p fixed, but the sample size, n → ∞ , and (b) both n and p tend to infinity. The theoretical results shed light onto the differences between the solutions based on the new penalty functions and those based on existing convex and non convex Bridge penalty functions. Our theory also shows that both Laplace and arctan penalties satisfy the oracle property. Finally, we also present results from a brief simulations study illustrating the performance of Laplace and arctan penalties based on the gradient descent optimization algorithm. [ABSTRACT FROM AUTHOR]

The estimation of the slope (b -value) of the frequency–magnitude distribution of earthquakes is based on a formula derived by Aki decades ago, assuming a continuous exponential distribution. However, as the magnitude is usually provided with a limited resolution, its distribution is not continuous but discrete. In the literature, this problem was initially solved by an empirical correction (due to Utsu) to the minimum magnitude, and later by providing an exact formula such as that by Tinti and Mulargia, based on the geometric distribution theory. A recent paper by van der Elst showed that the b -value can be estimated also by considering the magnitude differences (which are proven to follow an exponential discrete Laplace distribution) and that in this case the estimator is more resilient to the incompleteness of the magnitude data set. In this work, we provide the complete theoretical formulation including (i) the derivation of the means and standard deviations of the discrete exponential and Laplace distributions; (ii) the estimators of the decay parameter of the discrete exponential and trimmed Laplace distributions and (iii) the corresponding formulas for the parameter b. We deduce (iv) the standard 1 σ intervals for the estimated b. Moreover, we are able (v) to quantify the error associated with the Utsu minimum-magnitude correction. Furthermore, we have discussed the formulas to produce statistically independent magnitude differences. We tested extensively the b -value estimators on simulated synthetic data sets including complete catalogues as well as catalogues affected by a strong incompleteness degree such as aftershock sequences where the incompleteness is made to vary from one event to the next. We have also analysed the real aftershock sequence of the 30/10/2016 Norcia (central Italy) to integrate the finding of the simulations. To judge the performance of the various estimators we have introduced an index p that can be seen as a non-parametric extension of the Student's t index. The main outcomes of this paper are that (1) the b -value estimators devised for continuous magnitude data are not adequate for binned magnitudes, (2) for complete data sets, estimators based on magnitudes and on magnitude differences provide substantially equivalent results, (3) for incomplete magnitude data sets, estimators based on magnitude differences provide better results and (4) for incomplete aftershock sequences there is no evidence that methods based on positive magnitude differences are superior than other methods using differences. This conclusion is further confirmed by our analysis of the above-mentioned Norcia seismic sequence. This last finding contrasts with the van der Elst's claim that the so called |${{b}_ + }$| method is the most adequate to treat real aftershock sequences. [ABSTRACT FROM AUTHOR]

In recent years, multi-regional clinical trials (MRCTs) have increased in popularity in the pharmaceutical industry due to their ability to accelerate the global drug development process. To address potential challenges with MRCTs, the International Council for Harmonisation released the E17 guidance document which suggests the use of statistical methods that utilize information borrowing across regions if regional sample sizes are small. We develop an approach that allows for information borrowing via Bayesian model averaging in the context of a joint analysis of survival and longitudinal data from MRCTs. In this novel application of joint models to MRCTs, we use Laplace's method to integrate over subject-specific random effects and to approximate posterior distributions for region-specific treatment effects on the time-to-event outcome. Through simulation studies, we demonstrate that the joint modeling approach can result in an increased rejection rate when testing the global treatment effect compared with methods that analyze survival data alone. We then apply the proposed approach to data from a cardiovascular outcomes MRCT. [ABSTRACT FROM AUTHOR]

A semi-analytical solution is proposed for the biohe- at equation, which includes the epidermis, dermis, and hypodermis layers in the presence of a surface pulsed heat source. A switching time sur- face heating/cooling source, which has therapeutic applications in hu- man tissue burning, is used. The interface temperature is calculated by matching the temperature and heat ux between two adjacent layers. A high-performance computing algorithm is designed and implemented by combining semigroups theory, Laplace transform, and convolution opera- tors in each layer. It is proved that the proposed solution is consistent, convergent and stable. The reliability, performance and eficiency of semi- analytical solutions are compared using the bioheat transfer module of COMSOL software based on standard finite element methods. Numerical results for three difierent medical examples are given. In uences of blood pressure on temperature along the layered skin for difierent switching and final times are discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Riverbed elevations play a crucial role in sediment transport and flow resistance, making it essential to understand and quantify their effects. This knowledge is vital for various fields, including river engineering and stream ecology. Previous observations have revealed that fluctuations in the bed surface can exhibit both multifractal and monofractal behaviors. Specifically, the probability distribution function (PDF) of elevation increments may transition from Laplace (two‐sided exponential) to Gaussian with increasing scales or consistently remain Gaussian, respectively. These differences at the finest timescale lead to distinct patterns of bedload particle exchange with the bed surface, thereby influencing particle resting times and streamwise transport. In this paper, we utilize the fractional Laplace motion (FLM) model to analyze riverbed elevation series, demonstrating its capability to capture both mono‐ and multi‐fractal behaviors. Our focus is on studying the resting time distribution of bedload particles during downstream transport, with the FLM model primarily parameterized based on the Laplace distribution of increments PDF at the finest timescale. Resting times are extracted from the bed elevation series by identifying pairs of adjacent deposition and entrainment events at the same elevation. We demonstrate that in cases of insufficient data series length, the FLM model robustly estimates the tail exponent of the resting time distribution. Notably, the tail of the exceedance probability distribution of resting times is much heavier for experimental measurements displaying Laplace increments PDF at the finest scale, compared to previous studies observing Gaussian PDF for bed elevation. Plain Language Summary: The evolution of riverbed elevations is difficult to describe due to its highly variable and strongly non‐linear nature. Understanding and quantifying the dynamics of riverbed elevations are important for river engineering and stream ecology, and serve as the basis for numerical models of predicting sediment transport as well as interpreting stratigraphy from the past records. Through laboratory experiments, we have observed that the form of elevation increment PDF can change from Laplace to Gaussian as the timescale increases. This phenomenon is successfully modeled in this paper for the first time by the fractional Laplace motion, which essentially generates bed elevation series for the evolution of bed surface height at a certain spatial location of the bed. This series contains information on how long a bedload particle can rest (resting time) in the riverbed before it can be re‐entrained to move downstream, the determination of which by other means (e.g., particle‐tracking measurements) is challenging. By extracting resting times embedded in this bed elevation series, we obtain statistics (i.e., the tail behavior of the resting time distribution) that are key for correctly modeling the transport of bedload particles, and more specifically, that can help us to understand the anomalous bedload diffusion process. Key Points: The Fractional Laplace motion (FLM) can be used to describe the evolution of the distribution of bed elevation increments over different timescalesCorrectly predicting bed elevation fluctuations at the finest timescale is critical for estimating resting times for bedload tracer transportFLM model provides a means of robustly estimating the tail exponent of the resting time distribution in case of insufficient data series length [ABSTRACT FROM AUTHOR]

Abstract The molecular movement in single particle tracking (SPT) experiments shows a crucial role of diffusion in many biological processes such as signaling, cellular organization, transport mechanisms, and more. The SPT analysis detects not only classical Brownian motion but diffusion with other features. These include directed diffusion and confined motion. The behavior remains a challenging problem for several reasons. Due to the action of many physical processes, random trajectories of cellular molecules are segmented in different diffusive modes. Often their study requires sophisticated algorithms for the analysis of statistical properties. In this paper we consider the segment analysis for trajectories of G proteins and coupled-receptors in living cells. Their movement is often transient and switches among free diffusion, confined diffusion, directed diffusion, and immobility. Moreover, the confined segments can have both Gaussian and non-Gaussian statistics. The types of alternation of diffusive modes along the trajectories of G proteins and coupled-receptors are analyzed.

Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.

The main finding of this study is the derivation of a new probability distribution that reveals interesting properties, especially with various asymmetry and kurtosi; behavior. We call this distribution the asymmetric generalized error distribution (AGED). AGED is a new contribution to the field of statistical theory, offering more flexible probability density functions, cumulative distribution functions and hazard functions than the base distribution. The AGED also includes normal uniform, Laplace, asymmetric Laplace, and generalized error distribution (GED as special cases. The mathematical and statistical features of the distribution are derived and discussed. Estimators of the parameters of the distribution are obtained using the maximum likelihood approach. In a simulation study, random samples are generated from the new probability distribution to illustrate wha ideal data looks like. Using real data from diverse applications such as health industry, and cybersecurity domains, the performance of the new distribution is compared to that of other distributions. The new distribution is found to be #AKC better fit for the data, showing great adaptability in the context of real data analysis. We expect the distribution to be applied to many more real data, anc the findings of the study can be used as a basis for future research in the field. [ABSTRACT FROM AUTHOR]

We consider how a population of N haploid individuals responds to directional selection on standing variation, with no new variation from recombination or mutation. Individuals have trait values z 1 , ... , z N , which are drawn from a distribution ψ ; the fitness of individual i is proportional to e z i . For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance V 0 , and show that for large N , there is a scaling limit which depends on a single parameter N V 0 . When selection is weak relative to drift (N V 0 ≪ 1), the variance decreases exponentially at rate 1 / N , and the expected ultimate gain in log fitness (scaled by V 0 ), is just N V 0 , which is the same as Robertson's (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift (N V 0 ≫ 1), the ultimate gain can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time t ∼ 1 / V 0 of an allele with value z is P (z) , with mean P ¯ , the winning allele is the fittest of N P ¯ survivors drawn from a distribution ψ P / P ¯. The expected ultimate change is ∼ 2 log (1. 15 N V 0 ) for a Gaussian distribution, and ∼ − 1 2 log 0. 36 N V 0 − log − log 0. 36 N V 0 for a Laplace distribution. This approach also predicts the variability of the process, and its dynamics; we show that in the strong selection regime, the expected genetic variance decreases as ∼ t − 3 at large times. We discuss how these results may be related to selection on standing variation that is spread along a linear chromosome. [ABSTRACT FROM AUTHOR]

In this study, we introduce a matrix variate skew Laplace distribution as a variance-mean mixture of the matrix variate normal and the scale inverse gamma distribution. The proposed distribution is a generalization of the multivariate skew Laplace distribution studied by [1]. We explore some distributional properties of the proposed distribution such as the probability density function and the characteristic function. Also, we study the estimation of the parameters and give an EM algorithm to obtain the estimates of the parameters. Then, we give a small simulation study to illustrate the performance of the proposed EM algorithm for finding the estimates. [ABSTRACT FROM AUTHOR]

Despite the increasing importance of high‐dimensional varying coefficient models, the study of their Bayesian versions is still in its infancy. This paper contributes to the literature by developing a sparse empirical Bayes formulation that addresses the problem of high‐dimensional model selection in the framework of Bayesian varying coefficient modelling under Gaussian process (GP) priors. To break the computational bottleneck of GP‐based varying coefficient modelling, we introduce the low‐cost computation strategy that incorporates linear algebra techniques and the Laplace approximation into the evaluation of the high‐dimensional posterior model distribution. A simulation study is conducted to demonstrate the superiority of the proposed Bayesian method compared to an existing high‐dimensional varying coefficient modelling approach. In addition, its applicability to real data analysis is illustrated using yeast cell cycle data. [ABSTRACT FROM AUTHOR]

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 prior work, 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. [ABSTRACT FROM AUTHOR]

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L^{p}-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures. [ABSTRACT FROM AUTHOR]

High‐breakdown‐point estimators of multivariate location and shape matrices, such as the MM‐estimator with smoothed hard rejection and the Rocke S‐estimator, are generally designed to have high efficiency for Gaussian data. However, many phenomena are non‐Gaussian, and these estimators can therefore have poor efficiency. This article proposes a new tunable S‐estimator, termed the Sq‐estimator, for the general class of symmetric elliptical distributions, a class containing many common families such as the multivariate Gaussian, t‐, Cauchy, Laplace, hyperbolic, and normal inverse Gaussian distributions. Across this class, the Sq‐estimator is shown to generally provide higher maximum efficiency than other leading high‐breakdown estimators while maintaining the maximum breakdown point. Furthermore, the Sq‐estimator is demonstrated to be distributionally robust, and its robustness to outliers is demonstrated to be on par with these leading estimators while also being more stable with respect to initial conditions. From a practical viewpoint, these properties make the Sq‐estimator broadly applicable for practitioners. These advantages are demonstrated with an example application—the minimum‐variance optimal allocation of financial portfolio investments. [ABSTRACT FROM AUTHOR]

The paper deals with the stochastic analysis of a two identical unit cold standby system model. As soon as a unit fails it is attended first by assistant repairman, who completes its repair to make it as good as new with fixed known probability p. Otherwise, with probability q( =1-p), the unit repaired by assistant repairman goes to expert repairman so as to make it as good as new. Both the repairmen are always available with the system and they work their jobs simultaneously. The failure time and repair time by assistant repairman are correlated random variables having their joint distribution as Bivariate exponential. Various measures of system effectiveness useful to system designers and operation managers are obtained by using regenerative point technique. The MTSF and steady state profit have also been studied through graphs in respect of parameter alpha for three different values of repair rate by expert repairman and two different values of correlation coefficient between failure and repair times and important conclusions are drawn. [ABSTRACT FROM AUTHOR]

Investing in stocks and shares is a common strategy to pursue potential gains while considering future financial needs, such as retirement and children's education. Effectively managing investment risk requires thoroughly analyzing stock market returns and making informed predictions. Traditional models often utilize normal variance distributions to describe these returns. However, stock market returns often deviate from normality, exhibiting skewness, higher kurtosis, heavier tails, and a more pronounced center. This paper investigates the Laplace distribution and its generalized forms, including asymmetric Laplace, skewed Laplace, and the Kumaraswamy Laplace distribution, for modeling stock market returns. Our analysis involves a comparative study with the widely-used Variance-Gamma distribution, assessing their fit with the weekly returns of the S&P 500 Index and its eleven business sectors, drawing parallel inferences from international stock market indices like IBOVESPA and KOSPI for emerging and developed economies, as well as the 20+ Years Treasury Bond ETFs and individual stocks across varied time horizons. The empirical findings indicate the superior performance of the Kumaraswamy Laplace distribution, which establishes it as a robust alternative for precise return predictions and efficient risk mitigation in investments. [ABSTRACT FROM AUTHOR]

This research endeavour-investigates the enhanced adaptation of the Laplace-based variational iteration method (VIM) tailored specifically for tackling the Duffing Equation. This is accomplished by incorporating the Lagrange multiplier as a strategic tool to effectively address the inherent natural frequency within the Duffing Equation. Using a meticulous comparative analysis, here are juxtapose the analytical outcomes generated by the modified VIM approach with the numerical solution obtained through the application of the renowned Runge-Kutta Fehlberg method (RKF45), implemented by using the powerful mathematical software, MAPLE. Furthermore, by exploring the profound influence of diverse initial conditions on the resulting solution, a diverse array of distinct graphical representations is presented. [ABSTRACT FROM AUTHOR]

In practical radar systems, changes in the target aspect toward the radar will result in glint noise disturbances in the measurement data. The glint noise has heavy-tailed characteristics and cannot be perfectly modeled by the Gaussian distribution widely used in conventional tracking algorithms. In this article, we investigate the challenging problem of tracking a time-varying number of maneuvering targets in the context of glint noise with unknown statistics. By assuming a set of models for the possible motion modes of each single-target hypothesis and leveraging the multivariate Laplace distribution to model measurement noise, we propose a robust interacting multi-model multi-Bernoulli mixture filter based on the variational Bayesian method. Within this filter, the unknown noise statistics are adaptively learned while filtering and the predictive likelihood is approximately calculated by means of the variational lower bound. The effectiveness and superiority of our proposed filter is verified via computer simulations. [ABSTRACT FROM AUTHOR]

Finite mixture of linear regression (FMLR) models are an efficient tool to fit the unobserved heterogeneous relationships. The parameter estimation of FMLR models is usually based on the normality assumption, but it is very sensitive to outliers. Meanwhile, the traditional robust methods often need to assume a specific error distribution, and are not adaptive to dataset. In this paper, a robust estimation procedure for FMLR models is proposed by assuming that the error terms follow an asymmetric exponential power distribution, including normal distribution, skew-normal distribution, generalized error distribution, Laplace distribution, asymmetric Laplace distribution, and uniform distribution as special cases. The proposed method can select the suitable loss function from a broad class in a data driven fashion. Under some conditions, the asymptotic properties of proposed method are established. In addition, an efficient EM algorithm is introduced to implement the proposed methodology. The finite sample performance of the proposed approach is illustrated via some numerical simulations. Finally, we apply the proposed methodology to analyze a tone perception data. [ABSTRACT FROM AUTHOR]

Unidirectional liquid transport (UDLT) has been widely used in various fields as an important process for transferring both mass and energy. However, UDLT driven by a structural gradient has been witnessed for a long time only in wettable liquids. For nonwettable liquids, UDLT can hardly proceed merely by a structural gradient. Herein, we propose an asymmetrically concave structured surface (AMC-surface), featuring tip-to-base periodically arranged pyramid-shaped concave structures with a certain degree of overlap, which enables the UDLT of both wettable and nonwettable liquids. For wettable liquids, the capillary force along each corner leads to the UDLT pointing toward the base side of the concave pyramid, while for nonwettable liquids, the UDLT is attributable to the static liquid pressure overwhelming the repulsive Laplace pressure induced by the asymmetric grooves and overlapping part. As a result, both wettable and nonwettable liquids transport spontaneously and unidirectionally on the AMC-surface with no energy input. Moreover, the concave structure endows good mechanical stability and can be easily prepared using a facile nail-punching approach over a large area. We also demonstrated its application in a continuous chemical reaction in a confined area. We envision that the unique UDLT behavior on the as-developed AMC-surface will shed new light on the programmable manipulation of various liquids. [ABSTRACT FROM AUTHOR]

According to the Ray-Knight description, the Brownian local time in some conditional probability space is a diffusion process with respect to spatial variable. This diffusion has a local time. Thus the following definition of local time of the original Brownian local time seems to be natural: such a process is called the second-order Brownian local time. The paper studies the Laplace transform of the distribution of the Brownian local time of the second order. [ABSTRACT FROM AUTHOR]

Finite mixture of linear regression (FMLR) models are among the most exemplary statistical tools to deal with various heterogeneous data. In this paper, we introduce a new procedure to simultaneously determine the number of components and perform variable selection for the different regressions for FMLR models via an exponential power error distribution, which includes normal distributions and Laplace distributions as special cases. Under some regularity conditions, the consistency of order selection and the consistency of variable selection are established, and the asymptotic normality for the estimators of non-zero parameters is investigated. In addition, an efficient modified expectation-maximization (EM) algorithm and a majorization-maximization (MM) algorithm are proposed to implement the proposed optimization problem. Furthermore, we use the numerical simulations to demonstrate the finite sample performance of the proposed methodology. Finally, we apply the proposed approach to analyze a baseball salary data set. Results indicate that our proposed method obtains a smaller BIC value than the existing method. [ABSTRACT FROM AUTHOR]

Lithium-ion batteries are currently used as a key energy source in various industrial facilities, electronics, and automotive industries. However, due to the frequent charging and discharging of batteries, overcharging and overdischarging can occur, leading to fire and safety accidents as well as additional financial damages due to equipment failure. Therefore, accurately estimating the battery state of charge (SOC) is very important. In this paper, the robustness of estimation models was analyzed in relation to data collected amidst sensor failures. This analysis was especially pertinent during the battery SOC estimation process, when voltage and current sensors were prone to failure. The impact of these sensor failures on the accuracy and reliability of the SOC estimation models was rigorously scrutinized. Normal data was trained as training data, and Gaussian distribution, Laplace and chi-square combined distribution, Add bias distribution were employed as the test data. Herein, multilayer neural network, long short-term memory, gated recurrent unit, gradient boosting machine (GBM) were used as neural networks, the failure signal processing performance of each estimation algorithm was compared and analyzed, and the failure diagnosis was performed using support vector machine and GBM. [ABSTRACT FROM AUTHOR]

Composite quantile regression methods have been shown to be effective techniques in improving the prediction accuracy. In this article, we propose a Bayesian weighted composite quantile regression estimation procedure to estimate unknown regression coefficients and autoregressive parameters in the linear regression models with autoregressive errors. A Bayesian joint hierarchical model is established using the working likelihood of the asymmetric Laplace distribution. Adaptive Lasso-penalized type priors are used on regression coefficients and autoregressive parameters of the model to conduct inference and variable selection simultaneously. A Gibbs sampling algorithm is developed to simulate the parameters from the posterior distributions. The proposed method is illustrated by some simulation studies and analyzing a real data set. Both simulation studies and real data analysis indicate that the proposed approach performs well. [ABSTRACT FROM AUTHOR]

The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support $\mathbb{T} = {\mathbb{R}^ + }$ T = R + by ${\mathbb{T}} = {\mathbb{N}}\backslash \{ 0\} $ T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions. [ABSTRACT FROM AUTHOR]

Modeling longitudinal and survival data jointly offers many advantages such as addressing measurement error and missing data in the longitudinal processes, understanding and quantifying the association between the longitudinal markers and the survival events, and predicting the risk of events based on the longitudinal markers. A joint model involves multiple submodels (one for each longitudinal/survival outcome) usually linked together through correlated or shared random effects. Their estimation is computationally expensive (particularly due to a multidimensional integration of the likelihood over the random effects distribution) so that inference methods become rapidly intractable, and restricts applications of joint models to a small number of longitudinal markers and/or random effects. We introduce a Bayesian approximation based on the integrated nested Laplace approximation algorithm implemented in the R package R-INLA to alleviate the computational burden and allow the estimation of multivariate joint models with fewer restrictions. Our simulation studies show that R-INLA substantially reduces the computation time and the variability of the parameter estimates compared with alternative estimation strategies. We further apply the methodology to analyze five longitudinal markers (3 continuous, 1 count, 1 binary, and 16 random effects) and competing risks of death and transplantation in a clinical trial on primary biliary cholangitis. R-INLA provides a fast and reliable inference technique for applying joint models to the complex multivariate data encountered in health research. [ABSTRACT FROM AUTHOR]

This research delves into the exploration of a statistical testing approach grounded in Laplace transform techniques specifically tailored for the New Better than Used Laplace transform order (NBUL) class of life distributions. The developed test exhibits versatility, accommodating both complete and censored data, and critical values are systematically calculated for its application. Beyond the methodological presentation, our study investigates the test's statistical power and explores Pitman's asymptotic efficiency concerning various alternative distributions. Comparative analyses with other tests within the same class contribute to a comprehensive understanding of the proposed approach. To demonstrate practical applicability, we apply this novel testing technique to authentic engineering and medical datasets. The outcomes of these applications serve as illustrative examples, showcasing the effectiveness and relevance of the proposed methodology in real-world scenarios. [ABSTRACT FROM AUTHOR]

A family of parsimonious contaminated shifted asymmetric Laplace mixtures is developed for unsupervised classification of asymmetric clusters in the presence of outliers and noise. A series of constraints are applied to a modified factor analyzer structure of the component scale matrices, yielding a family of twelve models. Application of the modified factor analyzer structure and these parsimonious constraints makes these models effective for the analysis of high-dimensional data by reducing the number of free parameters that need to be estimated. A variant of the expectation-maximization algorithm is developed for parameter estimation with convergence issues being discussed and addressed. Popular model selection criteria like the Bayesian information criterion and the integrated complete likelihood (ICL) are utilized, and a novel modification to the ICL is also considered. Through a series of simulation studies and real data analyses, that includes comparisons to well-established methods, we demonstrate the improvements in classification performance found using the proposed family of models. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp - Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp (Rd) and a time weighted Lr (Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space. [ABSTRACT FROM AUTHOR]

Seismic b-value is one of the most important parameters for seismological research and seismic hazards assessment, while the accuracy of the b-value largely depended on the completeness of seismic catalog. This article compares eight methods for estimating the minimum magnitude of completeness (Mc). The results indicate that the modified maximum curvature method (MMAXC), exhibits greater stability and accuracy, closely approximating the standard M[sub c] obtained from the synthetic seismic catalogs. We then calculate the b-value using the instrumental seismic catalog from 2000-2023 in the eastern Tibetan Plateau. The results indicate that the five major earthquakes occur in regions with lower b-value. In addition, the temporal evolution of b-value before and after major earthquakes exhibits a common trend of decreasing before earthquakes, and increasing after earthquakes, which may reflect the stress accumulation and release during earthquakes. Combining the results of maximum shear strain rate and b-value, we identify five regions characterized by low b-value and high shear strain rate, indicating a higher potential seismic hazard in the future. [ABSTRACT FROM AUTHOR]

We present a novel depth from focus technique. Following prior work, our pipeline starts with a focal stack and an estimation of the amount of defocus as given by, for instance, the ring difference filter. To improve robustness to outliers while avoiding to rely on costly nonlinear optimizations, we propose an original scheme that linearly scans the profile over a fixed size window, searching for the best peak within each window using a linearized least-squares Laplace regression. As a post-process, depth estimates with low confidence are reconstructed though an adaptive moving least squares filter. We show how to objectively evaluate the performance of our approach by generating synthetic focal stacks from which the reconstructed depth maps can be compared to ground truth. Our results show that our method achieves higher accuracy than previous nonlinear Laplace regression technique, while being orders of magnitude faster. [ABSTRACT FROM AUTHOR]

We investigate the stochastic unstable nonlinear Schrödinger equation through bi-random sources. Specifically, we solve this equation via Itô sense, with the parameter following Laplace and Gumbel distributions. We provide vital stochastic solutions in applied sciences. We employ He's semi-inverse technique in order to provide these solutions in a unified way. Actually, this is the first time that the same model has been taken into account in these circumstances. In order to investigate the real relevance of the stochastic unstable nonlinear Schrödinger equation, we provide the simulations for some of the collected solutions using the appropriate parameter settings provided by the MATLAB software. Finally, our renewed drive might expand to incorporate further emerging natural science models. [ABSTRACT FROM AUTHOR]

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered. [ABSTRACT FROM AUTHOR]

Regression model is a popular and well-acknowledged technique for finding a relationship between random phenomena. In this context, the normality assumption on the error terms is one of the challenging issues in practical studies because many phenomena in the real world are prone to skewness, peakness, and outliers. In this paper, a new generalization of nonlinear regression models is postulated by considering asymmetric Laplace distribution on the error terms to cover the drawbacks of the normal-based model in accommodating skew data with peakness and mild outliers. We present three stochastic representations of the model which enables us to develop an expectation–maximization (EM) algorithm to computationally obtain the maximum-likelihood (ML) parameter estimates. The observed information matrix is computed by adopting an information-based approach. The Bayesian analysis for the proposed model is also investigated. Finally, experiments on the simulation and real-world datasets illustrate some computational and robust aspects of the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, we first develop exact likelihood inference for Laplace distribution based on a generalized Type-I hybrid censored sample (Type-I HCS). We derive explicit expressions for the maximum likelihood estimators (MLEs) of the location and scale parameters. We then derive the joint moment generating function (MGF) of the MLEs, and use it to obtain the exact distributions and moments of the MLEs. Using an analogous approach, we extend the results to a generalized Type-II hybrid censored sample (Type-II HCS) next. Finally, we present a numerical example to illustrate all the results established here. [ABSTRACT FROM AUTHOR]

The discrete Weibull model can be adapted to capture different levels of dispersion in the count data. This paper takes into account the direct relationship between explanatory variables and the median of discrete Weibull response variable. Additionally, it provides the Bayesian estimate of the discrete Weibull regression model using the random walk Metropolis algorithm. The prior distributions of the coeficient predictors were carried out based on the uniform non-informative, normal and Laplace distributions. The performance of the Bayes estimators was also compared with the maximum likelihood estimator in terms of the mean square error and the coverage probability through the Monte Carlo simulation study. Meanwhile, a real data set was analyzed to show how the proposed model and the methods work in practice. [ABSTRACT FROM AUTHOR]

Snapping Shrimps (SSs) live in a warm ocean except the North and South Poles, and they are characterized by generating strong shock waves underwater using large claws. Shock waves generated by these SSs are used for marine noise research as a signal and as a noise source, because they cause a decrease in the Signal-to-Noise Ratio (SNR), acting as one of the disruptors in fields such as sonar for target detection and underwater communication. A state-of-the-art technique to detect Snapping Shrimp Noise (SSN) is Linear Prediction (LP) analysis. Using the feature where SSN occurs for a very short time, the SSN interval was detected based on the phenomenon where the residuals appear large in the SSN interval when the LP analysis is used. In this paper, we propose an SSN interval detection technique using the Likelihood Ratio (LR) as a follow-up study to the LP-analysis-based method for further performance improvements. The proposed method was used to analyze the statistical distribution characteristics of the LP residual of SSNs compared to Gaussian, Laplace, and Gamma distributions through the Goodness-Of-Fit test. Based on this, the statistical-model-based LRs of the three distributions were computed to detect the SSN interval. Comparing the proposed method with the state-of-the-art method, the proposed method achieved 0.0620, 0.0675, and 0.0662 improvements in Gaussian, Laplace, and Gamma distributions in the Receiver Operating Characteristic curve and Area Under Curve, respectively. The study results confirmed that the proposed method can operate effectively in the marine acoustic environment. This can help find accurate intervals for the automatic labeling of or reduction in SSN. [ABSTRACT FROM AUTHOR]