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The continuous iteration of an exponential-type function allows to describe a whole range of growths between logarithmical and exponential. This principle has been applied to a class of lifetime di...

Mixture distributions have received considerable attention in life applications. This paper presents a finite Laplace mixture model with two components. We discuss the model properties and derive the parameters estimations using the method of moments and maximum likelihood estimation. We study the relationship between the parameters and the shape of the proposed distribution. The simulation study discusses the effectiveness of parameters estimations of Laplace mixture distribution.

In this paper, we will first derive explicit expressions for the predictive maximum likelihood estimators (PMLEs) for Laplace distribution based on a time-constrained life-testing experimen...

We provide bibliographical connections and extensions of several representations of the classical Laplace distribution, discussed recently in the study of Ding and Blitzstein. Beyond presenting rel...

A class of multivariate laws as an extension of univariate Weibull distribution is presented. A well known representation of the asymmetric univariate Laplace distribution is used as the starting point. This new family of distributions exhibits some similarities to the multivariate normal distribution. Properties of this class of distributions are explored including moments, correlations, densities and simulation algorithms. The distribution is applied to model bivariate exchange rate data. The fit of the proposed model seems remarkably good. Parameters are estimated and a bootstrap study performed to assess the accuracy of the estimators.

We apply the result of conditional independence of blocked ordered data established by Iliopoulos and Balakrishnan, together with the algorithm of Huffer and Lin, to obtain exact expressions for th...

In this study, we present a new family of distributions through generalization of the extended bimodal-normal distribution. This family includes several special cases, like the normal, Birnbaum-Saunders, Student’s t, and Laplace distribution, that are developed and defined using stochastic representation. The theoretical properties are derived, and easily implemented Monte Carlo simulation schemes are presented. An inferential study is performed for the Laplace distribution. We end with an illustration of two real data sets.

The search of appropriate models for describing the currency return distribution is one of the main interests not only in finance, but also in the more recent trans-disciplinary econophysics research field. Such a search is recently focusing on cryptocurrencies, due to their proliferation. Although there is no agreement of what theoretical models are the most appropriate, there is a general consensus that the cryptocurrency return distribution is highly-peaked and heavy-tailed, with a large excess kurtosis and the tail distribution decreases as the inverse cubic power-law. With these requirements, the Laplace distribution can be considered as a valid candidate model. However, there are two limitations to be taken into account: 1) the Laplace tail distribution does not decrease as the inverse cubic power-law and 2) the excess kurtosis is fixed at 3. To make the tailedness of the Laplace distribution more flexible, still maintaining its peculiar symmetric shape, we introduce the Laplace scale mixture (LSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional Laplace 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 θ governing the tail behavior of the resulting LSM. For illustrative purposes, we consider different mixing distributions; they give rise to LSMs having a closed-form probability density function where the Laplace distribution is obtained as a special case under a convenient choice of θ . We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered LSMs. Interestingly, we show how the influence of observations associated with large scaled absolute distances is reduced (downweighted), with respect to the nested Laplace distribution, in the estimation phase. We fit these models to the returns of 4 cryptocurrencies, considering several classical symmetric distributions for comparison. The analysis shows how the proposed models represent a valid alternative to the considered competitors in terms of AIC, BIC and likelihood-ratio tests, but also in reproducing the larger empirical excess kurtosis and in resembling the empirical inverse cubic power-law decrease of the tail distribution.

High dimensional statistics deals with the challenge of extracting structured information from complex model settings. Compared with the growing number of frequentist methodologies, there are rather few theoretically optimal Bayes methods that can deal with very general high dimensional models. In contrast, Bayes methods have been extensively studied in various nonparametric settings and rate optimal posterior contraction results have been established. This paper provides a unified approach to both Bayes high dimensional statistics and Bayes nonparametrics in a general framework of structured linear models. With the proposed two-step model selection prior, we prove a general theorem of posterior contraction under an abstract setting. The main theorem can be used to derive new results on optimal posterior contraction under many complex model settings including stochastic block model, graphon estimation and dictionary learning. It can also be used to re-derive optimal posterior contraction for problems such as sparse linear regression and nonparametric aggregation, which improve upon previous Bayes results for these problems. The key of the success lies in the proposed two-step prior distribution. The prior on the parameters is an elliptical Laplace distribution that is capable to model signals with large magnitude, and the prior on the models involves an important correction factor that compensates the effect of the normalizing constant of the elliptical Laplace distribution.

Repeated measures data describe multiple measurements taken from the same experimental unit under the different treatment conditions. In particular, researches with repeated measures data in various fields such as health and behavioral sciences, education, and psychology have an important role in applied statistics. There are many methods used to analyze the results of research designs planned with these measurements. The most important difference between these methods is the assumptions on which the models are based. One of the most important assumptions needed by classical methods is the normality assumption. Many methods are valid under the assumption of normality. However, it is not always possible to hold this assumption in applications. For this reason, in the analysis of repeated measures data, different distributions are necessary that can provide flexibility beyond the normal distribution, especially in cases where the assumption of normality does not hold. In this study, it is proposed to use Multivariate Laplace distribution (MLD) which is an alternative distribution in cases where normality assumption does not hold by examining the multivariate variance analysis model (MANOVA). Under MLD assumption, the parameter estimates for the Two-way Repeated Measures MANOVA model are carried out with the maximum likelihood (ML) estimation and ML estimates are obtained via the EM Algorithm.

The functionals defining the entropies of Shannon, Tsallis, Renyi are considered. It is shown that the entropies of Tsallis and Renyi, defined for continuous random variables, approach the Shannon entropy as the parameter of extensiveness approaches unity. The concept of relative parametric sensitivity, defined by the elasticity of entropy with respect to the parameters of the distribution law, is introduced. The expressions for the parametric elasticity of the Shannon entropy for the normal distribution, Laplace distribution, Cauchy distribution, logistic distribution, loglogistic distribution, Rayleigh distribution, exponential distribution, lognormal distribution, Pareto distribution, Weibull distribution, gamma distribution are obtained. The general expressions for determining the elasticity with respect to the extensivity parameter for the entropies of Tsallis and Renyi are obtained. The general conditions for determining the elasticity for the entropies of Tsallis and Renyi with respect to one of the parameters of the distribution density are obtained.

The present paper offers a new extension to the Ishita distribution called Size-Biased Ishia distribution (SBID). Various structural statistical properties of this distribution are derived such as the jth moment, moment generating function, the coefficients of variation, skewness and kurtosis. Also, the distribution of order statistics, harmonic mean, mode, reliability analysis, maximum likelihood estimation are provided, as well as the Fisher’s information, generalized and Renyi entropies are derived. The main advantage of using sized-based distributions appears when the sample are recorded with unequal probabilities. Accordingly, the superiority of the SBI distribution is illustrated to ball bearings data. It is shown that the SBID is the most appropriate model for this data set as compared to Rama distribution, Ishita distribution and Marshall–Olkin Esscher Transformed Laplace distribution. We believe that the SBID is an alternative distribution to lifetime data analysis.

Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. In this paper, we introduce a q-Esscher transformed Laplace distribution, which is a stretched model for Esscher transformed Laplace distribution, obtained by introducing a new pathway parameter q, which facilitates a slow transition to the Esscher transformed Laplace distribution as q → 1. This pathway model can be obtained by optimizing Mathai’s generalized entropy with more general setup, which is a generalization of various entropy measures due to Shannon and others. The various properties of the q-Esscher transformed Laplace distribution are studied and its applications are discussed.

In this paper, upon using the known expressions for the Best Linear Unbiased Estimators (BLUEs) of the location and scale parameters of the Laplace distribution based on a progressively Type-II right censored sample, we derive the exact moment generating function (MGF) of the linear combination of standard Laplace order statistics. By using this MGF, we obtain the exact density function of the linear combination. This density function is then utilized to develop exact marginal confidence intervals (CIs) for the location and scale parameters through some pivotal quantities. Next, we derive the exact density of the BLUEs-based quantile estimator and use it to develop exact CIs for the population quantile. A brief mention is made about the reliability and cumulative hazard functions and as to how exact CIs can be constructed for these functions based on BLUEs. A Monte Carlo simulation study is then carried out to evaluate the performance of the developed inferential results. Finally, an example is presented to illustrate the point and interval estimation methods developed here.

Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278--294], we obtain mom...

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics. © 2018, © 2018 Taylor & Francis Group, LLC.

The paper contains an introduction to the asymptotic theory of hypothesis testing and a review of recent results of the author and his students. We consider only the asymptotic approach, for which with increasing sample size n the test size is separated from zero, and the sequences of local alternatives, for which the power is separated from one. This paper focuses on asymptotically efficient tests when testing a simple hypothesis in the case of a one-parameter family. We study the difference between the powers of the best and asymptotically efficient tests. This difference is closely related to the notion of asymptotic test deficiency. We consider the formula for the limiting deviation of the power of asymptotically optimal test from the power of the best test in the case of Laplace distribution. Due to the irregularity of the Laplace distribution, this deviation is of order n −1/2, in contrast to the usual regular families for which this order is n −1. We also study the Bayesian settings and the case of increasing parameter dimension.

We consider exact procedures for testing the equality of means (location parameters) of two Laplace populations with equal scale parameters based on corresponding independent random samples. The test statistics are based on either the maximum likelihood estimators or the best linear unbiased estimators of the Laplace parameters. By conditioning on certain quantities we manage to express their exact distributions as mixtures of ratios of linear combinations of standard exponential random variables. This allows us to find their exact quantiles and tabulate them for several sample sizes. The powers of the tests are compared either numerically or by simulation. Exact confidence intervals for the difference of the means corresponding to those tests are also constructed. The exact procedures are illustrated via a real data example.

We develop a random partition procedure based on a Dirichlet process prior with Laplace distribution. Gibbs sampling of a Laplace mixture of linear mixed regressions with a Dirichlet process is implemented as a random partition model when the number of clusters is unknown. Our approach provides simultaneous partitioning and parameter estimation with the computation of classification probabilities, unlike its counterparts. A full Gibbs-sampling algorithm is developed for an efficient Markov chain Monte Carlo posterior computation. The proposed method is illustrated with simulated data and one real data of the energy efficiency of Tsanas and Xifara (Energy and Buildings, 49, 560-567, 2012).

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown leads to a nonparametric deconvolution problem. We begin by studying the rates of convergence relative to the $$L^2$$ -norm and the Hellinger metric for the direct problem of estimating the sampling density, which is a mixture of Laplace densities with a possibly unbounded set of locations: the rate of convergence for the Bayes’ density estimator corresponding to a Dirichlet process prior over the space of all mixing distributions on the real line matches, up to a logarithmic factor, with the $$n^{-3/8}\log ^{1/8}n$$ rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the $$L^2$$ -norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any $$L^p$$ -Wasserstein distance, $$p\ge 1$$ , between the corresponding mixing distributions, provided their Laplace transforms are finite in some neighborhood of zero, we derive the rates of convergence in the $$L^1$$ -Wasserstein metric for the Bayes’ and maximum likelihood estimators of the mixing distribution. Merging in the $$L^1$$ -Wasserstein distance between Bayes and maximum likelihood follows as a by-product, along with an assessment on the stochastic order of the discrepancy between the two estimation procedures.

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.

International audience; We provide a new and simple characterization of the multivariate generalized Laplace distribution. In particular, this result implies that the product of a Gaussian matrix with independent and identically distributed columns by an independent isotropic Gaussian vector follows a symmetric multivariate generalized Laplace distribution.

In this article, we derive the joint Laplace transform of the sequential probability ratio test (SPRT) and the resulting stopped random walk process for the negative exponential model. The Laplace transform is derived by solving a related difference equation. This technique is novel because it only takes advantage of the Markov structure and does not rely on the typical martingale methods used for deriving the Laplace transform of other SPRTs. The joint Laplace transform provides the joint distribution of the SPRT and the associated stopped process, which is a new result. Even the marginal distributions were hitherto unknown.

A multivariate semi-α-Laplace distribution (denoted by Ms-αLaplace) is introduced and studied in this paper. It is more general than the multivariate Linnik and Laplace distributions proposed by Sabu and Pillai (1991) or Anderson (1992). The Ms-αLaplace distribution has univariate semi-α-Laplace (Pillai, 1985) as marginal distribution. Various characterization theorems of the Ms-αLaplace distribution based on the closure property of the normalized geometric sum are proved.

We study the first hitting time of integral functionals of time-homogeneous diffusions, and characterize their Laplace transforms through a stochastic time change. We obtain explicit expressions of the Laplace transforms for the geometric Brownian motion (GBM) and the mean-reverting GBM process. We also introduce a novel probability identity based on an independent exponential randomization and obtain explicit Laplace transforms of the price of arithmetic Asian options and other derivative prices that non-linearly depend on the integral diffusions. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.

This article considers the adaptive elastic net estimator for regularized mean regression from a Bayesian perspective. Representing the Laplace distribution as a mixture of Bartlett–Fejer kernels with a Gamma mixing density, a Gibbs sampling algorithm for the adaptive elastic net is developed. By introducing slice variables, it is shown that the mixture representation provides a Gibbs sampler that can be accomplished by sampling from either truncated normal or truncated Gamma distribution. The proposed method is illustrated using several simulation studies and analyzing a real dataset. Both simulation studies and real data analysis indicate that the proposed approach performs well.

A limit theorem is proved for doubly stochastically rarefied renewal processes. It is shown that under rather general conditions, as limit laws in limit theorems for mixed geometric random sums, there appear mixed exponential and mixed Laplace distributions. Some known and new properties of these distributions are reviewed. Also, some nonobvious properties of special representatives of these classes (the Weibull, Mittag-Leffler, Linnik, and other distributions) are described.

We propose a distribution called Odds Exponential Log Logistic Distribution (OELLD), which is an odds family of distribution. Its hazard rate is an increasing and decreasing function based on the value of the parameter. Explicit expressions for the ordinary moments, L-moments, quantile, generating functions, Bonferroni Curve, Lorenz Curve, Gini's index and order statistics are derived. The parameters of the proposed distribution are estimated by using maximum likelihood method and also illustrated by a lifetime data set.

The Laplace distribution can be compared against the normal distribution. The Laplace distribution has an unusual, symmetric shape with a sharp peak and tails that are longer than the tails of a normal distribution. It has recently become quite popular in modeling financial variables (Brownian Laplace motion) like stock returns because of the greater tails. The Laplace distribution is very extensively reviewed in the monograph (Kotz et al. in the laplace distribution and generalizations—a revisit with applications to communications, economics, engineering, and finance. Birkhauser, Boston, 2001). In this article, we propose a density-based empirical likelihood ratio (DBELR) goodness-of-fit test statistic for the Laplace distribution. The test statistic is constructed based on the approach proposed by Vexler and Gurevich (Comput Stat Data Anal 54:531–545, 2010). In order to compute the test statistic, parameters of the Laplace distribution are estimated by the maximum likelihood method. Critical values and power values of the proposed test are obtained by Monte Carlo simulations. Also, power comparisons of the proposed test with some known competing tests are carried out. Finally, two illustrative examples are presented and analyzed.

In this paper we propose two extensions of the Exponential model to describe income distributions. The Exponential ArcTan (EAT) and the composite EAT–Lognormal models discussed in this paper preserve key properties of the Exponential model including its capacity to model distributions with zero incomes. This is an important feature as the presence of zeros conditions the modelling of income distributions as it rules out the possibility of using many parametric models commonly used in the literature. Many researchers opt for excluding the zeros from the analysis, however, this may not be a sensible approach especially when the number of zeros is large or if one is interested in accurately describing the lower part of the distribution. We apply the EAT and the EAT–Lognormal models to study the distribution of incomes in Australia for the period 2001–2012. We find that these models in general outperform the Gamma and Exponential models while preserving the capacity of the latter to model zeros.

New results on expected distances for the asymmetric Laplace distribution are derived and applied to develop a new goodness-of-fit test for the asymmetric Laplace distribution based on the energy distance.

The Bayesian estimation of the scale parameter of a Laplace Distribution is obtained using two approximation techniques, like Normal approximation and Tierney and Kadane (T-K) approximation, under different informative priors.

We study occupancy counts for the critical nearest-neighbor branching random walk on the $d$-dimensional lattice, conditioned on non-extinction. For $d\geq 3$, Lalley and Zheng (2011) showed that the properly scaled joint distribution of the number of sites occupied by $j$ generation-$n$ particles, $j=1,2,\ldots$, converges in distribution as $n$ goes to infinity, to a deterministic multiple of a single exponential random variable. The limiting exponential variable can be understood as the classical Yaglom limit of the total population size of generation $n$. Here we study the second order fluctuations around this limit, first, by providing a rate of convergence in the Wasserstein metric that holds for all $d\geq3$, and second, by showing that for $d\geq 7$, the weak limit of the scaled joint differences between the number of occupancy-$j$ sites and appropriate multiples of the total population size converge in the Wasserstein metric to a multivariate symmetric Laplace distribution. We also provide a rate of convergence for this latter result., Comment: Ver2: 22 pages, minor revision; Ver1: 22 pages

The motion of a tagged degree of freedom can give important insight into the interactions present in a complex environment. We investigate the dynamics of a tagged particle in two non-equilibrium systems that consist of interacting run-and-tumble particles. The first one is an exactly solvable polymer model, the second is a two-dimensional lattice model, which is studied through simulations. We find that in both cases a tagged particle shows anomalous dynamics and non-Gaussian behaviour for times below the persistence time of the run-and-tumble motion. For later times, the dynamics of the tagged monomer becomes diffusive. In the lattice model, the anomalous behaviour persists and can, for intermediate densities, be well approximated by a Laplace distribution. We attribute this behaviour to the dynamically changing environment of the tagged particle, which we argue, is an essential ingredient to observe deviations from Gaussianity. We wish to acknowledge the work and eort that Carlo Vanderzande put into this and many other research projects. He passed away on 2 September 2019. We will always remember his kindness, his never ending enthusiasm for research and his extensive knowledge. We grieve the loss of a great friend and colleague.

International audience; A model forecasting ionospheric disturbances and its impact on GNSS positioning is proposed, called HAPEE (High lAtitude disturbances Positioning Error Estimator). It allows predicting ROTI index and corresponding Precise Point Positioning (PPP) error in Arctic region (i.e. latitudes > 50°). The model is forecasting for the next hour a probability of a disturbance index or PPP error to exceed a given threshold, from solar wind conditions measured at L1 Lagrange point. Or alternatively, it is forecasting a disturbance index level that is exceeded during the next hour for a given percentage of the time. The ROTI model has been derived from NMA network measurements, considering a database covering the years 2007 up to 2019. It is demonstrated that the statistical variability of the ROTI index is mainly following a lognormal distribution. The proposed model has been tested favorably on measurements performed using measurements from stations of the NMA network that were not used for the model derivation. It is also shown that the statistics of PPP error conditioned by ROTI is following a Laplace distribution. Then a new compound model has been proposed, based on a conditional probability combining ROTI distribution conditioned by solar wind conditions and error distributions conditioned by ROTI index level.

In this paper, a new five-parameter lifetime distribution called beta generalized linear exponential distribution (BGLED) is introduced. It includes at least 17 popular sub-models as special cases such as the beta linear exponential, the beta generalized exponential, and the exponentiated generalized linear distributions. Mathematical and statistical properties of the proposed distribution are discussed in details. In particular, explicit expression for the density function, moments, asymptotics distributions for sample extreme statistics, and other statistical measures are obtained. The estimation of the parameters by the method of maximum-likelihood is discussed and the finite sample properties of the maximum-likelihood estimators (MLEs) are investigated numerically. A real data set is used to demonstrate its superior performance fit over several existing popular lifetime models.

Gibrat (1931) initiated the study of the distribution of firms’ profit rates, suggesting the distribution was log-normal. Although initial empirical work supported that finding, a consensus has developed in the literature that the distribution of firm profit rates is best approximated by the Laplace distribution. Using a richer database than prior studies and testing for more theoretical distributions, we find that the distribution of firm profit rates is best approximated by the heavier-tailed Cauchy distribution.

In this paper, a new class of goodness of fit tests for exponential distribution is proposed. The tests use the equidistribution characterizations of exponential distribution. Based on the U-empirical Laplace transforms of equidistributed statistics, test statistics of the integral type are formed. They are U-statistics with estimated parameters. Their asymptotic properties are derived. Two families of exponentiality tests from this class, based on two selected characterizations, are presented. The approximate Bahadur efficiency is used to assess their quality. Finally, their simulated powers are calculated and the tests are compared with different exponentiality tests.

This paper introduces a Bayesian approach for composite quantile regression employing the skewed Laplace distribution for the error distribution. We use a two-level hierarchical Bayesian model for coefficient estimation and future selection which assumes a prior distribution that favors sparseness. An efficient Gibbs sampling algorithm is developed to update the unknown quantities from the posteriors. The proposed approach is illustrated via simulation studies and two real datasets. Results indicate that the proposed approach performs quite good in comparison to the other approaches.

Coefficient of tail dependence measures the strength of dependence in the tail of a bivariate distribution and it has been found useful in the risk management. In this paper, we derive the upper tail dependence coefficient for a random vector following the skew Laplace distribution and the skew Cauchy distribution, respectively. The result shows that skew Laplace distribution is asymptotically independent in upper tail, however, skew Cauchy distribution has asymptotic upper tail dependence.

A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.