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The Kullback–Leibler information number criterion is a useful criterion for determining an optimal power transformation to approximate the target distribution. When the normal distribution with a fixed variance is set as the target distribution, this criterion may be used to show the asymptotic optimality of the square-root transformation. In particular, when the Box–Cox transformation corresponding to the square-root transformation is applied, the Kullback–Leibler information number of the transformed gamma distribution with the scale parameter equal to the fixed variance converges to zero asymptotically as the shape parameter approaches infinity. That provides a stronger result on the mode of convergence of the variance-stabilizing power transformation on the gamma distribution. Moreover, an analogous result may be obtained by setting the Laplace distribution with a fixed scale parameter as the target distribution. This result utilizes a novel application of the asymptotic expansion for the normalized upper incomplete gamma function at the transition point. [ABSTRACT FROM AUTHOR]

In this article, we consider an integral Stein equation for Laplace distribution and solve it using the semigroup approach. Next, we derive regularity estimates for the solution of the Laplace Stein equation. Finally, we apply these estimates to obtain a convergence rate for Laplace approximation of the random geometric sums. We also compare our rate with the existing literature. [ABSTRACT FROM AUTHOR]

Abstract Geometric dispersion models, characterized by their v-functions, are recently introduced arising from geometric sums of exponential dispersion models and they exhibit many potential applications. In this paper, we classify all the real quadratic v-functions. Up to affinity, there are only six types of such models with unbounded domain: asymmetric Laplace, geometric and the four remaining ones are obtained by the exponential mixtures of Poisson, gamma, negative binomial and generalized hyperbolic secant distributions. Further, we find the seventh one which is geometric hybrid distribution, purely a quadratic v-function on bounded domain and, classically steep as well as unbounded ones but not geometric-steep. [ABSTRACT FROM AUTHOR]

Compound random measures (CoRM’s) are a flexible and tractable framework for vectors of completely random measure. In this paper, we provide conditions to guarantee the existence of a CoRM. Furthermore, we prove some interesting properties of CoRM’s when exponential scores and regularly varying Lévy intensities are considered. [ABSTRACT FROM AUTHOR]

Assuming that the error terms follow a multivariate Laplace distribution, we propose a robust estimation procedure for mixture of multivariate linear regression models in this paper. Using the fact that the multivariate Laplace distribution is a scale mixture of the multivariate standard normal distribution, an efficient EM algorithm is designed to implement the proposed robust estimation procedure. The performance of the proposed algorithm is thoroughly evaluated by some simulation and comparison studies. [ABSTRACT FROM AUTHOR]

In this paper, we provide an answer to the following question: In the particular case of a non-differentiable likelihood, is the formula for the BIC model selection criterion the same? More precisely, we obtain the Laplace method for a sum-of-absolute-values function and we deduce that the usual BIC formula with penalty in l o g (n) remains the same in the context of a selection of explanatory variables by least absolute value regression. [ABSTRACT FROM AUTHOR]

We provide a new and simple characterization of the multivariate generalized Laplace distribution. In particular, our characterization implies that the product of a Gaussian matrix with independent and identically distributed columns and an independent isotropic Gaussian vector follows a symmetric multivariate generalized Laplace distribution. [ABSTRACT FROM AUTHOR]

We study the first exit from a general open set for a one-dimensional Lévy process, where the Lévy measure is proportional to a two-sided matrix-exponential distribution. Under appropriate conditions on the Lévy measure, we obtain an explicit solution for the joint distribution of the first-exit time and the position of the Lévy process upon first exit, in terms of the zeros and poles of the corresponding Laplace exponent. The present result complements several earlier works on the use of exit sets for Lévy processes with algebraically similar Laplace exponents, where exits from open intervals are the main focus. [ABSTRACT FROM AUTHOR]

A simple method is proposed for ‘blunting’ cusped distributions, i.e. removing the cusp. No additional parameters are required. The method is applied to the asymmetric Laplace distribution, to the van Dorp and Kotz double-sided power distribution, and to the double-sided asymmetric Pareto distribution, and some properties of the blunted distributions are derived. An example of fitting the blunted asymmetric Pareto distribution to data is given. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic process called generalized fractional Laplace motion (GFLM) is introduced. This process is obtained by superposition of n th-order fractional Brownian motion ( n -FBM) as outer process and gamma process as inner process. It is shown that n -FBM process has long-range-dependence (LRD), persistence of signs LRD and persistence of magnitudes LRD properties. Distributional properties of GFLM such as probability density function, moments, covariance structure are established. The fractional partial differential equation governed by the GFLM density is obtained. Finally, it is shown that GFLM has LRD property for all H ∈ ( n − 1 , n ) , similar to the n -FBM case. [ABSTRACT FROM AUTHOR]

This article provides the domain of existence of the Laplace transform of infinitely divisible negative multinomial distributions, defined by their probability generating functions. It makes it possible to construct all of these distributions. We give examples in dimensions 2 and 3. We give a construction and a simulation of negative multinomial distributions in dimension 2. [ABSTRACT FROM AUTHOR]

A test based on the ratio of the sample mean absolute deviation and the sample standard deviation is proposed for testing the Laplace distribution hypothesis. The asymptotic null distribution for this test statistic is found to be normal. The use of Anderson–Darling test based on a data transformation is also discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

By incorporating the Expectation–maximization (EM) algorithm into composite asymmetric Laplace distribution (CALD), an iterative weighted least square estimator for the linear composite quantile regression (CQR) models is derived. Two selection methods for the number of composite quantiles via redefined AIC and BIC are developed. Finally, the proposed procedures are illustrated by some simulations. [ABSTRACT FROM AUTHOR]

This note introduces the generalized Laplace distribution having two scale parameters and one location parameter (for which the ordinary Laplace distribution is a special case). This distribution is then characterized by a constant regression of a certain polynomial statistic on the sample mean in the sense of such characterizations initiated by Laha and Lukacs (1960). [ABSTRACT FROM AUTHOR]

Autoregressive (AR) models are an important tool in the study of time series data. However, the standard AR model only allows for unimodal marginal and conditional densities, and cannot capture conditional heteroscedasticity. Previously, the Gaussian mixture AR (GMAR) model was considered to remedy these shortcomings by using a Gaussian mixture conditional model. We introduce the Laplace mixture (LMAR) model that utilizes a Laplace mixture conditional model, as an alternative to the GMAR model. We characterize the LMAR model and provide conditions for stationarity. An MM (minorization–maximization) algorithm is then proposed for maximum pseudolikelihood (MPL) estimation of an LMAR model. Conditions for asymptotic inference and a rule for model selection for the MPL estimator are considered. An example analysis of data arising from the calcium imaging of a zebrafish brain is performed. [ABSTRACT FROM AUTHOR]

Under some regularity conditions, the paper provides an alternative proof for the revised moment conditions proposed by Hong and Tamer (2003) in the nonlinear least squares regression model, when the covariates are measured with Laplace error. The asymptotic normality of the revised moment estimates is developed. The choice of optimal weight functions is also discussed and a nearly optimal weight function is identified. Moreover, a simulation extrapolation estimation procedure is suggested when the estimating equations based on the revised moment conditions are difficult to solve. Simulation studies are conducted to evaluate the finite performance of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a set of Tweedie-type formulae are derived for a Kotz-type multivariate Laplace distribution. These formulae can be used as a powerful regression calibration tool to construct more efficient bias-corrected estimation procedures in statistical models, when some variables are contaminated by measurement errors with the Kotz-type multivariate Laplace distribution. [ABSTRACT FROM AUTHOR]

Abstract: Robust estimation procedures for linear and mixture linear errors-in-variables regression models are proposed based on the relationship between the least absolute deviation criterion and maximum likelihood estimation in a Laplace distribution. The finite sample performance of the proposed procedures is evaluated by simulation studies. [Copyright &y& Elsevier]

Abstract: A dimension reduction method for model-based clustering via a finite mixture of shifted asymmetric Laplace distributions is introduced. The approach is based on existing work within the Gaussian paradigm and relies on identification of a reduced subspace. This subspace contains linear combinations of the original data, ordered by importance using the associated eigenvalues. This clustering approach is illustrated on simulated and real data, where it performs favourably compared to its Gaussian analogue. [Copyright &y& Elsevier]

Abstract: A moment method for the three parameter multivariate asymmetric Laplace distribution is considered. It is based on the star product of both the coskewness and cokurtosis tensors. It improves on an earlier moment method by Visk (2009) and shows that the covariance matrix of this multivariate distribution is functionally dependent upon coskewness and cokurtosis. [Copyright &y& Elsevier]

Abstract: In this paper, a new class of skew distributions containing both unimodal as well as bimodal distributions with flexible hazard rate behavior is introduced. Several distributional properties of this class are investigated. A characterization, a generating method and parameter estimation are also presented. Finally, the model is examined using real data sets. [Copyright &y& Elsevier]

Abstract: The Laplace distribution is one of the earliest distributions in probability theory. For the first time, based on this distribution, we propose the so-called beta Laplace distribution, which extends the Laplace distribution. Various structural properties of the new distribution are derived, including expansions for its moments, moment generating function, moments of the order statistics, and so forth. We discuss maximum likelihood estimation of the model parameters and derive the observed information matrix. The usefulness of the new model is illustrated by means of a real data set. [Copyright &y& Elsevier]

Abstract: We introduce the notion of random self-decomposability and discuss its relation to the concepts of self-decomposability and geometric infinite divisibility. We present its connection with time series autoregressive schemes with a regression coefficient that randomly turns on and off. In particular, we provide a characterization of random self-decomposability as well as that of marginal distributions of stationary time series that follow this scheme. Our results settle an open question related to the existence of such processes. [ABSTRACT FROM AUTHOR]

Abstract: The Hermite polynomials can be represented to the moments of normal distribution by the work of Withers [2000. A simple expression for the multivariate Hermite polynomials. Statist. Probab. Lett. 47, 165–169]. This paper generally shows certain combinatorial polynomials and orthogonal polynomials are also the moments of random variables, such as Bernoulli polynomials, Euler polynomials, Gegenbauer polynomials. [Copyright &y& Elsevier]

In this paper, a nonparametric estimator for the regression function is constructed when the covariates are contaminated with the multivariate Laplace measurement error. The proposed estimator is based upon a simple relationship between the regression function and the conditional expectation of the regression function given the proxy data, as well as the second derivative of this expectation. Large sample properties of the proposed estimator, including the consistency and asymptotic normality, are established. The theoretical optimal bandwidth based on asymptotic integrated mean squared error is derived, and a data-driven bandwidth selector is recommended. Finite sample performance of the proposed estimator is evaluated by a simulation study. [ABSTRACT FROM AUTHOR]

We obtain the probability density function (pdf) of a single order statistic (OS) and the joint pdf of two OS based on a random sample Z i i = 1 n from Z = U X − (1 − U) Y , where U , X and Y are independent with U ∼ B e r n o u l l i β . We drive formulas for calculating the moments and the product moments of OS based on Z i i = 1 n. We apply the obtained results to the important class of Double distributions. This class includes the Double Exponential, the Double Logistic and the Double Pareto distributions. [ABSTRACT FROM AUTHOR]