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Increasing the parameter of a distribution helps to capture the skewness and peakedness characteristic in the data sets. This allows a more realistic modeling of data arising from different real life situations. In this paper, we modified Laplace distribution using the exponentiation method. The study proved that the modified Laplace distribution (MLD) is a probability density function. Some of the basic statistical properties of the modified Laplace distribution are obtained. We applied the proposed modified Laplace distribution on two life datasets and simulated data. Parameters of the distributions were estimated using method of maximum likelihood estimation. The study compared the modified Laplace distribution with Laplace distribution and Generalized error distribution using Schwartz Criteria (SC) measure of fitness. The results obtained revealed that the modified Laplace distribution has a better fit than the Laplace and Generalized error distributions and can be used for more realistic modeling of data arising from different real life situations. The simulation results obtained shows that as the sample size increases, the Biasedness and Root Mean Square Error (RMSE) of the proposed modified Laplace distribution reduces.

The geometric sums have been arisen from the necessity to resolve practical problems in ruin prob- ability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of characteristic functions, the necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution are shown. In special case, obtained result is a weak limit theorem for geometric sums of independent and identically distributed random variables which has been well-known as the second central limit theorem. Furthermore, based on the obtained results of this paper, the analogous results for the domains of geometric attraction of exponential distribution and Linnik distribution can be established. More generally, we may extend results to the domain of geometric attraction of geometrically strictly stable distributions.

This paper proposes a statistical approach to model the movement of a ball in soccer. The proposed work uses ball movement data from ISSIA dataset to put forward the soccer ball movement model. Here, ball displacements in x and y directions are identified to follow Laplace distribution and their probability density function leads to joint probability density function of x and y from which a novel probability density function is proposed for the resultant displacement of ball for normalized data. For solving non-converging integral, numerical method of Simpson's 3/8th Rule is used to obtain converging finite solution of probability density function of re-sultant displacement. After un-normalizing, the most probable displacement is obtained. Angular displacement required for ball position prediction is the difference between angular displacements of previous two consecutive frames. Thus the resultant radial and angular displacement is combined for estimating ball position in similar test video sequences.

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.

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.

Lower and upper estimates are obtained for deviations of the limit of a selectedmean from estimated mathematical expectations when rounded data are processed. Different cases of error distribution are considered: normal, Simpson (triangle), and Laplace (double exponential)distributions.

Tensor analysis has reached a celebrity status in the areas of machine learning, computer vision, and artificial intelligence. Completing and recovering tensor is an important problem for tensor analysis. It involves recovering a tensor from either a subset of its entries or the whole entries contaminated by noise. Classical tensor completion/recovery methods are based mainly on l2-norm (Gaussian distribution noise) models, and are sensitive to noise of large magnitude. While l1-norm (Laplacian distribution noise) based methods are robust to noise of large magnitude, they do not deal with dense noise effectively. In this paper, we present a novel Cauchy tensor decomposition method for simultaneously recovering and completing a low rank tensor with both missing data and complex noise. We utilize the Cauchy distribution to model noise and derive the objective function of Cauchy tensor decomposition under the maximum likelihood estimation (MLE) framework. Then we developed a robust tensor decomposition framework that used first-order optimization approaches to optimize the objective function. Extensive numerical experiments are conducted and show that our method is able to successfully recover and complete tensors with large/dense noise and missing data. We further demonstrate the usefulness of Cauchy tensor decomposition on three real-world applications, image inpainting, traffic data process and foreground/background separation. The experimental results show that the method is applicable to a wide range of problems.

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.

For a < r < b, the approach of Li and Zhou (2014) is adopted to find joint Laplace transforms of occupation times over intervals (a, r) and (r, b) for a time homogeneous diffusion process before it first exits from either a or b. The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions on the joint Laplace transforms of occupation times for Brownian motion with drift, Brownian motion with alternating drift and skew Brownian motion, respectively.

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

In this paper,we introduce a new circular distribution called Wrapped Holla and Bhattacharya’s (HB) Skewed Laplace distribution.We obtain explicit form for the density and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for estimation of parameters. The methods are applied to a data set which consists of the wind directions of a Black Mountain ACT and compared with that of wrapped variance gamma distribution (WvG) and generalised von Mises (GvM) 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.

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.

We characterize some natural exponential families in R 2 via diagonal elements of quadratic variance function. Our proofs rely on Laplace transforms, and along the way, we obtain some new characterizations for Laplace transforms of probability measures in certain classes of functions.

We present an asymptotically optimal importance sampling for Monte Carlo simulation of the Laplace transform of exponential Brownian functionals which plays a prominent role in many disciplines. To this end we utilize the theory of large deviations to reduce finding an asymptotically optimal importance sampling measure to solving a calculus of variations problem. Closed-form solutions are obtained. In addition we also present a path to the test of regularity of optimal drift which is an issue in implementing the proposed method. The performance analysis of the method is provided through the Dothan bond pricing model.

The simulation of a free-space optical (FSO) communication channel in the presence of strong turbulence typically requires the generation of channel states with a K or gamma–gamma distribution and a predefined autocorrelation function. In this paper, we propose a simple and effective simulator of the strong-turbulence FSO channel that addresses the influence of the temporal covariance effect. Specifically, the proposed simulator provides K and gamma–gamma distributed values with the exponential autocorrelation function and a prescribed correlation time. This simulator is based on the numerical solution of the first-order stochastic differential equation. The simulated channel states are generated by a simple discrete-time differential equation and the simulator performance is analyzed in the paper.

This paper introduces a new Laplace transform inversion method designed specifically for when the target function is a probability distribution function. In particular, we use fixed point theory and Mann type iterative algorithms to provide a means by which to estimate and sample from the target probability distribution.

Exact solutions of the bicomponent Smoluchowski’s equation with a composition-dependent additive kernelK(va,vb;va′,vb′)=α(va+va′)+(vb+vb′)are derived by using the Laplace transform for any initial particle size distribution. The exact solution for an exponential initial distribution is then used to analyse the effects of parameterαon mixing degree of such bicomponent mixtures and the conditional distribution of the first component for particles with given mass. The main finding is that the conditional distribution of large particles at larger time is a Gaussian function which is independent of the parameterα.

The weighted exponential distribution has been generalized by many authors in recent years. Here, we introduce a new generalization, called the weighted half exponential power distribution, that can be used to model negative or positive skewed data. Various structural properties of the distribution are derived. The method of maximum likelihood is used to estimate the parameters of the distribution. Finally, a real data application is presented to compare the goodness-of-t of the various generalized weighted exponential distributions. Mathematics Subject Classication: 60E05, 62F10

In this communication, a convenient Laplace transform of the bivariate supremum and the last time the supremum is attained, is established when the underlying Levy process is subordinate Brownian motion with drift. Explicit integral representations of the Laplace transform of the joint supremum and the last time it occurred are derived in terms of the Levy–Khintchine exponent of the subordinator Laplace exponent. As an example, a subordinator with exponential Levy measure is exploited.

The representations of record ranges via sums of independent identically distributed exponential random variables are obtained for asymmetrical Laplace distributions. This result generalizes the corresponding relations for record values in the cases of exponential and negative exponential distributions

Under certain conditions, a symmetric unimodal continuous random variable ξ can be represented as a scale mixture of a standard Normal distribution Z, that is, ξ=WZ, where the mixing distribution W is independent of Z. It is well known that if the mixing distribution is inverse Gamma, then ξ has Student’s t distribution. However, it is less well known that if the mixing distribution is Gamma, then ξ has a Laplace distribution. Several existing proofs of the latter result rely on complex calculus or nontrivial change of variables in integrals. We offer two simple and intuitive proofs based on representation and moment generating functions. As a byproduct, our proof by representation makes connections to many existing results in statistics. Supplementary materials for this article are available online.