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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...

This paper investigates a robust identification solution for the nonlinear state-space model in which the outputs are polluted by unknown outliers. The problem of outliers is frequently encountered...

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.

Delta Modulation (DM) is a simple waveform coding algorithm used mostly when timely data delivery is more important than the transmitted data quality. While the implementation of DM is fairly simple and inexpensive, it suffers from several limitations, such as slope overload and granular noise, which can be overcome using Adaptive Delta Modulation (ADM). This paper presents novel 2-digit ADM with six-level quantization using variable-length coding, for encoding the time-varying signals modelled by Laplacian distribution. Two variants of quantizer are employed, distortion-constrained quantizer which is optimally designed for minimal mean-squared error (MSE), and rate-constrained quantizer, which is suboptimal in the minimal MSE sense, but enables minimal loss in SQNR for the target bit rate. Experimental results using real speech signal are provided, indicating that the proposed configuration outperforms the baseline ADM algorithms, including Constant Factor Delta Modulation (CFDM), Continuously Va...

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...

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.

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.

SYNOPTIC ABSTRACTKozubowski and Podgorski (2003) have discussed properties, characterizations, and estimation of parameters of skew log Laplace distribution (SLLD). In this article, classical optimum tests for scale parameter of SLLD are derived. The most powerful (MP) test is obtained for scale parameter when shape parameters are known. Wald’s sequential probability ratio test (SPRT) is obtained, and its properties are studied. The likelihood ratio tests (LRT) for scale parameter are derived when the shape parameters are known and unknown. Finally, the SPRT and LRT are illustrated to the real life data.

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.

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.

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.

In this paper, we introduce a bivariate weighted exponential distribution based on the generalized exponential distribution. This class of distributions generalizes the bivariate distribution with weighted exponential marginals (BWE). We derive different properties of this new distribution. It is a four-parameter distribution, and the maximum-likelihood estimator of unknown parameters cannot be obtained in explicit forms. One data set has been re-analyzed and it is observed that the proposed distribution provides better fit than the BWE distribution.

SYNOPTIC ABSTRACTThe Two-Piece Double Exponential Distribution is known to be very useful in modeling currency rates, interest rates, share price indices, and so on. Two characterization properties of this distribution are derived in this article. These properties are also applicable to the symmetric Laplace distribution and they reduce to well-known characterizations of the exponential distribution as special cases.

In this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient ...

Three new weighted rank correlation coefficients are proposed which are sensitive to both agreement on top and bottom rankings. The first one is based on the weighted rank correlation coefficient proposed by Maturi and Abdelfattah [13], the second and the third are based on the order statistics and the quantiles of the Laplace distribution, respectively. The limiting distributions of the new correlation coefficients under the null hypothesis of no association between the rankings are presented, and a summary of the exact and approximate quantiles for these coefficients is provided. A simulation study is performed to compare the performance of Kendall's tau, Spearman's rho, and the new weighted rank correlation coefficients in detecting the agreement on the top and the bottom rankings simultaneously. Finally, examples are given for illustration purposes, including a real data set from financial market indices.

In this paper, Vasicek [A test for normality based on sample entropy. J R Stat Soc Ser B. 1976;38:54–59] entropy estimator is modified using paired ranked set sampling (PRSS) method. Also, two goodness-of-fit tests using PRSS are suggested for the inverse Gaussian and Laplace distributions. The new suggested entropy estimator and goodness-of-fit tests using PRSS are compared with their counterparts using simple random sampling (SRS) via Monte Carlo simulations. The critical values of the suggested tests are obtained, and the powers of the tests based on several alternatives hypotheses using SRS and PRSS are calculated. It turns out that the proposed PRSS entropy estimator is more efficient than the SRS counterpart in terms of root mean square error. Also, the proposed PRSS goodness-of-fit tests have higher powers than their counterparts using SRS for all alternative considered in this study.

We present an algorithm for multivariate robust Bayesian linear regression with missing data. The iterative algorithm computes an approximative posterior for the model parameters based on the variational Bayes (VB) method. Compared to the EM algorithm, the VB method has the advantage that the variance for the model parameters is also computed directly by the algorithm. We consider three families of Gaussian scale mixture models for the measurements, which include as special cases the multivariate t distribution, the multivariate Laplace distribution, and the contaminated normal model. The observations can contain missing values, assuming that the missing data mechanism can be ignored. A Matlab/Octave implementation of the algorithm is presented and applied to solve three reference examples from the literature. acceptedVersion

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing t...

In this article, we proposed a new three parameter lifetime distribution motivated mainly by lifetime issues, which generalizes the Exponential Poisson distribution proposed by Cancho et al. (2011). We derive various standard mathematical properties of the proposed model including a formal proof of its probability density function and hazard rate function. The inference via the maximum likelihood approach is discussed. The performance of the maximum likelihood estimators, the likelihood ratio test and its power are studied by simulation. Finally, the proposed model is fitted to two real data sets and it is compared with several models.

The problem of estimation of R = P(Y < X) have been used in the paper. Let X has exponential distribution mixing with exponential distribution with parameters β and θ and Y independently of X has exponential distribution with parameter λ. By using a prior guess or estimate R0, different shrinkage estimators of R are derived. Then the performance of the estimators are discussed. Finally, we compare these results with Baklizei and Dayyeh (2003) approaches.

A new lifetime distribution is proposed and studied. The Harris extended exponential is obtained from a mixture of the exponential and Harris distributions, which arises from a branching process. Several structural properties of the new distribution are discussed, including moments, generating function and order statistics. The new distribution can model data with increasing or decreasing failure rate. The shape of the hazard rate function is controlled by one of the added parameters in an uncomplicated manner. An application to a real dataset illustrates the usefulness of the new distribution.

In this paper, we first present explicit expressions for the maximum likelihood estimators (MLEs) of the location and scale parameters of the Laplace distribution based on a Type-II right censored sample under different cases. Next, we derive the joint moment generating function of the MLEs of the two parameters and use it to obtain the bias and mean squared error of the MLEs for all the cases. We then derive the exact density functions of the MLEs and utilize them to develop exact confidence intervals for the parameters. Next, a Monte Carlo simulation study is carried out to evaluate the performance of the developed inferential results. Finally, some examples are presented to illustrate the point and interval estimation methods.

The new class of weighted exponential (WE) distributions obtained by Gupta and Kundu (2009) by implementing Azzalini's method to the exponential distribution. In this study, we generalize the WE distribution to a new class of generalized weighted exponential (GWE) distribution. Several statistical and reliability properties of this new class of distribution are obtained. Estimation and inference procedure for distribution parameters are investigated. Finally, we show that the proposed model can provide better fit than the recent class of weighted exponential by using two real data examples.

WOS: 000383971600018 Skew-symmetric distributions have been discussed by several research-ers. In this article we construct a skew-symmetric Laplace distribution, which is the generalization of distribution given by Ali etal. (2009) and Nekoukhou and Alamatsaz (2012). This new distribution contains more parameters, and this induces flexibility properties, such as unimodality or bimodality. We study on some properties of this distribution. In the last section we also provide an application with a real data. Concerning example has recently been discussed by Nekoukhou etal. (2013) to apply to their model. We compare the behavior of our distribution to their distribution on this example.

Dear Editor,In 2004 the Communications in Statistics—Theory and Methods, vol. 33, pp. 1733–1753, published an article (Reed and Jorgensen, 2004) authored by William J. Reed, which studied the Norma...

In this article, a transmuted linear exponential distribution is developed that generalizes the linear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al. Some statistical properties of the proposed distribution such as moments, quantiles, and the failure rate function are investigated. The maximum likelihood estimators of unknown parameters are also discussed and a real data analysis is carried out to illustrate the superiority of the proposed distribution.

Bivariate and multivariate exponential distributions are widely applied in several areas such as reliability, queueing systems or hydrology. A frequently used bivariate exponential distribution is the Moran–Downton distribution. Because of this reason, this paper proposes a goodness-of-fit test for this distribution. The test statistic exploits the analytically convenient formula of its characteristic function. Large sample properties of the proposed test such as consistency against fixed and local alternatives are studied. The finite sample performance is numerically studied. Finally, an application of this distribution to hydrological data is presented.

The generalized normal Laplace distribution has been used in financial modeling because of its skewness and excess kurtosis. To estimate its parameters, we use a method based on minimizing the quadratic distance between the real and imaginary parts of the empirical and theoretical characteristic functions. The quadratic distance estimator (QDE) obtained is shown to be robust, consistent, and with an asymptotically normal distribution. The goodness-of-fit test statistics presented follow an asymptotic chi-square distribution. The performance of the QDE is illustrated by simulation results and an application to financial data.

Recently, Gupta and Kundu [R.D. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (2009), pp. 621–634] have introduced a new class of weighted exponential (WE) distributions, and this can be used quite effectively to model lifetime data. In this paper, we introduce a new class of weighted Marshall–Olkin bivariate exponential distributions. This new singular distribution has univariate WE marginals. We study different properties of the proposed model. There are four parameters in this model and the maximum-likelihood estimators (MLEs) of the unknown parameters cannot be obtained in explicit forms. We need to solve a four-dimensional optimization problem to compute the MLEs. One data set has been analysed for illustrative purposes and finally we propose some generalization of the proposed model.

We study the discrete-time model of Lopez-Ruiz, Lopez and Calbet, describing the evolution of a wealth distribution under random pairwise exchanges of wealth among agents. This requires the analysis of the behaviour of iterations of a non-linear operator defined on a space of probability distributions. We prove that, as conjectured by Lopez-Ruiz, Lopez and Calbet, starting from a general wealth distribution, the wealth distribution converges to the exponential equilibrium distribution. The proof employs a special metric defined on spaces of probability distributions through their Laplace transforms.

Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures are derived. Three real-data applications are described to show superior performance versus at least five of the known lifetime models.

In this work we construct exact prediction intervals for order statistics from the Laplace (double exponential) distribution. We consider both the one- and two-sample prediction cases. The intervals are based on certain pivotal quantities that employ the corresponding maximum-likelihood predictors and the predictive maximum-likelihood estimators of the unknown parameters. Similar to Iliopoulos and Balakrishnan [Exact likelihood inference for Laplace distribution based on Type-II censored samples. J. Statist. Plann. Inference. 2011;141:1224–1239], we express the distributions of the pivotal quantities as mixtures of ratios of linear combinations of independent standard exponential random variables. Since these distributions are in closed form we solve numerically the corresponding equations and obtain their exact quantiles. Tables containing selected quantiles of the pivotal quantities are provided. Numerical examples are also given for illustration purposes.

In modelling financial return time series and time-varying volatility, the Gaussian and the Student-t distributions are widely used in stochastic volatility (SV) models. However, other distributions such as the Laplace distribution and generalized error distribution (GED) are also common in SV modelling. Therefore, this paper proposes the use of the generalized t (GT) distribution whose special cases are the Gaussian distribution, Student-t distribution, Laplace distribution and GED. Since the GT distribution is a member of the scale mixture of uniform (SMU) family of distribution, we handle the GT distribution via its SMU representation. We show this SMU form can substantially simplify the Gibbs sampler for Bayesian simulation-based computation and can provide a mean of identifying outliers. In an empirical study, we adopt a GT–SV model to fit the daily return of the exchange rate of Australian dollar to three other currencies and use the exchange rate to US dollar as a covariate. Model implementation re...

A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based representation for samples of k-records which apparently has not been known so far. Moreover, sufficient conditions for the validity of this representation in location and scale family settings are given. The result is illustrated by considering generalized order statistics based on logistic, Laplace, and extreme value distributions.

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.

This article considers the two-piece normal-Laplace (TPNL) distribution, a split skew distribution consisting of a normal part, and a Laplace part. The distribution is indexed by three parameters, representing location, scale, and shape. As illustrated with several examples, the TPNL family of distributions provides a useful alternative to other families of asymmetric distributions on the real line. However, because the likelihood function is not well behaved, standard theory of maximum-likelihood (ML) estimation does not apply to the TPNL family. In particular, the likelihood function can have multiple local maxima. We provide a procedure for computing ML estimators, and prove consistency and asymptotic normality of ML estimators, using non standard methods.

We propose a class of goodness-of-fit tests for the gamma distribution that utilizes the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity, the test statistics approach limit values related to the first non zero component of Neyman's smooth test for the gamma law. The new tests are compared with other omnibus tests for the gamma distribution.

We show that the two bivariate exponential distributions constructed in Bladt and Nielsen[ 5 ] have the maximum and minimum correlation coefficients for any given order. We also generalize their constructions to the case where the matrix representations of the two (marginal) exponential distributions have different orders and show that the new constructions also have the maximum and minimum correlation coefficients. Our main tool is a majorization result for a special set of PH-generators.

In the article, we consider the AR(1) time series model when X t has two parameter inverse–gamma distribution IG2(a, b), a ∈ (0, 1/2], b > 0. It will be shown that Laplace transform of two parameter inverse gamma distribution is Kratzel function. Using suitable approximation procedure Laplace transform of IG2(a, b) will be approximated when transformation argument is large. It will be shown that innovation process has continuous distribution. Finally, model's parameters have been estimated.

A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.

This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.

Laplace motion is a Levy process built upon Laplace distributions. Non Gaussian stochastic fields that are integrals with respect to this process are considered and methods for their model fitting are discussed. The proposed procedures allow for inference about the parameters of the underlying Laplace distributions. A fit of dependence structure is also addressed. The importance of a convenient parameterization that admits natural and consistent estimation for this class of models is emphasized. Several parameterizations are introduced and their advantages over one another discussed. The proposed estimation method targets the standard characteristics: mean, variance, skewness and kurtosis. Their sample equivalents are matched in the closest possible way as allowed by natural constraints within this class. A simulation study and an example of potential applications conclude the article.

This article proposes a sparse conditional random field (SCRF) model to exploit contextual information for classification problems, and select relevant features to prevent overfitting derived from excessively large numbers of features. The sparsity arises from the use of Laplacian priors on the parameters of CRFs, which encourages the parameter estimates to be either significantly large or exactly zero. Since the Laplacian distribution is nondifferentiable at the origin, we developed a simple but efficient sub-gradient-based training algorithm to compute a maximum a posteriori (MAP) point estimate of the CRF parameters. We used SCRF for the classification between urban and non-urban areas in an optical remote sensing image database. The results attest to the accuracy, sparsity and effectiveness of the proposed model.