Skew Laplace Distribution Research Articles

An alternative discrete skew Laplace distribution

In this paper, an alternative discrete skew Laplace distribution is proposed, which is derived by using the general approach of discretizing a continuous distribution while retaining its survival function. The distribution’s properties are explored and it is compared to a Laplace distribution on integers recently proposed in the literature. The issues related to the sample estimation of its parameters are discussed, with a particular focus on the maximum likelihood method and large-sample confidence intervals based on Fisher’s information matrix; a modified version of the method of moments is presented along with the method of proportion, which is particularly suitable for such a discrete model. Two hypothesis tests are suggested. A Monte Carlo simulation study is carried out to assess the statistical properties of these inferential techniques. Applications of the proposed model to real data are given as well.

Exponentially Smoothing the Skewed Laplace Distribution for Value‐at‐Risk Forecasting

ABSTRACTValue‐at‐risk (VaR) is a standard measure of market risk in financial markets. This paper proposes a novel, adaptive and efficient method to forecast both volatility and VaR. Extending existing exponential smoothing as well as GARCH formulations, the method is motivated from an asymmetric Laplace distribution, where skewness and heavy tails in return distributions, and their potentially time‐varying nature, are taken into account. The proposed volatility equation also involves novel time‐varying dynamics. Back‐testing results illustrate that the proposed method offers a viable, and more accurate, though conservative, improvement in forecasting VaR compared to a range of popular alternatives. Copyright © 2013 John Wiley & Sons, Ltd.

Alpha–Skew–Laplace distribution

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.

A Fade-Level Skew-Laplace Signal Strength Model for Device-Free Localization with Wireless Networks

Device-free localization (DFL) is the estimation of the position of a person or object that does not carry any electronic device or tag. Existing model-based methods for DFL from RSS measurements are unable to locate stationary people in heavily obstructed environments. This paper introduces measurement-based statistical models that can be used to estimate the locations of both moving and stationary people using received signal strength (RSS) measurements in wireless networks. A key observation is that the statistics of RSS during human motion are strongly dependent on the RSS level” during no motion. We define fade level and demonstrate, using extensive experimental data, that changes in signal strength measurements due to human motion can be modeled by the skew-Laplace distribution, with parameters dependent on the position of person and the fade level. Using the fade-level skew-Laplace model, we apply a particle filter to experimentally estimate the location of moving and stationary people in very different environments without changing the model parameters. We also show the ability to track more than one person with the model.

Forecasting Value-at-Risk using nonlinear regression quantiles and the intra-day range

Value-at-Risk (VaR) is commonly used for financial risk measurement. It has recently become even more important, especially during the 2008-09 global financial crisis. We pro-pose some novel nonlinear threshold conditional autoregressive VaR (CAViaR) models that incorporate intra-day price ranges. Model estimation and inference are performed using the Bayesian approach via the link with the Skewed-Laplace distribution. We examine how a range of risk models perform during the 2008-09 fnancial crisis, and evaluate how the crisis afects the performance of risk models via forecasting VaR. Empirical analysis is conducted on five Asia-Pacific Economic Cooperation stock market indices as well as two exchange rate series. We examine violation rates, back-testing criteria, market risk charges and quantile loss function values to measure and assess the forecasting performance of a variety of risk models. The proposed threshold CAViaR model, incorporating range information, is shown to forecast VaR more eficiently than other models, across the series considered, which should be useful for financial practitioners.

Wrapped Skew Laplace Distribution on Integers:A New Probability Model for Circular Data

In this paper we propose a new family of circular distributions, obtained by wrapping discrete skew Laplace distribution on Z = 0, ±1, ±2, around a unit circle. In contrast with many wrapped distributions, here closed form expressions exist for the probability density function, the distribution function and the characteristic function. The properties of this new family of distribution are studied.

Bayesian Time-Varying Quantile Forecasting for Value-at-Risk in Financial Markets

Recently, advances in time-varying quantile modeling have proven effective in financial Value-at-Risk forecasting. Some well-known dynamic conditional autoregressive quantile models are generalized to a fully nonlinear family. The Bayesian solution to the general quantile regression problem, via the Skewed-Laplace distribution, is adapted and designed for parameter estimation in this model family via an adaptive Markov chain Monte Carlo sampling scheme. A simulation study illustrates favorable precision in estimation, compared to the standard numerical optimization method. The proposed model family is clearly favored in an empirical study of 10 major stock markets. The results that show the proposed model is more accurate at Value-at-Risk forecasting over a two-year period, when compared to a range of existing alternative models and methods.

Inference for grouped data with a truncated skew-Laplace distribution

The skew-Laplace distribution has been used for modelling particle size with point observations. In reality, the observations are truncated and grouped (rounded). This must be formally taken into account for accurate modelling, and it is shown how this leads to convenient closed-form expressions for the likelihood in this model. In a Bayesian framework, “noninformative” benchmark priors, which only require the choice of a single scalar prior hyperparameter, are specified. Conditions for the existence of the posterior distribution are derived when rounding and various forms of truncation are considered. The main application focus is on modelling microbiological data obtained with flow cytometry. However, the model is also applied to data often used to illustrate other skewed distributions, and it is shown that our modelling compares favourably with the popular skew-Student models. Further examples with simulated data illustrate the wide applicability of the model.

A family of skew-symmetric-Laplace distributions

Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we shall introduce another more general class of skew distributions, specially related to the Laplace distribution. This new class contains some previously known skew distributions. We shall investigate different characteristics of members of this class such as its moments, thus generalizing a result of Umbach (Stat Probab Lett 76:507–512, 2006), limiting behavior, moment generating function, unimodality and reveal its natural occurrence as the distribution of some order statistics. In addition, we will generalize a result of Aryal and Rao (Nonlinear Anal 63:639–646, 2005) in connection with truncated skew-Laplace distribution and study its certain stochastic orderings. Some illustrative examples are also provided.

Skew-Laplace and Cell-Size Distribution in Microbial Axenic Cultures: Statistical Assessment and Biological Interpretation

We report a skew-Laplace statistical analysis of both flow cytometry scatters and cell size from microbial strains primarily grown in batch cultures, others in chemostat cultures and bacterial aquatic populations. Cytometry scatters best fit the skew-Laplace distribution while cell size as assessed by an electronic particle analyzer exhibited a moderate fitting. Unlike the cultures, the aquatic bacterial communities clearly do not fit to a skew-Laplace distribution. Due to its versatile nature, the skew-Laplace distribution approach offers an easy, efficient, and powerful tool for distribution of frequency analysis in tandem with the flow cytometric cell sorting.

Application of the Truncated Skew Laplace Probability Distribution in Maintenance System

)( ) ( ) ( G x g x f λ = , where g (.) and G (.), respectively, denote the pdf and the cdf of the Laplace distribution. When the skew Laplace distribution is truncated on the left at 0 it is called it the truncated skew Laplace (TSL) distribution. This article provides a comparison of TSL distribution with twoparameter gamma model and the hypoexponential model, and an application of the subject model in maintenance system is studied.

Bayesian Causal Effects in Quantiles: Accounting for Heteroscedasticity

Testing for Granger non-causality over varying quantile levels could be used to measure and infer dynamic linkages, enabling the identification of quantiles for which causality is relevant, or not. However, dynamic quantiles in financial application settings are clearly affected by heteroscedasticity, as well as the exogenous and endogenous variables under consideration. GARCH-type dynamics are added to the standard quantile regression model, so as to more robustly examine quantile causal relations between dynamic variables. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between quantile regression and the skewed-Laplace distribution, is designed for estimation and inference of the quantile causal relations, simultaneously estimating and accounting for heteroscedasticity. Dynamic quantile linkages for the international stock markets in Taiwan and Hong Kong are considered over a range of quantile levels. Specifically, the hypothesis that these stock returns are Granger-caused by the US market and/or the Japanese market is examined. The US market is found to significantly and positively Granger-cause both markets at all quantile levels, while the Japanese market effect was also significant at most quantile levels, but with weaker effects.

Bayesian causal effects in quantiles: Accounting for heteroscedasticity

Testing for Granger non-causality over varying quantile levels could be used to measure and infer dynamic linkages, enabling the identification of quantiles for which causality is relevant, or not. However, dynamic quantiles in financial application settings are clearly affected by heteroscedasticity, as well as the exogenous and endogenous variables under consideration. GARCH-type dynamics are added to the standard quantile regression model, so as to more robustly examine quantile causal relations between dynamic variables. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between quantile regression and the skewed-Laplace distribution, is designed for estimation and inference of the quantile causal relations, simultaneously estimating and accounting for heteroscedasticity. Dynamic quantile linkages for the international stock markets in Taiwan and Hong Kong are considered over a range of quantile levels. Specifically, the hypothesis that these stock returns are Granger-caused by the US market and/or the Japanese market is examined. The US market is found to significantly and positively Granger-cause both markets at all quantile levels, while the Japanese market effect was also significant at most quantile levels, but with weaker effects.

Use of grain size distribution parameters to explain variation in small grain screenings of wheat in multi-environment trials involving new cultivars

The proportion of small grain screenings is a worldwide problem for wheat growers in water-limited environments and is known to be influenced by both genetic and environmental factors including crop management. This paper follows our previous finding that parameters μ (reflecting point of majority mass of the grain size distribution) and 1/α (reflecting the slope of grain size distribution up to the majority point) of a skew-Laplace distribution can be used to differentiate the tendency of cultivars to produce small grains. The variability of these parameters over environments (15 Cultivar × Time of sowing and 2 Cultivar × Plant population × Nitrogen rates experiments) was investigated in order to indicate their utility in breeding cultivars and managing wheat crops. The estimated values of both μ and 1/α varied over experiments but the relative cultivar ranking remained similar. Of the genetic and management factors, cultivar had more effect on variance of μ than of 1/α, while applied nitrogen level significantly influenced 1/α. We conclude that: (i) cultivars can be grouped according to their μ and 1/α values; (ii) cultivars owing their lower screenings to high 1/α need a careful optimising of nitrogen inputs, while breeding for high μ is likely to provide stability against small grain screenings; (iii) both the parameters were associated with kernel weight and grains/spike but not grains/m2; and (iv) future research establishing the relationship of 1/α with specific kernel position within the plant will be useful in proposing an ideal ear and plant structure for minimising small grain screenings. When using parameters of the suggested grain size distribution, we suggest the following relevant points be considered: (i) 1/α is not a linear slope, it is related to ‘skewness’; (ii) the peak at majority mass (MM) reflects the tendency to concentrate grain mass and therefore provides an opportunity to increase skewness (1/α) despite a lower μ and a lower mean grain weight; (iii) there is a point up to which skewness of grain size distribution can sufficiently protect against high screenings and is highly dependent upon the value of μ; and (iv) 1/α is a potentially useful parameter nonetheless, because in practice, grain weight may be reduced as number of grains/area is increased in pursuit of high yield through breeding and agronomic practices.

An alternative multivariate skew Laplace distribution: properties and estimation

A special case of the multivariate exponential power distribution is considered as a multivariate extension of the univariate symmetric Laplace distribution. In this paper, we focus on this multivariate symmetric Laplace distribution, and extend it to a multivariate skew distribution. We call this skew extension of the multivariate symmetric Laplace distribution the “multivariate skew Laplace (MSL) distribution” to distinguish between the asymmetric multivariate Laplace distribution proposed by Kozubowski and Podgorski (Comput Stat 15:531–540, 2000a) Kotz et al. (The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Chap. 6. Birkhauser, Boston, 2001) and Kotz et al. (An asymmetric multivariate Laplace Distribution, Working paper, 2003). One of the advantages of (MSL) distribution is that it can handle both heavy tails and skewness and that it has a simple form compared to other multivariate skew distributions. Some fundamental properties of the multivariate skew Laplace distribution are discussed. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of the multivariate skew Laplace distribution is given. Some examples are provided to demonstrate the modeling strength of the skew Laplace distribution.

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.

Small grain screenings in wheat—using the grain size distribution for predicting cultivar responses

Small grain is a concern for wheat growers in water-limited environments worldwide. Following from our preliminary results that cultivars can differ for small grain screenings despite similarity of average grain weight in wheat, grain size distribution was investigated with the aim of identifying alternative parameters that could more reliably discriminate cultivars for their tendency to screenings without actually subjecting breeding material to harsh environments. Pre-cleaned harvest samples (using screen with 1.5-mm-wide holes) of 5 cultivars from 2 field cultivar × time of sowing experiments were divided into 5 grain-width fractions using 4 sieves with holes from 2.5 to 3.4-mm wide. The experiments conducted in the Western Australian wheatbelt experienced a dry finish in 2000 and a relatively wetter year in 2003. The grain that was separated into each width fraction was weighed and the average weight of grains for each fraction (called the size-specific weight) was derived from a subsample. The weight of grain (expressed as a percentage of the whole) collected between the 2.8 and 3.1-mm screens (fraction F3) was highly and negatively correlated with small grain screenings (r = –0.85; P < 0.001), thus strengthening the perception that screenings percentages may follow a certain form of grain size distribution. Among various grain size distributions, the skew-Laplace distribution fitted on grain weight was found to be the most suitable to characterise the cultivars. This provides a general approach in the examination of a grain size distribution, which is invariant to combinations of sieve sizes that different individuals may choose. We conclude that: (i) parameters µ and 1/α of the skew-Laplace distribution based on grain weight separated into each fraction can be used as a selection tool in predicting propensity for small grain screenings in wheat without having to subject breeding material to harsh conditions; (ii) higher values for both µ and 1/α are desirable but if the cultivar has small µ, a high 1/α is required to reduce screenings; (iii) breeders could consider the grain size distribution in addition to grain weight as a breeding objective; and (iv) millers should be looking for higher average grain weight and higher weight per unit grain length especially at small width fractions, to maximise flour yield. Hence, we anticipate that future research into aspects of starch packaging such as grain shape and endosperm density will optimise grain size distribution, grain weight per unit grain length, and eventually the flour yield per unit land area.

Reliability model using truncated skew-Laplace distribution

A random variable X follows the skew-Laplace distribution if its pdf is f ( x ) = 2 g ( x ) G ( λ x ) , where g ( · ) and G ( · ) denote pdf and cdf of the standard Laplace distribution. We study different characteristics of truncated from the left at zero skew-Laplace distribution. The reliability of this model is compared with a two-parameter Gamma model.

Skew-Laplace distribution in Gram-negative bacterial axenic cultures: new insights into intrinsic cellular heterogeneity

The application of flow cytometry and skew-Laplace statistical analysis to assess cellular heterogeneity in Gram-negative axenic cultures is reported. In particular, fit to the log-skew-Laplace distribution for cellular side scatter or 'granulosity' is reported, and a number of theoretical and applied issues are considered in relation to the biological significance of this fit.

On the skew Laplace distribution

A random variable X is said to have the skew-Laplace distribution if its pdf is f(x) = 2g(x)G(λx), where g(·) and G(·), respectively, denote the pdf and the cdf of the Laplace distribution. This distribution – in spite of its simplicity – appears not to have been studied in detail. The only work that appears to give some details of this distribution is Gupta et. al [Random Operators and Stochastic Equations, Vol. 10 (2002), pp. 133–140], where expressions for the expectation, variance, skewness and the kurtosis of X are given. But these expressions appear to contain some errors. In this paper, we provide a comprehensive description of the mathematical properties of X. The properties derived include the k th moment, variance, skewness, kurtosis, moment generating function, characteristic function, cumulant generating function, the k th cumulant, hazard rate function, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon's entropy, cumulative residual entropy and the asymptotic distribution of the extreme order statistics. We also consider estimation and simulation issues.