scholarly journals The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey’shTransformation as a Special Case

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Georg M. Goerg
Keyword(s):  
Heavy Tails ◽  
Random Variable ◽  
R Package ◽  
Heavy Tail ◽  
Cumulative Distribution ◽  
Lambert W Function ◽  
Inverse Transformation ◽  
Heavy Tailed ◽  
Probability Density Function Pdf ◽  
Special Case

I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables. The tail behavior of thisheavy tail Lambert  W × FXrandom variable depends on a tail parameterδ≥0: forδ=0,Y≡X, forδ>0 Yhas heavier tails thanX. ForXbeing Gaussian it reduces to Tukey’shdistribution. The Lambert W function provides an explicit inverse transformation, which can thus remove heavy tails from observed data. It also provides closed-form expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield analytic expression for Tukey’shpdf and cdf. Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology. The R packageLambertWimplements most of the introduced methodology and is publicly available onCRAN.

scholarly journals Inhomogeneous phase-type distributions and heavy tails

2019 ◽  
Vol 56 (4) ◽  
pp. 1044-1064 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Mogens Bladt
Keyword(s):  
Real World ◽  
Heavy Tails ◽  
Heavy Tail ◽  
Transition Rates ◽  
Modelling Framework ◽  
Inhomogeneous Phase ◽  
Fire Insurance ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Heavy Tailed

AbstractWe extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.

scholarly journals Improved algorithms for rare event simulation with heavy tails

2006 ◽  
Vol 38 (2) ◽  
pp. 545-558 ◽  
Author(s):  
Søren Asmussen ◽  
Dirk P. Kroese
Keyword(s):  
Monte Carlo ◽  
Relative Error ◽  
Heavy Tails ◽  
Numerical Study ◽  
Rare Event ◽  
Random Variable ◽  
Event Simulation ◽  
Conditional Monte Carlo ◽  
Heavy Tailed ◽  
Geometric Random Variable

The estimation of P(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

scholarly journals Improved algorithms for rare event simulation with heavy tails

2006 ◽  
Vol 38 (02) ◽  
pp. 545-558 ◽  
Author(s):  
Søren Asmussen ◽  
Dirk P. Kroese
Keyword(s):  
Monte Carlo ◽  
Relative Error ◽  
Heavy Tails ◽  
Numerical Study ◽  
Rare Event ◽  
Random Variable ◽  
Event Simulation ◽  
Conditional Monte Carlo ◽  
Heavy Tailed ◽  
Geometric Random Variable

The estimation of P(S n >u) by simulation, where S n is the sum of independent, identically distributed random varibles Y 1 ,…,Y n , is of importance in many applications. We propose two simulation estimators based upon the identity P(S n >u)=nP(S n >u, M n =Y n ), where M n =max(Y 1 ,…,Y n ). One estimator uses importance sampling (for Y n only), and the other uses conditional Monte Carlo conditioning upon Y 1 ,…,Y n−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

scholarly journals Statistical Analysis of Ratio of Random Variables and Its Application in Performance Analysis of Multihop Wireless Transmissions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Edis Mekić ◽  
Mihajlo Stefanović ◽  
Petar Spalević ◽  
Nikola Sekulović ◽  
Ana Stanković
Keyword(s):  
Performance Analysis ◽  
Communication Systems ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Wireless Communication Systems ◽  
Multihop Wireless ◽  
Ratio Of Random Variables ◽  
Probability Density Function Pdf

The distributions of random variables are of interest in many areas of science. In this paper, the probability density function (PDF) and cumulative distribution function (CDF) of ratio of products of two random variables and random variable are derived. Random variables are described with Rayleigh, Nakagami-m, Weibull, andα-μdistributions. An application of obtained results in performance analysis of multihop wireless communication systems in different transmission environments described in detail. The proposed mathematical analysis is also complemented by various graphically presented numerical results.

scholarly journals A Technique for Computing the PDFs and CDFs of Nonnegative Infinitely Divisible Random Variables

2011 ◽  
Vol 48 (01) ◽  
pp. 217-237 ◽  
Author(s):  
Mark S. Veillette ◽  
Murad S. Taqqu
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Cumulative Distribution Function ◽  
Stable Distribution ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Probability Density Function Pdf

We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-λX = e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.

scholarly journals A Technique for Computing the PDFs and CDFs of Nonnegative Infinitely Divisible Random Variables

2011 ◽  
Vol 48 (1) ◽  
pp. 217-237 ◽  
Author(s):  
Mark S. Veillette ◽  
Murad S. Taqqu
Keyword(s):  
Distribution Function ◽  
Laplace Transform ◽  
Cumulative Distribution Function ◽  
Stable Distribution ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Probability Density Function Pdf

We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-λX = e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.

scholarly journals Gaussian processes with skewed Laplace spectral mixture kernels for long-term forecasting

2021 ◽  
Author(s):  
Kai Chen ◽  
Twan van Laarhoven ◽  
Elena Marchiori
Keyword(s):  
Gaussian Processes ◽  
Multivariate Time Series ◽  
Heavy Tail ◽  
Lottery Ticket ◽  
Spectral Densities ◽  
Heavy Tailed ◽  
Long Term Forecasting ◽  
The Time Domain ◽  
Spectral Mixture

AbstractLong-term forecasting involves predicting a horizon that is far ahead of the last observation. It is a problem of high practical relevance, for instance for companies in order to decide upon expensive long-term investments. Despite the recent progress and success of Gaussian processes (GPs) based on spectral mixture kernels, long-term forecasting remains a challenging problem for these kernels because they decay exponentially at large horizons. This is mainly due to their use of a mixture of Gaussians to model spectral densities. Characteristics of the signal important for long-term forecasting can be unravelled by investigating the distribution of the Fourier coefficients of (the training part of) the signal, which is non-smooth, heavy-tailed, sparse, and skewed. The heavy tail and skewness characteristics of such distributions in the spectral domain allow to capture long-range covariance of the signal in the time domain. Motivated by these observations, we propose to model spectral densities using a skewed Laplace spectral mixture (SLSM) due to the skewness of its peaks, sparsity, non-smoothness, and heavy tail characteristics. By applying the inverse Fourier Transform to this spectral density we obtain a new GP kernel for long-term forecasting. In addition, we adapt the lottery ticket method, originally developed to prune weights of a neural network, to GPs in order to automatically select the number of kernel components. Results of extensive experiments, including a multivariate time series, show the beneficial effect of the proposed SLSM kernel for long-term extrapolation and robustness to the choice of the number of mixture components.

scholarly journals A general stochastic model for bivariate episodes driven by a gamma sequence

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Charles K. Amponsah ◽  
Tomasz J. Kozubowski ◽  
Anna K. Panorska
Keyword(s):  
Stochastic Model ◽  
Joint Distribution ◽  
Pareto Distribution ◽  
Integral Transforms ◽  
Bivariate Distribution ◽  
Conditional Distributions ◽  
General Stochastic Model ◽  
Heavy Tailed ◽  
Discrete Pareto Distribution ◽  
Special Case

AbstractWe propose a new stochastic model describing the joint distribution of (X,N), where N is a counting variable while X is the sum of N independent gamma random variables. We present the main properties of this general model, which include marginal and conditional distributions, integral transforms, moments and parameter estimation. We also discuss in more detail a special case where N has a heavy tailed discrete Pareto distribution. An example from finance illustrates the modeling potential of this new mixed bivariate distribution.

Growth Optimal Investment Strategy: The Impact of Reallocation Frequency and Heavy Tails

2012 ◽  
Vol 13 (2) ◽  
pp. 228-240 ◽  
Author(s):  
G. Bamberg ◽  
A. Neuhierl
Keyword(s):  
Heavy Tails ◽  
Geometric Mean ◽  
Asymptotic Optimality ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Risky Asset ◽  
Optimal Investment Strategy ◽  
Optimality Properties ◽  
Heavy Tailed ◽  
The Impact

Abstract The strategy to maximize the long-term growth rate of final wealth (maximum expected log strategy, maximum geometric mean strategy, Kelly criterion) is based on probability theoretic underpinnings and has asymptotic optimality properties. This article reviews the allocation of wealth in a two-asset economy with one risky asset and a risk-free asset. It is also shown that the optimal fraction to be invested in the risky asset (i) depends on the length of the basic return period and (ii) is lower for heavy-tailed log returns than for light-tailed log returns.

scholarly journals On the control of the difference between two Brownian motions: a dynamic copula approach

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Thomas Deschatre
Keyword(s):  
Cumulative Distribution Function ◽  
Survival Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Gaussian Copula ◽  
Brownian Motions ◽  
Dynamic Copula ◽  
Copula Approach ◽  
The Difference ◽  
The Right

AbstractWe propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for is the same for Markovian pairs and all pairs of Brownian motions, that is with φ being the cumulative distribution function of a standard Gaussian random variable.

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