scholarly journals From moment explosion to the asymptotic behavior of the cumulative distribution for a random variable

2016 ◽  
Vol 61 (3) ◽  
pp. 489-508
Author(s):  
S M Aly ◽  
S M Aly
Keyword(s):  
Asymptotic Behavior ◽  
Random Variable ◽  
Cumulative Distribution

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.

3 Power and sample size for ABE in the 2× 2 design Here we give formulae for the calculation of the power of the TOST procedure assuming that there are in total n subjects in the 2× 2 trial. Suppose that the null hypotheses given below are to be tested using a 100(1 − α)% two-sided confidence interval and a power of (1 − β) is required to reject these hypotheses when they are false. H :µ ≤− ln 1.25 H :µ ≥ ln 1.25. Let us define x to be a random variable that has a noncentral t-distribution with df degrees of freedom and noncentrality parameter nc, i.e., x ∼ t(df,nc). The cumulative distribution function of x is defined as CDF(t,df,nc) = Pr(x≤ t). Assume that the power is to be calculated using log(AUC). If σ is the common within-subject variance for T and R for log(AUC), and n/2 subjects are allocated to each of the sequences RT and TR, then 1−β = CDF(t , df,nc )−CDF(t , df,nc ) (7.1) where √ n(log(µ )− log(0.8)) nc = 2σ √ n(log(µ )− log(1.25)) nc = 2σ √ (n− 2)(nc −nc = ) df 2t and t is the 100(1 − α)% point of the central t-distribution on n− 2 degrees of freedom. Some SAS code to calculate the power for an ABE 2 × 2 trial is given below, where the required input variables are α, σ value of the ratio µ and n, the total number of subjects in the trial.

2003 ◽  
pp. 361-361
Keyword(s):  
Distribution Function ◽  
Confidence Interval ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Noncentrality Parameter ◽  
Cumulative Distribution ◽  
Total N ◽  
The Common ◽  
Input Variables ◽  
T Distribution

scholarly journals On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
Author(s):  
Jalal Chachi
Keyword(s):  
Probability Theory ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Distribution Characteristics ◽  
Fuzzy Random Variables ◽  
Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

Series expansions for the all-time maximum of α-stable random walks

2016 ◽  
Vol 48 (3) ◽  
pp. 744-767
Author(s):  
Clifford Hurvich ◽  
Josh Reed
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Series Expansions ◽  
Stable Random Variables ◽  
The Mean ◽  
The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

COMPARISON OF APPROXIMATIONS FOR COMPOUND POISSON PROCESSES

2015 ◽  
Vol 45 (3) ◽  
pp. 601-637 ◽  
Author(s):  
Raffaello Seri ◽  
Christine Choirat
Keyword(s):  
Random Variable ◽  
Cumulative Distribution ◽  
Claim Amount ◽  
Approximation Methods ◽  
Total Claim Amount ◽  
Aggregate Claim ◽  
Time Period ◽  
Total Claim ◽  
The Difference ◽  
Object Of Study

AbstractIn this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

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.

THE ASYMPTOTIC BEHAVIOR OF SEMI-INVARIANTS FOR LINEAR STOCHASTIC SYSTEMS

2002 ◽  
Vol 02 (02) ◽  
pp. 281-294
Author(s):  
G. N. MILSTEIN
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Linear System ◽  
Characteristic Function ◽  
Stochastic Systems ◽  
Random Variable ◽  
Characteristic Functions ◽  
Linear Stochastic Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment

The asymptotic behavior of semi-invariants of the random variable ln |X(t,x)|, where X(t,x) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(p). Namely, it is obtained that the nth semi-invariant is asymptotically proportional to the time t with the coefficient of proportionality g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t,x)| in a neighborhood of the origin of the complex plane is controlled by the extension g(iz) of g(p).

2. Relative Distribution Methods

1998 ◽  
Vol 28 (1) ◽  
pp. 53-97 ◽  
Author(s):  
Mark S. Handcock ◽  
Martina Morris
Keyword(s):  
Reference Group ◽  
Density Ratio ◽  
Random Variable ◽  
Wage Distribution ◽  
Cumulative Distribution ◽  
Outcome Variable ◽  
Relative Distribution ◽  
Statistical Framework ◽  
Exploratory Data ◽  
Nonparametric Statistical

We present an outline of relative distribution methods, with an application to recent changes in the U.S. wage distribution. Relative distribution methods are a nonparametric statistical framework for analyzing data in a fully distributional context. The framework combines the graphical tools of exploratory data analysis with statistical summaries, decomposition, and inference. The relative distribution is similar to a density ratio. It is technically defined as the random variable obtained by transforming a variable from a comparison group by the cumulative distribution function (CDF) of that variable for a reference group. This transformation produces a set of observations, the relative data, that represent the rank of the original comparison value in terms of the reference group's CDF. The density and CDF of the relative data can therefore be used to fully represent and analyze distributional differences. Analysis can move beyond comparisons of means and variances to tap the detailed information inherent in distributions. The analytic framework is general and flexible, as the relative density is decomposable into the effect of location and shape differences, and into effects that represent both compositional changes in covariates, and changes in the covariate-outcome variable relationship.

scholarly journals SYSTEM RELIABILITY AND WEIGHTED LATTICE POLYNOMIALS

2008 ◽  
Vol 22 (3) ◽  
pp. 373-388 ◽  
Author(s):  
Alexander Dukhovny ◽  
Jean-Luc Marichal
Keyword(s):  
System Reliability ◽  
Cumulative Distribution Function ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Special Cases ◽  
Indicator Variables ◽  
Lattice Polynomials ◽  
Lattice Polynomial

The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in the case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of “indicator” variables. A connection is studied between Y and order statistics of the set of arguments.

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