Marcinkiewicz laws with infinite moments

2010 ◽  
Vol 127 (1-2) ◽  
pp. 64-84 ◽  
Author(s):  
Z. S. Szewczak
Keyword(s):  
Infinite Moments

scholarly journals A Method for Confidence Intervals of High Quantiles

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 70
Author(s):  
Mei Ling Huang ◽  
Xiang Raney-Yan
Keyword(s):  
Confidence Intervals ◽  
Computational Algorithm ◽  
Geometric Mean ◽  
Asymptotic Properties ◽  
Quantile Estimation ◽  
Heavy Tailed Distributions ◽  
High Quantiles ◽  
High Quantile ◽  
Heavy Tailed ◽  
Infinite Moments

The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. This paper proposes a new estimator for high quantiles based on the geometric mean. The new estimator has good asymptotic properties as well as it provides a computational algorithm for estimating confidence intervals of high quantiles. The new estimator avoids difficulties, improves efficiency and reduces bias. Comparisons of efficiencies and biases of the new estimator relative to existing estimators are studied. The theoretical are confirmed through Monte Carlo simulations. Finally, the applications on two real-world examples are provided.

LÉVY FLIGHTS: CHAOTIC, TURBULENT, AND RELATIVISTIC

Fractals ◽  
1995 ◽  
Vol 03 (03) ◽  
pp. 491-497 ◽  
Author(s):  
MICHAEL F. SHLESINGER ◽  
JOSEPH KLAFTER ◽  
GERT ZUMOFEN
Keyword(s):  
Random Walk ◽  
Turbulent Diffusion ◽  
Random Variables ◽  
Lévy Flight ◽  
Scale Invariant ◽  
Lévy Flights ◽  
Levy Flight ◽  
Levy Flights ◽  
Mathematical Investigation ◽  
Infinite Moments

Lévy flights were introduced through the mathematical investigation of the algebra of random variables with infinite moments. For many years Lévy flights remained an abstract topic in mathematics. Mandelbrot recognized that the Lévy flight prescription had a deep connection to scale-invariant fractal random walk trajectories. We review the utility of Lévy flights in several physics topics involving chaotic and turbulent diffusion and introduce the scaling to describe trajectories in relativistic turbulence.

scholarly journals Empirical cumulant function based parameter estimation in stable laws

2019 ◽  
Vol 22 (2) ◽  
pp. 311-338 ◽  
Author(s):  
Annika Krutto
Keyword(s):  
Closed Form ◽  
Selection Rule ◽  
Empirical Characteristic Function ◽  
Challenging Problem ◽  
Positive Real ◽  
Infinitely Divisible ◽  
Asymptotically Normal ◽  
Cumulant Function ◽  
Independent Identically Distributed ◽  
Infinite Moments

Stable distributions are a subclass of infinitely divisible distributions that form the only family of possible limiting distributions for sums of independent identically distributed random variables. A challenging problem is estimating their parameters because many have densities with no explicit form and infinite moments. To address this problem, a class of closed-form estimators, called cumulant estimators, has been introduced. Cumulant estimators are derived from the logarithm of empirical characteristic function at two arbitrary distinct positive real arguments. This paper extends cumulant estimators in two directions: (i) it is proved that they are asymptotically normal and (ii) a sample based rule for selecting the two arguments is proposed. Extensive simulations show that under the provided selection rule, the closed-form cumulant estimators generally outperform the well-known algorithmic methods.

scholarly journals Lévy Processes Linked to the Lower-Incomplete Gamma Function

2021 ◽  
Vol 5 (3) ◽  
pp. 72
Author(s):  
Luisa Beghin ◽  
Costantino Ricciuti
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Gamma Function ◽  
Lévy Processes ◽  
Fractional Power ◽  
Levy Processes ◽  
Incomplete Gamma Function ◽  
Governing Equations ◽  
Infinite Moments ◽  
Fractional Power Of Operators

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

scholarly journals A Weighted Estimation for Risk Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Mei Ling Huang ◽  
Ke Zhao
Keyword(s):  
Pareto Distribution ◽  
Risk Model ◽  
Estimation Method ◽  
Risk Models ◽  
Semiparametric Method ◽  
The Usa ◽  
Heavy Tailed ◽  
Truncated Pareto ◽  
Weighted Estimation ◽  
Infinite Moments

We propose a weighted estimation method for risk models. Two examples of natural disasters are studied: hurricane loss in the USA and forest fire loss in Canada. Risk data is often fitted by a heavy-tailed distribution, for example, a Pareto distribution, which has many applications in economics, actuarial science, survival analysis, networks, and other stochastic models. There is a difficulty in the inference of the Pareto distribution which has infinite moments in the heavy-tailed case. Firstly this paper applies the truncated Pareto distribution to overcome this difficulty. Secondly, we propose a weighted semiparametric method to estimate the truncated Pareto distribution. The idea of the new method is to place less weight on the extreme data values. This paper gives an exact efficiency function, L1-optimal weights and L2-optimal weights of the new estimator. Monte Carlo simulation results confirm the theoretical conclusions. The two above mentioned examples are analyzed by using the proposed method. This paper shows that the new estimation method is more efficient by mean square error relative to several existing methods and fits risk data well.

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