Exact distribution of the product and the quotient of two stable Lévy random variables

Rathie, Pushpa N.; Ozelim, Luan Carlos de Sena Monteiro; Otiniano, Cira E. G.

doi:10.1016/j.cnsns.2015.11.012

Cited by 13 publications

(11 citation statements)

References 9 publications

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“…First differences of heavytailed variables will also be heavy-tailed, but will always be two-sided. Quotients of two heavy-tailed variables may yield different types of variables (Rathie et al 2016), but in our case, for both labor productivity, q i,t = y i,t /l i,t , and value added growth, y i,t = y i,t −y i,t−1 y i,t−1 , we obtain another heavy-tailed distribution. In fact, these quotients themselves are fitted very well by Lévy alpha-stable distributions with tail indices α < 2.…”

Section: Distributions Of Firm Level Variablesmentioning

confidence: 58%

Levels of structural change

2021

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We investigate structural change in the PR China during a period of particularly rapid growth 1998-2014. For this, we utilize sectoral data from the World Input-Output Database and firm-level data from the Chinese Industrial Enterprise Database. Starting with correlation laws known from the literature (Fabricant’s laws), we investigate which empirical regularities hold at the sectoral level and show that many of these correlations cannot be recovered at the firm level. For a more detailed analysis, we propose a multi-level framework, which is validated empirically. For this, we perform a robust regression, since various input variables at the firm-level as well as the residuals of exploratory OLS regressions are found to be heavy-tailed. We conclude that Fabricant’s laws and other regularities are primarily characteristics of the sectoral level which rely on aspects like infrastructure, technology level, innovation capabilities, and the knowledge base of the relevant labor force. We illustrate our analysis by showing the development of some of the larger sectors in detail and offer some policy implications in the context of development economics, evolutionary economics, and industrial organization.

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Section: Distributions Of Firm Level Variablesmentioning

confidence: 58%

Levels of structural change

2021

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“…We have decided to consider a power series approximation for the α-stable density by the so-called Fox's H-function or the H-function, involving Mellin-Barnes integrals, which is a generalization of the Gfunction of Meijer. For more details on the H-function, we refer the reader to [23] and [29]. The H-function is defined by means of a Mellin-Barnes type integral…”

Section: Bayesian Estimationmentioning

confidence: 99%

A Generalization of the Ornstein-Uhlenbeck Process: Theoretical Results, Simulations and Parameter Estimation

Stein,

Lopes,

Medino

2021

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In this work, we study the class of stochastic process that generalizes the Ornstein-Uhlenbeck processes, hereafter called by Generalized Ornstein-Uhlenbeck Type Process and denoted by GOU type process. We consider them driven by the class of noise processes such as Brownian motion, symmetric α-stable Lévy process, a Lévy process, and even a Poisson process. We give necessary and sufficient conditions under the memory kernel function for the time-stationary and the Markov properties for these processes. When the GOU type process is driven by a Lévy noise we prove that it is infinitely divisible showing its generating triplet. Several examples derived from the GOU type process are illustrated showing some of their basic properties as well as some time series realizations. These examples also present their theoretical and empirical autocorrelation or normalized codifference functions depending on whether the process has a finite or infinite second moment. We also present the maximum likelihood estimation as well as the Bayesian estimation procedures for the so-called Cosine process, a particular process in the class of GOU type processes. For the Bayesian estimation method, we consider the power series representation of Fox's H-function to better approximate the density function of a random variable α-stable distributed. We consider four goodness-of-fit tests for helping to decide which Cosine process (driven by a Gaussian or an α-stable noise) best fit real data sets. Two applications of GOU type model are presented: one based on the Apple company stock market price data and the other based on the cardiovascular mortality in Los Angeles County data.

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“…Determining distributions of the functions of random variables is a very crucial task and this problem has been attracted a number of researchers because there are numerous applications in Risk Management, Finance, Economics, Science, and, many other areas, see, for example, (Donahue 1964;Ly et al 2016;Nadarajah and Espejo 2006;Springer 1979). Basically, the distributions of an algebraic combination of random variables including the sum, product, and quotient are focused on some common distributions along with the assumptions of independence or correlated through Pearson's coefficient or dependence via multivariate normal joint distributions (Arnold and Brockett 1992;Bithas et al 2007;Cedilnik et al 2004;Hinkley 1969;Macalos and Arcede 2015;Marsaglia 1965;Matović et al 2013;Mekićet al 2012;Nadarajah and Espejo 2006;Kotz 2006a, 2006b;Pham-Gia et al 2006;Pham-Gia 2000;Rathie et al 2016;Sakamoto 1943). Regarding ratio, it often appears in the problems of constructing statistics used in hypothesis testing and estimating issues.…”

Section: Introductionmentioning

confidence: 99%