Sum, product and ratio of Pareto and gamma variables

Nadarajah, Saralees

doi:10.1080/00949650902926175

Cited by 3 publications

(6 citation statements)

References 38 publications

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“…Different classes of distributions are also considered in the literature and the product of such random variables is analyzed; see, e.g., [34,[41][42][43][44][45][46] and references therein. General properties of the products of subexponential and long-tailed class were studied by [47].…”

Section: State Of the Artmentioning

confidence: 99%

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

et al. 2022

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Multivariate modelling of economics data is crucial for risk and profit analyses in companies. However, for the final conclusions, a whole set of variables is usually transformed into a single variable describing a total profit/balance of company’s cash flows. One of the possible transformations is based on the product of market variables. Thus, in this paper, we study the distribution of products of Pareto or Student’s t random variables that are ubiquitous in various risk factors analysis. We review known formulas for the probability density functions and derive their explicit forms for the products of Pareto and Gaussian or log-normal random variables. We also study how the Pareto or Student’s t random variable influences the asymptotic tail behaviour of the distribution of their product with the Gaussian or log-normal random variables and discuss how the dependency between the marginal random variables of the same type influences the probabilistic properties of the final product. The theoretical results are then applied for an analysis of the distribution of transaction values, being a product of prices and volumes, from a continuous trade on the German intraday electricity market.

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Section: State Of the Artmentioning

confidence: 99%

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

et al. 2022

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“…As one can see, the two-dimensional Pareto distribution defined by the PDF in Eq. (20) does not cover the case with uncorrelated marginal random variables. In the case when X and Y are independent and are described by the Pareto distributions of the PDFs given in (21) with a X , θ X > 0 and a Y , θ Y > 0, respectively, then the PDF of the vector (X, Y ) is given by the product of the marginal PDFs, f X (x) and f Y (y).…”

Section: Pareto Distributionmentioning

confidence: 99%

“…In Fig. 2 we plot the PDF and the corresponding distribution tail of the random variable Z that is a product of marginal random variables from the two-dimensional Pareto distribution described by the PDF (20) for different values of the a parameter. Recall that the correlation (if exists) is directly related to the shape parameter a, i.e.…”

Section: Pareto Distributionmentioning

confidence: 99%

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On the distribution of the product of two continuous random variables with an application to electricity market transactions. Finite and infinite-variance case

Adamska¹,

Bielak²,

Janczura³

et al. 2021

Preprint

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In this paper we study the distribution of a product of two continuous random variables. We derive formulas for the probability density functions and moments of the products of the Gaussian, log-normal, Student's t and Pareto random variables. In all cases we analyze separately independent as well as correlated random variables. Based on the theoretical results we use the general maximum likelihood approach for the estimation of the parameters of the product random variables and apply the methodology for a real data case study. We analyze a distribution of the transaction values, being a product of prices and volumes, from a continuous trade on the German intraday electricity market.

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“…However, one can also find the analysis related to the product of random variables coming from different classes of distributions, see e.g. Gaussian and Laplace distributions Kotz, 2005a, 2011), Gamma and Weibull distributions (Nadarajah and Kotz, 2006), Gamma and Beta distributions (Nadarajah and Kotz, 2005b) or Pareto and Gamma distributions (Nadarajah, 2010). For other references we refer the readers to Shakil and Kibria (2007); Nadarajah and Kotz (2016); Idrizi (2014).…”

Section: Introductionmentioning

confidence: 99%

Dependence structure for the product of bi-dimensional finite-variance VAR(1) model components. An application to the cost of electricity load prediction errors

Janczura¹,

Puć²,

Bielak³

et al. 2022

Preprint

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In this paper we analyze the product of bi-dimensional VAR(1) model components. For the introduced time series we derive general formulas for the autocovariance function and study its properties for different cases of cross-dependence between the VAR(1) model components. The theoretical results are then illustrated in the simulation study for two types of bivariate distributions of the residual series, namely the Gaussian and Student's t. We also show a possible practical application of the obtained results based on the data from the electricity market.

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Sum, product and ratio of Pareto and gamma variables

Abstract: The exact distributions of X + Y , XY and X/(X + Y ) are studied when X and Y are independent Pareto and gamma random variables. Applications are discussed, to real problems in clinical trials, computer networks and economics.

Cited by 3 publications

References 38 publications

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

On the distribution of the product of two continuous random variables with an application to electricity market transactions. Finite and infinite-variance case

Dependence structure for the product of bi-dimensional finite-variance VAR(1) model components. An application to the cost of electricity load prediction errors

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