On the convolution of Pareto and gamma distributions

Nadarajah, Saralees; Kotz, Samuel

doi:10.1016/j.comnet.2007.03.003

Cited by 10 publications

(16 citation statements)

References 21 publications

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“…That means that as n increases, the next term approaches the same size as the nth term, such that the absolute sum of terms is not bounded above, and the order of addition of the original series determines what the total sum is, making changes in order of summation yield different, i.e., ambiguous, results. If a limiting term ratio is less than 1, for example 1 2 , the series is absolutely convergent, e.g., the limiting infinite sum ratio of 1 2 for some eventual term is, in binary arithmetic, 0:111111. . .…”

Section: Appendixmentioning

confidence: 99%

“…Although there are multiple possible GPC models and different nomenclatures used to describe them, a natural classification would arise from Pareto distribution classification, types I through IV, and the Lomax distribution, a type II subtype, which is the classification scheme of reference [4] and the Mathematica computer language [5]. 1 Convolution was first introduced to pharmacokinetics in 1933 by Gehlen who used the convolution of two exponential distributions, EDCðt; b; bÞ ¼ be Àb x Ã be Àb…”

Section: Introductionmentioning

confidence: 99%

“…hðtÞ; which was used to derive a Lomax gamma-Pareto distribution [1]. The relevance of this is that the GPC type I and Lomax GPC derivations are similar.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A series acceleration algorithm for the gamma-Pareto (type I) convolution and related functions of interest for pharmacokinetics

Wesolowski

Alcorn

Tucker

2021

J Pharmacokinet Pharmacodyn

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The gamma-Pareto type I convolution (GPC type I) distribution, which has a power function tail, was recently shown to describe the disposition kinetics of metformin in dogs precisely and better than sums of exponentials. However, this had very long run times and lost precision for its functional values at long times following intravenous injection. An accelerated algorithm and its computer code is now presented comprising two separate routines for short and long times and which, when applied to the dog data, completes in approximately 3 min per case. The new algorithm is a more practical research tool. Potential pharmacokinetic applications are discussed. Graphic abstract

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Section: Appendixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A series acceleration algorithm for the gamma-Pareto (type I) convolution and related functions of interest for pharmacokinetics

Wesolowski

Alcorn

Tucker

2021

J Pharmacokinet Pharmacodyn

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“…Pareto's distributions and their close relatives and generalizations provide a very flexible family of fat‐tailed distributions, which may be used to model income distributions as well as a wide variety of other social and economic distributions. As a consequence, many generalizations of the Pareto distribution have appeared in the literature such as the lognormal‐Pareto distribution by Cooray and Ananda, 1 the two‐parameter Weibull‐Pareto Composite distribution by Cooray, 2 the beta‐Pareto distribution by Akinsete et al, 3 and the beta exponentiated Pareto distribution by Zea et al 4 In addition, Nadarajah and Kotz 5 studied the convolution of Pareto and gamma distributions, and specifically, they studied the application of such type of convolution in analyzing the interarrival times between on‐traffic (off‐traffic). Another recent generalization of the Pareto distribution is the WDP proposed by Alzaaterh et al 6 The probability density function and the cumulative distribution function (CDF) of the WDP are given by

f false(x false) = \frac{β c}{x} {({}, β \log false(x false/ θ false))}^{c - 1} \exp ({}, - {false(β \log false(x false/ θ false) false)}^{c})

and

F false(x false) = 1 - \exp ({}, - {((), β \log false(x false/ θ false))}^{c}),

where x > θ ; c , β , θ >0.…”

Section: Introductionmentioning

confidence: 99%

Parameter estimation methods for the Weibull‐Pareto distribution

Dey¹,

Alzaatreh²,

Ghosh

2019

Comp and Math Methods

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Recently there has been a growing interest to study various estimation procedures to estimate the model parameters of the Weibull Pareto distribution (WPD). It is observed that the proposed distribution can be used quite effectively to model skewed data. They also proposed a modification of the maximum likelihood method (MML) to estimate the parameters of the WPD. The proposed methods showed a better performance from the standard maximum likelihood (ML) method. However, when the shape parameter c>>1, the modified ML method showed unforgettable large bias and standard deviation. In this article, we address some new properties and propose different methods of estimation of the unknown parameters of a WPD from the frequentist point of view. We briefly describe different frequentist approaches, namely, moments estimators, L‐moments estimators, percentile‐based estimators, least squares estimators, method of maximum product of spacings estimators, method of Cramér‐von‐Mises estimators, method of Anderson‐Darling estimators. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and compare them with modified maximum likelihood estimation from the work of Alzaatreh et al (2013a). A real data set have been analyzed for illustrative purposes. Finally, bootstrap confidence intervals are obtained for the parameters of the model based on a real data set.

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“…Also, the inter arrival times between on/off-traffic is the convolution of the Pareto and Gamma random variables. For details see Nadarajah and Kotz [14]. The Pareto distribution is widely applied in different fields such as finance, insurance, physics, hydrology, geology, climatology, astronomy.…”

Section: Introductionmentioning

confidence: 99%