Geometrical properties of Pareto distribution

Abdel-All, Nassar H.; Mahmoud, M. A. W.; Abd-Ellah, H. N.

doi:10.1016/s0096-3003(02)00490-3

Cited by 13 publications

(5 citation statements)

References 12 publications

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“…These two regions were less affected by human activities as well as riverine input by comparing with PRE. Moreover, alkyl PAHs are thermodynamically unstable compared to the corresponding parent PAHs and largely result from the oil release or combustion of fuels at low to moderate temperatures (Abdell-All et al, 2003;Zhou et al, 2009). These compounds were widely distributed in the atmosphere and accumulated in the sediments by deposition (Lian et al, 2009).…”

Section: Sediment Pah Compositionmentioning

confidence: 99%

Characterizing the parent and alkyl polycyclic aromatic hydrocarbons in the Pearl River Estuary, Daya Bay and northern South China Sea: Influence of riverine input

Yuan

Wang

Lin

et al. 2015

Environmental Pollution

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Section: Sediment Pah Compositionmentioning

confidence: 99%

Characterizing the parent and alkyl polycyclic aromatic hydrocarbons in the Pearl River Estuary, Daya Bay and northern South China Sea: Influence of riverine input

Yuan

Wang

Lin

et al. 2015

Environmental Pollution

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“…It is worthy to note that the shortest path between θ 1 and θ 2 is the geodesic on the statistical manifold M . The matrix G ( θ ) is the Fisher information matrix (FIM) of the distribution with respect to θ whose elements can be calculated by

g_{i j} false(bold-italicθ false) = E_{θ} false[\partial_{i} ℓ_{θ} \partial_{j} ℓ_{θ} false] = - E_{θ} false[\partial_{i} \partial_{j} ℓ_{θ} false] = - \int \partial_{i} \partial_{j} ℓ false(bold-italicx false| bold-italicθ false) p false(bold-italicx false| bold-italicθ false) normald bold-italicθ,

where

ℓ_{θ} = ℓ_{θ} false(bold-italicx false| bold-italicθ false) = \ln p false(bold-italicx false| bold-italicθ false)

and E θ denotes the expectation with respect to the pdf p ( x | θ ) . It is easy to see that the squared differential FID between p ( x | θ ) and p ( x | θ + d θ ) can be represented as

normald s^{2} = \sum_{i j} g_{i j} false(bold-italicθ false) normald θ^{i} normald θ^{j} .

…”

Section: Statistical Manifoldmentioning

confidence: 99%

Geometric structure of the joint N‐voM distribution manifold and its applications to sensor networks

Sun

Zhang

2017

Math Methods in App Sciences

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In this paper, the geometric structure for normal distribution manifold, von Mises distribution manifold and their joint distribution manifold are firstly given by the metric, curvature, and divergence, respectively. Furthermore, the active detection with sensor networks is presented by a classical measurement model based on metric manifold, and the information resolution is presented for the range and angle measurements sensor networks. The preliminary analysis results introduced in this paper indicate that our approach is able to offer consistent and more comprehensive means to understand and solve sensor network problems containing sensors management and target detection, which are not easy to be handled by conventional analysis methods. ;40:6332-6347. XU ET AL. 6333manifold and information resolution to the analysis of sensor networks are explored to gain a better understanding and more comprehensive investigation of sensor system issues for target detection and tracking based on the joint distribution manifold.The outline of this paper is organized as follows. In Section 2, the mathematical background about information geometry is stated comprehensively. The geometric structure about the voM distribution manifold is presented in Section 3. The applications about the joint distribution manifold are presented and analyzed in Section 4. Finally, some conclusions are given at Section 5. STATISTICAL MANIFOLDNow, we consider the parameterized family of probability distributions M = {p(x| )}, where x ∈ R m is a random variable and = ( 1 , · · · , n ) is a real vector parameter to specify a distribution.Definition 2.1. The probability distribution family M = {p(x| )} is called as a statistical model, if it satisfies the following regularity conditions:

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“…It has been noticed that from the recent literature that the Pareto distribution can be applied efficiently in various situations rather than other distributions to model data. e significance of Pareto distribution in forming various real phenomena is patent from the following revealed research and references mentioned therein, Abdel-All et al [31], Ismail [32], Sankaran and Nair [33], and Nadarajah and Kotz [34]. Inspired by the wide real-life applications of mixture distributions, the main objective of this study is to develop a new three-component mixture of Pareto distributions (TCMPD) for lifetime data modeling under type I mixture.…”

Section: Introductionmentioning

confidence: 99%

On Estimation of Three-Component Mixture of Distributions via Bayesian and Classical Approaches

Tahir

Almanjahie

Abid

et al. 2021

Mathematical Problems in Engineering

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In this study, we model a heterogeneous population assuming the three-component mixture of the Pareto distributions assuming type I censored data. In particular, we study some statistical properties (such as various entropies, different inequality indices, and order statistics) of the three-component mixture distribution. The ML estimation and the Bayesian estimation of the mixture parameters have been performed in this study. For the ML estimation, we used the Newton Raphson method. To derive the posterior distributions, different noninformative priors are assumed to derive the Bayes estimators. Furthermore, we also discussed the Bayesian predictive intervals. We presented a detailed simulation study to compare the ML estimates and Bayes estimates. Moreover, we evaluated the performance of different estimates assuming various sample sizes, mixing weights and test termination times (a fixed point of time after which all other tests are dismissed). The real-life data application is also a part of this study.

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Geometrical properties of Pareto distribution

Cited by 13 publications

References 12 publications

Characterizing the parent and alkyl polycyclic aromatic hydrocarbons in the Pearl River Estuary, Daya Bay and northern South China Sea: Influence of riverine input

Characterizing the parent and alkyl polycyclic aromatic hydrocarbons in the Pearl River Estuary, Daya Bay and northern South China Sea: Influence of riverine input

Geometric structure of the joint N‐voM distribution manifold and its applications to sensor networks

On Estimation of Three-Component Mixture of Distributions via Bayesian and Classical Approaches

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