One-Dimensional Stable Distributions

Золотарев, В. М.

doi:10.1090/mmono/065

Cited by 1,189 publications

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“…Thus the value of 3 for a is slightly too large to characterize properly the power decay of the total ejecta deposits. Though there exists no definitive value for a that includes continuous and discontinuous ejecta, we believe that values < 3 (e.g., 2.50, 2.67) would improve the fit significantly based on the graphic observation of the probability distribution function for varying a [Zolotarev, 1986;Samorodnitsky and Taqqu, 1994]. Although there are no common explicit solutions for the anomalous diffusion model with varying a, solutions do exist for specific cases (e.g., a = 2.5 or a-2.67 [Zolotarev, 1986]).…”

Section: 445mentioning

confidence: 98%

Compositional gradients across mare‐highland contacts: Importance and geological implication of lateral transport

Li¹,

Mustard²

2000

J. Geophys. Res.

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Observations of mixing across mare-highland contacts using high-resolution Clementine data allow a new assessment of the relative importance of vertical versus lateral mass transport on the Moon at least at mare-highland contact. We analyze mare-highland contacts in the Grimaldi, Orientale, and Fecunditatis basins and Tsiolkovsky crater through image-based nonlinear spectral mixture modeling of Clementine ultraviolet-visible (UV-VIS) light spectrometer multispectral data. The mare in these regions differ in ages, but they are characterized by having simple geological contacts. The symmetric distribution of mare and highland soil constituents across their geological contact indicates that lateral transport is more efficient than deep vertical transport during the formation of the observed mixing zones. Observations across the lunar limb with the Clementine UV-VIS filters show that there is a stray light component of scattered light across high-contrast albedo boundaries. However, the magnitude and spatial properties of scattered light do not significantly affect the nature of compositional boundaries determined here. Since repetitive meteorite bombardment governs lateral transport across the geological contacts and is a random process, a stochastic model is developed to describe this lateral transport. Analysis of the power decay law for crater ejecta thickness indicates that the high-velocity ejecta travels long distances and follows a -3 power decay law. This results in an infinite variance, which requires an anomalous diffusion model since a classical diffusion model would be invalid. Mathematical modeling supports this result where it is shown that classical diffusion produces a relatively poor fit to profiles of material abundance, while the anomalous diffusion model fits the profiles adequately. These results indicate that high-velocity ejecta dominates the formation of the observed compositional gradients, while the ejecta near the crater rim is relatively less important. On the basis of these ejecta distributions and the assumption of random impact cratering, we derive a relationship that can be used as an index for relative age dating of geological contacts as well as for investigating cratering rates.

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Section: 445mentioning

confidence: 98%

Compositional gradients across mare‐highland contacts: Importance and geological implication of lateral transport

Li¹,

Mustard²

2000

J. Geophys. Res.

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“…For a detailed review of properties of stable distributions the reader is referred to, e.g., the monographs by Zolotarev (1986) and Uchaikin and Zolotarev (1999).…”

Section: Notations and Distributional Assumptionsmentioning

confidence: 99%

EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS

Ибрагимов

2007

Econ. Theory

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The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Among other results, the paper shows that the sample mean is the best linear unbiased estimator of the population mean for not extremely heavy-tailed populations in the sense of its peakedness properties. In addition, in such a case, the sample mean exhibits the property of monotone consistency and, thus, an increase in the sample size always improves its performance. However, efficiency of the sample mean in the sense of its peakedness decreases with the sample size if the sample mean is used to estimate the population center under extreme heavy-tailedness. The paper also provides applications of the main efficiency and majorization comparison results in the study of concentration inequalities for linear estimators.

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“…Here g(x; α, θ, λ) is the density function of the strictly stable law and g(y; β, 1, 1) is the density of the one-sided strictly stable law with the characteristic function (see [18])…”

Section: Fractional-stable Laws and Estimators Of Their Parametersmentioning

confidence: 99%

Дробно-Устойчивая Статистика Экспрессии Генов В Экспериментальных Данных Секвенирования Нового Поколения

Saenko¹

2016

Math.Biol.Bioinf.

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Abstract.As has been shown in the previous article [1] an application of class of the fractional-stable laws to the genes expression results obtained by DNA-microarrays leads to poor agreement between experimental and theoretical distributions. This difference can be explained by the imperfection of the technology of the gene expression determination. In this article the distributions of the gene expression obtained by Next Generation Sequence technology are investigated. In this technology the determination technique of the gene expression differs from the DNA-microarrays technology. This results to more qualitative results of an approximation. In particular, it is established that the probability density function of the gene expression has a form of shift-scale mixture of probability laws, where one of the components of the mixture is the fractional-stable distribution.

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One-Dimensional Stable Distributions

Cited by 1,189 publications

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Compositional gradients across mare‐highland contacts: Importance and geological implication of lateral transport

Compositional gradients across mare‐highland contacts: Importance and geological implication of lateral transport

EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS

Дробно-Устойчивая Статистика Экспрессии Генов В Экспериментальных Данных Секвенирования Нового Поколения

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