Parameter Estimation for Exponentially Tempered Power Law Distributions

Meerschaert, Mark M.; Roy, Parthanil; Shao, Qin

doi:10.1080/03610926.2011.552828

Cited by 38 publications

(47 citation statements)

References 35 publications

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“…p is the same power law index as above, but calculated via fitting to the truncated power law model, and M 0 is the characteristic mass for the exponential truncation. To calculate p , and M 0 , we use another MLE approach, following the work by Meerschaert et al (2012). In particular, from their equations 2.13-2.14, p and M 0 can be found by solving the following set of nonlinear equations,…”

Section: Statistical Methods For Quantifying the Initial Mass Functionmentioning

confidence: 99%

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The Mass and Size Distribution of Planetesimals Formed by the Streaming Instability. II. The Effect of the Radial Gas Pressure Gradient

Abod

Simon

et al. 2019

ApJ

118

107

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The streaming instability concentrates solid particles in protoplanetary disks, leading to gravitational collapse into planetesimals. Despite its key role in producing particle clumping and determining critical length scales in the instability's linear regime, the influence of the disk's radial pressure gradient on planetesimal properties has not been examined in detail. Here, we use streaming instability simulations that include particle self-gravity to study how the planetesimal initial mass function depends on the radial pressure gradient. Fitting our results to a power-law, dN/dM p ∝ M −p p , we find a single value p ≈ 1.6 describes simulations in which the pressure gradient varies by 2. An exponentially truncated power-law provides a significantly better fit, with a low mass slope of p ≈ 1.3 that weakly depends on the pressure gradient. The characteristic truncation mass is found to be ∼ M G = 4π 5 G 2 Σ 3 p /Ω 4 . We exclude the cubic dependence of the characteristic mass with pressure gradient suggested by linear considerations, finding instead a linear scaling. These results strengthen the case for a streaming-derived initial mass function that depends at most weakly on the aerodynamic properties of the disk and participating solids. A simulation initialized with zero pressure gradientwhich is not subject to the streaming instability-also yields a top-heavy mass function but with modest evidence for a different shape. We discuss the consistency of the theoretically predicted mass function with observations of Kuiper Belt planetesimals, and describe implications for models of early stage planet formation.

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Section: Statistical Methods For Quantifying the Initial Mass Functionmentioning

confidence: 99%

“…In both criteria, K is the number of parameters involved in the fit (K = 1 for the simple power law and K = 2 for the truncated power law), and ln(L) is the log-likelihood function, which is what is maximized in the MLE method (see, e.g., Meerschaert et al 2012;Li et al 2019. ln(L)…”

Section: Comparison Of Fitsmentioning

confidence: 99%

The Mass and Size Distribution of Planetesimals Formed by the Streaming Instability. II. The Effect of the Radial Gas Pressure Gradient

Abod

Simon

et al. 2019

ApJ

118

107

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“…As we increase the MAD K , the power‐law exponent gradually increases, which may be interpreted as proof of much faster tail decay than initially conjectured by the power laws. It often is argued that the power‐law behaviour does not extend indefinitely and that there is an upper bound that truncates the probability tail (Aban et al ., ; Chakrabarty and Samorodnitsky, ; Meerschaert et al ., ). As is evident in Tables , the power‐law exponent slowly increases as we step up the MAD K , and therefore the respective tails may gradually transition from a power‐law to an exponential decay.…”

Section: Empirical Analysismentioning

confidence: 97%

Agricultural Financial Risks Resulting from Extreme Events

Xouridas¹

2014

J Agricultural Economics

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Decision‐makers in the agricultural sector operate in a volatile and risky environment. The statistical assessment of agricultural commodity prices is necessary to deduce the stylised facts of agricultural markets and guide the action of market participants. This article examines the kurtosis values of 60 agricultural commodities and presents evidence that the distributions of their returns are fat‐tailed. We use power‐law distributions to model the tail returns and the possible time‐varying extreme event risks in commodity markets. Our results suggest that the usefulness of the value at risk and expected shortfall as risk management tools is questionable.

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“…Residence time for particles deposited at each elevation within the bed and also the overall residence time distributions were calculated from bed elevation data for both the no bedform case (Figure d) and the bedform case (Figure h). An exponentially tempered Pareto distribution (black curves),

P (T > t) = λ t^{- α} exp (- βt), t \geq t_{0} > 0,

was fitted to each of the overall residence time distributions where fitting parameters are ( α , β , γ ) and

λ = t_{0}^{α} \exp (β t_{0})

[ Meerschaert et al ., ]. Overall residence time distributions were computed by cumulating all residence times for each elevation level (Figures d and h, red curves) into a single set.…”

Section: Case Studiesmentioning

confidence: 99%

Sediment residence time distributions: Theory and application from bed elevation measurements

Voepel

Schumer

Hassan

2013

J. Geophys. Res. Earth Surf.

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[1] Travel distance and residence time probability distributions are the key components of stochastic models for coarse sediment transport. Residence time for individual grains is difficult to measure, and residence time distributions appropriate to field and laboratory settings are typically inferred theoretically or from overall transport characteristics. However, bed elevation time series collected using sonar transducers and lidar can be translated into empirical residence time distributions at each elevation in the bed and for the entire bed thickness. Sediment residence time at a given depth can be conceptualized as a stochastic return time process on a finite interval. Overall sediment residence time is an average of residence times at all depths weighted by the likelihood of deposition at each depth. Theory and experiment show that when tracers are seeded on the bed surface, power law residence time will be observed until a timescale set by the bed thickness and bed fluctuation statistics. After this time, the long-time (global) residence time distribution will take exponential form. Crossover time is the time of transition from power law to exponential behavior. The crossover time in flume studies can be on the order of seconds to minutes, while that in rivers can be days to years.

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Parameter Estimation for Exponentially Tempered Power Law Distributions

Cited by 38 publications

References 35 publications

The Mass and Size Distribution of Planetesimals Formed by the Streaming Instability. II. The Effect of the Radial Gas Pressure Gradient

The Mass and Size Distribution of Planetesimals Formed by the Streaming Instability. II. The Effect of the Radial Gas Pressure Gradient

Agricultural Financial Risks Resulting from Extreme Events

Sediment residence time distributions: Theory and application from bed elevation measurements

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