Thin Tails Research Articles

Broad plumes rooted at the base of the Earth's mantle beneath major hotspots

Plumes of hot upwelling rock rooted in the deep mantle have been proposed as a possible origin of hotspot volcanoes, but this idea is the subject of vigorous debate. On the basis of geodynamic computations, plumes of purely thermal origin should comprise thin tails, only several hundred kilometres wide, and be difficult to detect using standard seismic tomography techniques. Here we describe the use of a whole-mantle seismic imaging technique--combining accurate wavefield computations with information contained in whole seismic waveforms--that reveals the presence of broad (not thin), quasi-vertical conduits beneath many prominent hotspots. These conduits extend from the core-mantle boundary to about 1,000 kilometres below Earth's surface, where some are deflected horizontally, as though entrained into more vigorous upper-mantle circulation. At the base of the mantle, these conduits are rooted in patches of greatly reduced shear velocity that, in the case of Hawaii, Iceland and Samoa, correspond to the locations of known large ultralow-velocity zones. This correspondence clearly establishes a continuous connection between such zones and mantle plumes. We also show that the imaged conduits are robustly broader than classical thermal plume tails, suggesting that they are long-lived, and may have a thermochemical origin. Their vertical orientation suggests very sluggish background circulation below depths of 1,000 kilometres. Our results should provide constraints on studies of viscosity layering of Earth's mantle and guide further research into thermochemical convection.

An Overview of Generalized Gamma Mittag–Leffler Model and Its Applications

Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated with the Mittag–Leffler function. This family gives an extension to the generalized gamma family, opens up a vast area of potential applications and establishes connections to the topics of fractional calculus, nonextensive statistical mechanics, Tsallis statistics, superstatistics, the Mittag–Leffler stochastic process, the Lévi process and time series. Apart from examining the properties, the matrix-variate analogue and the connection to fractional calculus are also explained. By using the pathway model of Mathai, the model is further generalized. Connections to Mittag–Leffler distributions and corresponding autoregressive processes are also discussed.

Thin and heavy tails in stochastic programming

Optimization problems depending on a probability measure correspond to many applications. These problems can be static (single-stage), dynamic with finite (multi-stage) or infinite horizon, single- or multi-objective. It is necessary to have complete knowledge of the “underlying” probability measure if we are to solve the above-mentioned problems with precision. However this assumption is very rarely fulfilled (in applications) and consequently, problems have to be solved mostly on the basis of data. Stochastic estimates of an optimal value and an optimal solution can only be obtained using this approach. Properties of these estimates have been investigated many times. In this paper we intend to study one-stage problems under unusual (corresponding to reality, however) assumptions. In particular, we try to compare the achieved results under the assumptions of thin and heavy tails in the case of problems with linear and nonlinear dependence on the probability measure, problems with probability and risk measure constraints, and the case of stochastic dominance constraints. Heavy-tailed distributions quite often appear in financial problems [26] while nonlinear dependence frequently appears in problems with risk measures [22, 30]. The results we introduce follow mostly from the stability results based on the Wasserstein metric with the ``underlying $ {\cal L}_{1}$ norm. Theoretical results are completed by a simulation investigation.

Triggering Comet-Like Activity of Main Belt Comets

AbstractMain Belt Comets (MBCs) have attracted a great deal of interest since their identification as a new class of bodies by Hsieh and Jewitt in 2006. Much of this interest is due to the implication that MBC activity is driven by the sublimation of volatile material (presumed to be water-ice) presenting these bodies as probable candidates for the delivery of a significant fraction of Earth's water. Results of the studies of the dynamics of MBCs suggest that these objects might have formed in-situ as the remnants of the break-up of large icy asteroids. Simulations also show that collisions among MBCs and small objects could have played an important role in triggering the cometary activity of these bodies. Such collisions might have exposed sub-surface water-ice which sublimated and created thin atmospheres and tails around MBCs. In order to drive the effort of understanding the nature of the activation of MBCs, we have investigated these collision processes by simulating the impacts in detail using a smooth particle hydrodynamics (SPH) approach that includes material strength and fracture models. We have carried out simulations for a range of impact velocities and angles, allowing m-sized impactors to erode enough of an MBC's surface to expose volatiles and trigger its activation. Impact velocities were varied between 0.5 km/s and 5.3 km/s, and the projectile radius was chosen to be 1 m. As expected, we observe significantly different crater depths depending on the impact energy, impact angle, and MBC's material strength. Results show that for all values of impact velocity and angle, crater depths are only a few meters, implying that if the activity of MBCs is due to the sublimation of water-ice, ice has to exist in no deeper than a few meters from the surface. We present details of our simulations and discuss the implications of their results.

Stochastic Processes via the Pathway Model

After collecting data from observations or experiments, the next step is to analyze the data to build an appropriate mathematical or stochastic model to describe the data so that further studies can be done with the help of the model. In this article, the input-output type mechanism is considered first, where reaction, diffusion, reaction-diffusion, and production-destruction type physical situations can fit in. Then techniques are described to produce thicker or thinner tails (power law behavior) in stochastic models. Then the pathway idea is described where one can switch to different functional forms of the probability density function through a parameter called the pathway parameter. The paper is a continuation of related solar neutrino research published previously in this journal.

Pregibit: a family of binary choice models

The pregibit binary choice model is built on a distribution that allows symmetry or asymmetry and thick tails, thin tails, or no tails. Thus, the model is much more flexible than the traditional binary choice models: pregibit nests logit, approximately nests probit, loglog, cloglog, and gosset models and incorporates the linear probability model. Greater flexibility allows a more accurate estimation of the data-generating process, including asymmetric and thick/thin tails. We prove that the maximum likelihood estimator of the pregibit model is consistent and asymptotically normally distributed. A Monte Carlo analysis and two real-world examples show that probit and logit estimates may show misleading evidence in cases where a pregibit model is statistically preferred. One example concerns enrollment in post-secondary education in Belgium: The pregibit estimate of the enrollment gap between Belgian natives and foreign students is 50 % larger, and the type of high school (general, technical, catholic) is more influential. The second example examines the outcome of mortgage applications in the USA. Here, pregibit estimates assign a stronger role to variables that measure the financial strength of mortgage applicants and a weaker role to demographic characteristics including minority status. More importantly, the distribution of the disturbances proves to be seriously skewed: Pregibit indicates that even high-risk applicants (with a probit acceptance probability of nearly 0) have a positive probability of getting their mortgage application approved. Apparently, mortgage officers are more inclined to uncover reasons to make a mortgage deal than to send clients away empty-handed.

Singular Limits for Reaction-Diffusion Equations with Fractional Laplacian and Local or Nonlocal Nonlinearity

We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. Next, we show that such rescaling is also possible for models with non-local reaction terms, as selection-mutation models. However, to obtain a more relevant qualitative behavior for this second case, we introduce, in the second part of the paper, a second rescaling where we assume that the diffusion steps are small. In this way, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limit which describes the asymptotic dynamics of the solutions, similarly to the case of selection-mutation models with a classical Laplace term or an integral kernel with thin tails. However, the rescaling introduced here is very different from the latter cases. We extend these results to the multidimensional case.

An Empirical Characteristic Function Approach to Selecting a Transformation to Normality

In this paper, we study the problem of transforming to normality. We propose to estimate the transformation parameter by minimizing a weighted squared distance between the empirical characteristic function of transformed data and the characteristic function of the normal distribution. Our approach also allows for other symmetric target characteristic functions. Asymptotics are established for a random sample selected from an unknown distribution. The proofs show that the weight function <TEX>$t^{-2}$</TEX> needs to be modified to have thinner tails. We also propose the method to compute the influence function for M-equation taking the form of U-statistics. The influence function calculations and a small Monte Carlo simulation show that our estimates are less sensitive to a few outliers than the maximum likelihood estimates.

Comparison of the Tails of Market Return Distributions

Summary The aim of this study is to analyze the tails of the distributions of stock market returns and to compare the differences between them. It is a well-established fact that the vast majority of stock market return distributions exhibit fat tails (a bigger probability of extreme outcomes then in the case of the normal probability). Apart from that, there seems to be a popular opinion that most market returns are negatively skewed with a fatter left tail. The study utilizes two methods for comparing the tails of a distribution. A simple approached based on the sample kurtosis, with a modification that allows for the calculation of kurtosis separately for the right and the left tail of a single distribution and a more complex approach based on the maximum likelihood fitting of the Generalized Pareto Distribution to both tales of standardized return distributions. The second approach is based on the assumptions of the Extreme Value Theory (EVT) and the Pickands-Balkema-de Haan theorem. Both approaches provide similar conclusions. Results suggest that whether the left or the right tail of the return distribution is bigger varies from market to market. All four major equity indices of the Polish Warsaw Stock Exchange exhibited a fatter left tale. However, in the whole sample it was actually more common for the right tail to be heavier, with 12 indices out of 20 exhibiting a fatter right tail then the left. The sample kurtosis indicated that all stock market return’s distributions were heavy tailed, whereas the estimates of Generalized Pareto Distribution parameters did indicate standard or thin tails in two cases. Statistical tests indicate that the differences between the tails of stock market distributions are not statistically significant

The Future Has Thicker Tails than the Past: Model Error as Branching Counterfactuals

Ex ante forecast outcomes should be interpreted as counterfactuals (potential histories), with errors as the spread between outcomes. We reapply measurements of uncertainty about the estimation errors of the estimation errors of an estimation treated as branching counterfactuals. Such recursions of epistemic uncertainty have markedly different distributial properties from conventional sampling error, and lead to fatter tails in the projections than in past realizations. Counterfactuals of error rates always lead to fat tails, regardless of the probability distribution used. A mere .01% branching error rate about the STD (itself an error rate), and .01% branching error rate about that error rate, etc. (recursing all the way) results in explosive (and infinite) moments higher than 1. Missing any degree of regress leads to the underestimation of small probabilities and concave payoffs (a standard example of which is Fukushima). The paper states the conditions under which higher order rates of uncertainty (expressed in spreads of counterfactuals) alters the shapes the of final distribution and shows which a priori beliefs about conterfactuals are needed to accept the reliability of conventional probabilistic methods (thin tails or mildly fat tails).

Tail Length and Sexual Size Dimorphism (SSD) in Desmognathan Salamanders

Tails serve multiple functions in salamanders: energy storage, courtship, antipredator defense, and respiration. Differences in life history strategies between males and females may generate sexual size dimorphism (SSD) in tail length. However, the contribution of tail length to SSD is relatively unstudied. Because the basal member of the desmognathan salamander Phaeognathus hubrichti shows SSD in tail dimensions (females have long, thin tails compared to the shorter, thicker tails of males), we hypothesized that other desmognathans may follow the same pattern. We took standard morphological measurements, aged, and sexed 342 Desmognathus quadramaculatus (the largest species of Desmognathus) and 244 Desmognathus aeneus (one of the smallest species) to determine whether SSD in tail length followed the same pattern as in P. hubrichti. Neither Desmognathus species showed SSD in tail length and the trends were in opposite directions. Males of both species had significantly wider tail diameters than the females. Females had a significantly longer distance between limbs than did males. We also found that all three species differ in resource allocation to different body parts as they mature. This suggests that life history traits, and behavioral and ecological requirements, have a stronger influence on body size than does phylogeny in this group of salamanders.

Effects of carbofuran on survival and growth of Duttaphrynus melanostictus tadpoles

This study investigated the impact of a commonly used carbamate pesticide, carbofuran, on survival, growth, development and activity of larvae of the Asian Common Toad, Duttaphrynus melanostictus Schneider 1999. Continuous exposure trials of 15 days were conducted using tadpoles of Gosner stages 24-25 with a commercial grade carbofuran (Curaterr 3G). Four concentrations 50, 150, 250 and 500 ?gl-1 were tested and treatments and controls (without pesticides) were maintained in triplicate (n=56 per treatment or control). Results revealed that there were marked elevations in mortality in larvae exposed to all tested concentrations as compared to those in the controls. Mortality was, however, not dose-dependent, but depicted a hermetic response, where mortality at 500 ?gl-1(35 %) was much lower than that recorded at 250 ?gl-1 (62 %). The 15day LC50 value of D. melanostictus larvae was 190 ?gl- 1. In contrast to mortality, the effects on growth, development and activity although pronounced, were transient. For instance growth impairments were noted until day 7, but the body size of surviving tadpoles were comparable to those in the controls by the end of the trial. Exposure to carbofuran to concentrations of 150 ?gl-1 and above also caused several notable morphological aberrations such as swollen heads and thin tails. In particular, the fact that carbofuran induced high levels of mortality at the tested concentrations, which are below the field application levels, suggest that this pesticide may be detrimental to the survival of amphibians inhabiting agricultural landscapes in Sri Lanka.

Pregibit: A Family of Discrete Choice Models

The pregibit discrete choice model is built on a distribution that allows symmetry or asymmetry and thick tails, thin tails or no tails. Thus the model is much richer than the traditional models that are typically used to study behavior that generates discrete choice outcomes. Pregibit nests logit, approximately nests probit, loglog, cloglog and gosset models, and yields a linear probability model that is solidly founded on the discrete choice framework that underlies logit and probit.

The Illusion of Thin-Tails Under Aggregation

It is assumed that while portfolio theory fails with daily returns, that it would work with yearly returns, an standard argument recently repeated in Treynor (2011). This paper debunks the confusion that daily returns, when non-Gaussian but with finite variance can aggregate to thin tails. Alas, portfolio theory fails in both the short and the long run. The central limit theorem operates too slowly for economic data for us to use it and take portfolio theory with any degree of seriousness. The point is illustrated with a Monte Carlo simulation.

Fat Tails, Thin Tails, and Climate Change Policy

Climate policy is complicated by the considerable uncertainties concerning the benefits and costs of abatement. We do not even know the probability distributions for future temperatures and impacts, making benefit–cost analysis based on expected values challenging to say the least. There are good reasons to believe that those probability distributions are fat-tailed, which implies that if social welfare is based on the expectation of a constant relative risk aversion utility function, then we should be willing to sacrifice close to 100 percent of gross domestic product to reduce greenhouse gas emissions. I argue that unbounded marginal utility makes little sense and that once we put a bound on marginal utility, this implication of fat tails goes away: Expected marginal utility will be finite even if the distribution for outcomes is fat-tailed. Furthermore, depending on the bound on marginal utility, the index of risk aversion, and the damage function, a thin-tailed distribution can actually yield a higher expected marginal utility (and thus a greater willingness to pay for abatement) than a fat-tailed one.

Evolution of dispersal in explicitly spatial metacommunities

We apply an evolutionary game theoretic approach to the evolution of dispersal in explicitly spatial metacommunities, using a flexible parametric class of dispersal kernels, namely 2Dt kernels, and study the resulting evolutionary dynamics and outcomes. We observe strong selective pressure on mean dispersal distance (i.e., the first moment), and weaker, but significant, one on the shape of dispersal kernel (i.e., higher moments). We investigate the effects of landscape topology and spatial heterogeneity on the resulting ‘optimal’ dispersal kernels. The shape—importantly the tail structure—and stability of evolutionarily optimal dispersal strategies are strongly affected by landscape topology or connectivity. Specifically, the results suggest that the optimal dispersal kernels in the river network topology have heavier tails and are stable, while those in the direct topology, where organisms are allowed to travel directly from one location to another, have relatively thin tails and may be unstable. We also find that habitat spatial heterogeneity enables coexistence and controls spatial distribution of distinct groups of dispersal strategies and that alteration in topology alone may not be sufficient to change such coexistence. This work provides a tool to translate environmental changes such as global climate change and human intervention into changes in dispersal behavior, which in turn may lead to important alterations of biodiversity and biological invasion patterns.

First passage percolation on random graphs with finite mean degrees

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount. We analyze the configuration model with degree power-law exponent $\tau>2$, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent $\tau-1>1$, or has even thinner tails ($\tau=\infty$). In this model, the degrees have a finite first moment, while the variance is finite for $\tau>3$, but infinite for $\tau\in(2,3)$. We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to $\alpha\log{n}$, where $\alpha\in(0,1)$ for $\tau\in(2,3)$, while $\alpha>1$ for $\tau>3$. Here $n$ denotes the size of the graph. For $\tau\in (2,3)$, it is known that the graph distance between two randomly chosen connected vertices is proportional to $\log \log{n}$ [Electron. J. Probab. 12 (2007) 703--766], that is, distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [J. Math. Phys. 49 (2008) 125218] of showing that $\log{n}$ is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees ($\tau\in[1,2)$) is studied in [Extreme value theory, Poisson--Dirichlet distributions and first passage percolation on random networks (2009) Preprint] where it is proved that the hopcount remains uniformly bounded and converges in distribution.

Perpetuities with thin tails revisited

Perpetuities with thin tails revisited

Accounting for the U.S. Wealth Inequality with Heterogeneous Portfolio Performance

This paper accounts for the U.S. wealth inequality; we study ex ante identical households that face uninsured idiosyncratic shocks to their labor endowments and portfolio performances. The approach is motivated by numerous data sources indicating there is a great deal of inequality in households portfolio performance due to the under-diversification in portfolio equity. Thus, assuming all households face only aggregate risk in their portfolios (which is the common approach taken in the literature) may be too rough an approximation and lead to biased results about wealth accumulation. Our model results closely account for the U.S. wealth and earnings inequality, especially the top thin tails in the Lorenz curves. Collectively, incorporating the stock return inequality changes our inference about stochastic growth models, particularly those which account for the U.S. wealth distribution.

Perpetuities with thin tails revisited

We consider the tail behavior of random variables R which are solutions of the distributional equation $R\stackrel{d}{=}Q+MR$, where (Q, M) is independent of R and |M|≤1. Goldie and Grübel showed that the tails of R are no heavier than exponential and that if Q is bounded and M resembles near 1 the uniform distribution, then the tails of R are Poissonian. In this paper, we further investigate the connection between the tails of R and the behavior of M near 1. We focus on the special case when Q is constant and M is nonnegative.