Class Of Distribution Functions Research Articles

Multivariate tail conditional expectation for scale mixtures of skew-normal distribution

ABSTRACTIn practice, a financial or actuarial data set may be a skewed or heavy-tailed and this motivates us to study a class of distribution functions in risk management theory that provide more information about these characteristics resulting in a more accurate risk analysis. In this paper, we consider a multivariate tail conditional expectation (MTCE) for multivariate scale mixtures of skew-normal (SMSN) distributions. This class of distributions contains skewed distributions and some members of this class can be used to analyse heavy-tailed data sets. We also provide a closed form for TCE in a univariate skew-normal distribution framework. Numerical examples are also provided for illustration.

A New Numerical Technique for the Inverse Problem in Thomson Scattering and Its Application to Experimental and Synthetic Spectra

The inverse problem for Thomson scattering (TS), that is, finding the electron distribution function (EDF), not restricted to be Maxwellian or isotropic, from the observation of the scattered spectrum, is addressed. Based on previous results by the authors, a new parallel FORTRAN code, INVERT, has been developed that allows to estimate the free parameters of a wide class of distribution functions by fitting experimental or numerical (synthetic) spectra using a variant of the simplex method. The application of these techniques to the extraction of non-Maxwellian or anisotropic features in the electron distribution function is analyzed in detail. The performance of the new code on noisy synthetic spectra and its capabilities to quantitatively discriminate among several competing EDFs modeling data are discussed. The issues of uniqueness (or nonuniqueness) of the inverse problem in case of multiparameter distribution functions are discussed. In such cases, the prospects of multiple diagnostics synthesis, or having several simultaneous scattering chords to remove the ambiguity in the reconstruction of the EDF, are also discussed. Some comments on the requirements of a TS system able to detect nonthermal or anisotropic effects are also included.

A unifying approach to incentive compatibility in moral hazard problems

A new approach to moral hazard is presented. Once local incentive compatibility is satisfied, the problem of verifying global incentive compatibility is shown to be isomorphic to the problem of comparing two classes of distribution functions. Thus, tools from choice under uncertainty can be brought to bear on the problem. The approach allows classic justifications of the first-order approach (FOA) to be proven using the same unifying methodology. However, the approach is especially useful for analyzing higher-dimensional moral hazard problems. New and more tractable multi-signal justifications of the FOA are derived and implications for optimal monitoring are examined. The approach yields justifications of the FOA in certain settings where the action is multidimensional, as in the case when the agent is multitasking. Finally, a tractable multitasking model with richer predictions than the popular but simple linear-exponential-normal model is presented.

Numerical comparison of classical and permutation statistical hypothesis testing methods

The article is devoted to the classical problem of statistical hypothesis testing for the equality of two distributions. For normal distributions, Student’s test is optimal in many senses. However, in practice, distributions to be compared are often not normal and, generally speaking, unknown. When nothing is known about the distributions to be compared, one usually applies the nonparametric Kolmogorov–Smirnov test to solve this problem. In the present paper, methods are considered that are based on permutations and, in recent years, have attracted interest for their simplicity, universality, and relatively high efficiency. Methods of stochastic simulation are applied to the comparative analysis of the power of a few permutation tests and classical methods (such as the Kolmogorov–Smirnov test, Student’s test, and the Mann–Whitney test) for a wide class of distribution functions. Normal distributions, Cauchy distributions, and their mixtures, as well as exponential, Weibull, Fisher’s, and Student’s distributions are considered. It is established that, for many typical distributions, the permutation method based on the sum of the absolute values of differences is the most powerful one. The advantage of this method over other ones is especially large when one compares symmetric distributions with the same centers. Thus, this permutation method can be recommended for application in cases when the distributions to be compared are different from normal ones.

Limits of interval orders and semiorders

We study poset limits given by sequences of nite interval orders or, as a special case, nite semiorders. In the interval order case, we show that every such limit can be represented by a probability mea- sure on the space of closed subintervals of (0; 1), and we dene a subset of such measures that yield a unique representation. In the semiorder case, we similarly nd unique representations by a class of distribution functions.

Quantum Fourier transform, Heisenberg groups and quasi-probability distributions

This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl–Gabor–Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of the Wigner function from the marginal distributions via inverse Radon transform giving explicit formulae. We consider some applications of our approach to quantum information processing and quantum process tomography.

Changing Correlation and Equity Portfolio Diversification Failure for Linear Factor Models during Market Declines*

The paper considers a linear factor model (LFM) to study the behaviour of the correlation coefficient between various stock returns during a downturn. Changing correlation is related to the tail distribution of the driving factors, which is the market for Sharpe's one‐factor model. General classes of distribution functions are considered and asymptotic conditions found on the tails of the distribution, which determine whether diversification will succeed or fail during a market decline.

Generalized Chebyshev Inequalities for some Classes of Distribution Functions

Generalized Chebyshev Inequalities for some Classes of Distribution Functions

New Class of Level Statistics in Correlated Disordered Chains

We study the properties of the level statistics of 1D disordered systems with long-range spatial correlations. We find a threshold value in the degree of correlations below which in the limit of large system size the level statistics follows a Poisson distribution (as expected for 1D uncorrelated-disordered systems), and above which the level statistics is described by a new class of distribution functions. At the threshold, we find that with increasing system size, the standard deviation of the function describing the level statistics converges to the standard deviation of the Poissonian distribution as a power law. Above the threshold we find that the level statistics is characterized by different functional forms for different degrees of correlations.

Second-Order Renewal Theorem in the Finite-Means Case

Let F be a distribution function (d.f.) on $(0, \infty )$ and let~U be the renewal function associated with F. If F has a finite first moment~$\mu$, then it is well known that $U(t)$ asymptotically equals $t/\mu$. It is also well known that $U(t)-t/\mu $ asymptotically behaves as $S(t)/\mu, $ where~S denotes the integral of the integrated tail distribution~$F_1$ of~F. In this paper we discuss the rate of convergence of $U(t)-t/\mu -S(t)/\mu $ for a large class of distribution functions. The estimate improves earlier results of Geluk, Teugels, and Embrechts and Omey.

Bondesson's functions in reliability theory

Bondesson's functions in reliability theory are shown to be related to a recursive sequence of probability distributions. These are the ‘higher-order’ versions of the mean remaining lifetime in an equilibrium renewal process. Based on these functions, classes of distribution functions can be defined. This paper will investigate these classes and place Bondesson's work in the content of the other work done in reliability theory. Connections are made with the decreasing variance residual lifetime class and stochastic ordering. Copyright © 1999 John Wiley & Sons, Ltd.

On the Rate of Entropy Production for the Boltzmann Equation

We show that there exists a wide class of distribution functions (with moments of any order as close to their equilibrium values as we like) which can provide an abnormally low rate of entropy production. The result is valid for the Boltzmann equation with any cross section σ(|V|, θ) satisfying a mild restriction. The functions are constructed in an explicit form and we discuss some applications of our results.

Functional laws of the iterated logarithm for local times of recurrent random walks on Z2

We prove functional laws of the iterated logarithm for L n 0 , the number of returns to the origin, up to step n, of recurrent random walks on Z 2 with slowly varying partial Green's function. We find two distinct functional laws of the iterated logarithm depending on the scaling used. In the special case of finite variance random walks, we obtain one limit set for L n x 0/(log n log 3 n); 0 ≤ x ≤ 1, and a different limit set for L xn 0/(log nlog 3 n); 0 ≤ x ≤ 1. In both cases the limit sets are classes of distribution functions, with convergence in the weak topology.

One-sided nonparametric tolerance limits

It is well known that a single order statistic can sometimes be used as a one-sided nonparametric tolerance limit. Unfortunately, there are often minimum sample sizes below which these single-order-statistic limits do not exist. Also, the number of confidence levels for exact limits is equal to the sample size, and this can lead to further conservatism for small samples. This article presents one-sided nonparametric tolerance limits, based on two or more order statistics, which do not have these limitations. These limits are an extension of the work of Hanson and Koopmans (1964) on conservative one-sided tolerance limits for large classes of distribution functions. The extensions include greatly improved algorithms for determining the constants required by the tolerance limits, and a generalization to more than two order statistics.

Deformation of a magnetic dipole field by trapped particles

We investigate the equilibrium and stability of rigidly rotating quasi‐neutral magnetospheres of objects with an aligned magnetic dipole moment, e.g., planetary magnetospheres. We use the Vlasov theory to calculate the equilibria with emphasis on trapped particle populations. Based on a new approach for the description of the trapped particles within the framework of collisionless theory, we calculate self‐consistent axisymmetric magnetospheric models. For the purpose of demonstrating the capability of this approach, we choose a mathematically simple, but nevertheless physically relevant, class of distribution functions. By varying the plasma density, we calculate solution branches and investigate the bifurcation properties for typical cases. Furthermore, the stability of the solutions is tested by applying a sufficient stability criterion to them. We find that in our models there is always a maximum amount of plasma which can be confined in a rigidly rotating magnetosphere within a dipolelike magnetic field. The implications for possible scenarios of magnetospheric activity are discussed on the basis of these results.

A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations

In a class of distribution functions, including exponential, power function, Pareto, Lomax, and logistic distributions, a general recurrence relation for moments of order statistics is given. The validity of this identity for certain constants and some sequence of order statistics leads to characterizations of probability distributions. Several recurrence relations and characterization results known from the literature are particular cases of the theorems stated.

A monotone variational method for optimal probability distributions

LetI(F) be a functional whereF belongs to a class of distribution functions. In order to find an extremalF, a variational technique is developed which is based upon the fact that positive powers ofF remain probability distributions. The resulting necessary conditions derived from this technique reduce to the Euler-Lagrange equations under the condition that the extremal,F, has a continuous derivative. We apply this technique to certain quadratic functionals, and to the maximum entropy problem. This variational method applies generally to positive bounded, and to positive monotone extremals. This is illustrated by proving that the “stoploss” insurance policy is optimal under standard conditions.

The tail index of exchange rate returns

In the literature on the empirical distribution of foreign exchange rates there is now consensus that exchange rate yields are fat-tailed. Three problems, however, persist: (1) Which class of distribution functions is most appropriate? (2) Are the parameters of the distribution invariant over subperiods? (3) What are the effects of aggregation over time on the distribution? In this paper we employ extreme value theory to shed new light on these questions. We apply the theoretical results to EMS data.

Phase-space representations of the Bloch equation.

The authors discuss the phase-space representation of the Bloch equation and present analytic expressions for two generalized classes of distribution functions. Series solutions to order of {h bar}{sup 2} are given. The authors conclude that the Wigner quantum corrections are the simplest among the two general classes of distribution functions. 21 refs.

Breakdown of linear response theory in one-dimensional classical diffusion problems (charge transport)

The authors examine charge transport in one-dimensional disordered systems which can be described by a master equation. The diffusivity is not a meaningful quantity to study because in general the linear response assumption does not hold. The authors formulate a scaling theory for the current and examine three classes of distribution functions which lead to localised, quasilocalised and delocalised behaviour in the long-time limit. Whereas linear response is valid as t to infinity for delocalised motion (finite DC conductivity), interesting anomalies are found and predicted for the electric field dependence of the current in the first two situations.