Finite Mean Research Articles

On the convergence to statistical equilibrium for harmonic crystals

We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n arbitrary, d,n⩾1, and study the distribution μt of the solution at time t∈R. The initial measure μ0 has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt—resp. Ibragimov–Linnik type mixing condition. The main result is the convergence of μt to a Gaussian measure as t→∞. The proof is based on the long time asymptotics of the Green’s function and on Bernstein’s “room-corridors” method.

On linear regression and ratio-product estimation of a finite population mean

Summary. We consider a class of ratio-product estimators for estimating a finite population mean. The asymptotically optimum estimator in the class is identified, along with its approximate mean-square error. This estimator requires prior knowledge of the parameter C = pCy/Cx, where p is the correlation coefficient between the study variate y and the auxiliary variate x, and Cy and Cx are coefficients of variation of y and x respectively. If C is unknown in advance, then it can be replaced by its consistent estimate 0, with the resulting estimator known as an 'estimator based on the estimating optimum'. It is shown that, to the first order of approximation, both estimators have the same mean-square error, and that they are generally more efficient than the usual ratio and product estimators.

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern −1). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.

Optimal linear Bayes and empirical Bayes estimation and prediction of the finite population mean

In this article we investigate the estimation problem of the population mean of a finite population. Both point and interval estimators are of interest from Bayes and empirical Bayes point of views. Empirical Bayes analysis is concerned with the ‘current’ population mean, say γ m , when the sample data are available from other similar ( m−1) finite populations, Y 1, …, Y m−1 , as well as the data from the current population, Y m , where Y i=(Y i1, …, Y in i ) , i=1, …, m . Previous results on inference of γ m have assumed either a normal model or a posterior linearity condition in making Bayes inference which is the kernel of the empirical Bayes problem. They resulted in examination of linear estimators of the sample mean Y ̄ m=n m −1∑ j=1 n m Y mj . In this paper, we propose to investigate a generalizing idea which generates optimal linear Bayes estimators of γ m as functions of Y ̄ m . We develop optimal linear Bayes estimators of γ m under two Bayesian models. They are optimal in the sense of minimizing the mean squared error with respect to the underlying models. The corresponding empirical Bayes analogues are obtained by replacing the unknown hyperparameters by their respective consistent estimates as usual. The asymptotic optimality criterion is employed in order to measure the goodness of the proposed empirical Bayes estimators. Very promising Bayes and empirical Bayes two-sided confidence intervals and predictors of γ m are also discussed. A Monte Carlo study is conducted to evaluate the performance of the proposed estimators.

Empirical Testing Of The Infinite Source Poisson Data Traffic Model#

The infinite source Poisson model is a fluid queue approximation of network data transmission that assumes that sources begin constant rate transmissions of data at Poisson time points for random lengths of time. This model has been a popular one as analysts attempt to provide explanations for observed features in telecommunications data such as self-similarity, long range dependence and heavy tails. We survey some features of this model in cases where transmission length distributions have (a) tails so heavy that means are infinite, (b) heavy tails with finite mean and infinite variance and (c) finite variance. We survey the self-similarity properties of various descriptor processes in this model and then present analyses of four data sets which show that certain features of the model are consistent with the data while others are contradicted. The data sets are 1) the Boston University 1995 study of web sessions, 2) the UC Berkeley home IP HTTP data collected in November 1996, 3) traces collected in end of 1997 at a Customer Service Switch in Munich, and 4) detailed data from a corporate Ericsson WWW server from October 1998. #Research supported by the Gothenburg Stochastic Centre, by the EU TMR network ERB-FMRX-CT96-0095 on “Computational and statistical methods for the analysis of spatial data” and by the Knut and Aline Wallenburg Foundation. Sidney Resnick's research was also partially supported by NSF grant DMS-97-04982 and NSA Grant MDA904-98-1-0041 at Cornell University.

Conditional value‐at‐risk bounds for compound Poisson risks and anormal approximation

A considerable number of equivalent formulas defining conditional value‐at‐risk and expected shortfall are gathered together. Then we present a simple method to bound the conditional value‐at‐risk of compound Poisson loss distributions under incomplete information about its severity distribution, which is assumed to have a known finite range, mean, and variance. This important class of nonnormal loss distributions finds applications in actuarial science, where it is able to model the aggregate claims of an insurance‐risk business.

Multiscaling fractional advection‐dispersion equations and their solutions

The multiscaling fractional advection‐dispersion equation (ADE) is a multidimensional model of solute transport that encompasses linear advection, Fickian dispersion, and super‐Fickian dispersion. The super‐Fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions. The directions need not be orthogonal, so the model can be applied to irregular, noncontinuum fracture networks. The statistical model underlying multiscaling fractional dispersion is a continuous time random walk (CTRW) in which particles have arbitrary jump length distributions and finite mean waiting time distributions. The meaning of the parameters in a compound Poisson process, a subset of CTRWs, is used to develop a physical interpretation of the equation variables. The Green's function solutions are the densities of operator stable probability distributions, the limit distributions of normalized sums of independent, and identically distributed random vectors. These densities can be skewed, heavy‐tailed, and scale nonlinearly, resembling solute plumes in granular aquifers. They can also have fingers in any direction, resembling transport along discrete pathways such as fractures.

Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

We consider stochastic processes, \(S^t \equiv (S_x^t :x \in \mathbb{Z}^d ) \in \mathcal{S}_0^{\mathbb{Z}^d } \) with \(\mathcal{S}_0\) finite, in which spin flips (i.e., changes of S t x ) do not raise the energy. We extend earlier results of Nanda–Newman–Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.

Strong law of large numbers and mixing for the invariant distributions of measure-valued diffusions

Let M(R d) denote the space of locally finite measures on R d and let M 1( M(R d)) denote the space of probability measures on M(R d) . Define the mean measure π ν of ν∈ M 1( M(R d)) by π ν(B)= ∫ M(R d) η(B) dν(η), for B⊂R d. For such a measure ν with locally finite mean measure π ν , let f be a nonnegative, locally bounded test function satisfying 〈 f, π ν 〉=∞. ν is said to satisfy the strong law of large numbers with respect to f if 〈 f n , η〉/〈 f n , π ν 〉 converges almost surely to 1 with respect to ν as n→∞, for any increasing sequence { f n } of compactly supported functions which converges to f. ν is said to be mixing with respect to two sequences of sets { A n } and { B n } if ∫ M(R d) f(η(A n))g(η(B n)) dν(η)− ∫ M(R d) f(η(A n)) dν(η) ∫ M(R d) g(η(B n)) dν(η) converges to 0 as n→∞ for every pair of functions f, g∈ C b 1([0,∞)). It is known that certain classes of measure-valued diffusion processes possess a family of invariant distributions. These distributions belong to M 1( M(R d)) and have locally finite mean measures. We prove the strong law of large numbers and mixing for many such distributions.

Asymptotic independence and a network traffic model

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

Asymptotic independence and a network traffic model

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

On Two-Temperature Problem for Harmonic Crystals

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n \ge 1$. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_d\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution $\mu_t$ of the solution at time $t\in\R$. The main result is the convergence of $\mu_t$ to a Gaussian translation-invariant measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green function and on Bernstein's `room-corridor' argument. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density is $- (0,...,0,C(T_+ - T_-))$ with some positive constant $C>0$ what corresponds to Second Law.

Relativistic Mean-Field Theory with the Pion in Finite Nuclei

We study the possibility of the existence of a finite pion mean field in finite nuclei in the relativistic mean-field (RMF) theory. We are led to conjecture this existence by findings of the essential role of the pion in few-body systems and (p,n) experiments for spin-isospin excitations of medium and heavy nuclei. We carry out explicit calculations for various N=Z closed shell nuclei with a finite pion mean field in the RMF theory with the standard parameter set and pion-nucleon coupling in free space. We find that the finite pion mean field breaks the parity symmetry of the intrinsic single-particle states. We demonstrate the actual existence and elucidate the properties of the finite pion mean field, whose qualitative consequences for various observables are also briefly discussed.

A characterization of income distributions in terms of generalized Gini coefficients

Most commonly used parametric models for the size distribution of incomes possess only a few finite moments, and hence cannot be characterized by the sequence of their moments. However, all income distributions with a finite mean can be characterized by the sequence of first moments of the order statistics. This is an attractive feature since the generalized Gini coefficients of Kakwani (1980), Donaldson and Weymark (1980, 1983) and Yitzhaki (1983) are simple functions of expectations of sample minima. We present results which streamline these characterizations motivated by Aaberge (2000).

On asymptotic optimality in empirical Bayes credibility

In this paper we consider linear empirical Bayes estimation of the pure premiums in a Bühlmann–Straub model, with m risk contracts and n i observations from the ith contract, and provide sufficient conditions, analogous to the conditions of Norberg [Scandinavian Actuarial Journal (1980) 172–194], for a stricter notion of asymptotic optimality. The sufficient conditions are used in the paper to show asymptotic optimality of explicit non-parametric examples. Our results, among other things, strengthen the estimators proposed by Bühlmann and Straub (1970) and their generalization by Sundt (1983). They also strengthen the finite population mean estimators of Ghosh and Lahiri (1987) by showing the asymptotic optimality of their estimators under a weaker condition than the boundedness of the moments of fourth order.

Estimation of two ordered mean residual life functions

If X is a life distribution with finite mean then its mean residual life function (MRLF) is defined by M( x)= E[ X− x| X> x]. It has been found to be a very intuitive way of describing the aging process. Suppose that M 1 and M 2 are two MRLFs, e.g., those corresponding to the control and the experimental groups in a clinical trial. It may be reasonable to assume that the remaining life expectancy for the experimental group is higher than that of the control group at all times in the future, i.e., M 1( x)⩽ M 2( x) for all x. Randomness of data will frequently show reversals of this order restriction in the empirical observations. In this paper we propose estimators of M 1 and M 2 subject to this order restriction. They are shown to be strongly uniformly consistent and asymptotically unbiased. We have also developed the weak convergence theory for these estimators. Simulations seem to indicate that, even when M 1= M 2, both of the restricted estimators improve on the empirical (unrestricted) estimators in terms of mean squared error, uniformly at all quantiles, and for a variety of distributions.

A sharp inequality for the tail probabilities of sums of i.i.d. r.v.’s with dominatedly varying tails

Let F be a distribution function supported on (-∞, ∞) with a finite mean μ . In this note we show that if its tail F =1- F is dominatedly varying, then for any γ ＞max { μ , 0}, there exist C ( γ )＞0 and D ( γ )＞0 such that C ( γ ) nF ( x )≤ F n *( x ) ≤ D ( γ ) nF ( x )≤ F n *( x ) for all n ≥1 and all x ≥ γ n . This inequality sharpens a classical inequality for the subexponential distribution case.

Electronic Transport Properties of Liquid Less-Simple Metals

Electronic transport properties, namely the electrical resistivity and the thermoelectric power, of liquid less-simple metals, Zn, Cd, Hg, In, Tl, Sn, Pb, Sb, and Bi, are calculated using Ziman's theory. The effective electron–ion and ion–ion interactions are described by the Bretonnet-Silbert model. The static structure factors are evaluated by the self-consistent Variational Modified Hypernetted Chain (VMHNC) integral equation of liquids. The results for the electrical resistivity are fairly good when compared with experimental data, but the agreement improves significantly when the blurring of the Fermi surface due to the finite electron mean free path is accounted for. The values of the thermoelectric power, for most systems, have the same sign and order of magnitude as the experimental ones. The results for the thermopower also show that the contribution of the energy dependent term of the electron–ion potential is significant for the concerned systems.

Dynamics of cosmological perturbations in position space

We show that the linear dynamics of cosmological perturbations can be described by coupled wave equations, allowing their efficient numerical and, in certain limits, analytical integration directly in position space. The linear evolution of any perturbation can then be analyzed with the Green's function method. Prior to hydrogen recombination, assuming tight coupling between photons and baryons, neglecting neutrino perturbations, and taking isentropic (adiabatic) initial conditions, the obtained Green's functions for all metric, density, and velocity perturbations vanish beyond the acoustic horizon. A localized primordial cosmological perturbation expands as an acoustic wave of photon-baryon density perturbation with narrow spikes at its acoustic wave fronts. These spikes provide one of the main contributions to the cosmic microwave background radiation anisotropy on all experimentally accessible scales. The gravitational interaction between cold dark matter and baryons causes a dip in the observed temperature of the radiation at the center of the initial perturbation. We first model the radiation by a perfect fluid and then extend our analysis to account for finite photon mean free path. The resulting diffusive corrections smear the sharp features in the photon and baryon density Green's functions over the scale of Silk damping.

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