Distributions and moments for estimators of Gini index in an exponential distribution

In this paper, we propose several estimators of Gini index of the two-parameter exponential distribution and obtain distributions and moments of the proposed estimators. The proposed estimators are shown to cosistency and will be compared in terms of the mean squared error (MSE) through Monte Carlo method.     

Model selection based on Lorenz and concentration curves,Gini indices and convex order

In order to determine an appropriate amount of premium, statistical goodness-of-fit criteria must be supplemented with actuarial ones when assessing performance of a given candidate pure premium. In this paper, concentration curves and Lorenz curves are shown to provide actuaries with effective tools to evaluate whether a premium is appropriate or to compare two competing alternatives. The idea is to compare the premium income for sub-portfolios gathering low risks (identified as low by means of the premiums under consideration) to the true one, or equivalently, to the actual losses. Numerical illustrations performed on hypothetical data and real ones demonstrate the usefulness of the proposed approach.     

On the Gini measure decomposition

The purpose of this research is to introduce a new approach to the decomposition of the Gini measure in terms of concordance and discordance shares: a new kind of dependence, the Gini rank dependence (GRD), and its formal definition are provided.     

A source density function

The delta function initial condition solution of the generalized Feller equation gives rise to a four-parameter class of one-sided probability source density functions. Because of this origin, each of these functions represents the instantaneous density of an autonomous Markov diffusion process. As the source location parameter approaches zero the density class to be investigated here reduces to the hyper-Gamma class. Because of the immense flexibility in shape (as compared with the hyper-Gamma class) the source density class offers itself to a wide variety of applications in physics and statistics, especially if the observations are far removed from zero. Relative to applications, the natural and paramount concern must be directed toward the parameter estimation problem. This paper provides the analytical foundation for a numerical approach to its solution. The log-likelihood function for the source density class will be established. Based on it the maximum-likelihood equations will be derived. For practical numerical reasons, direct maximization of the log-likelihood function may be more successful than an attempt to solve the (three) maximum-likelihood equations. Therefore, the analytical properties of the log-likelihood function will be discussed, as they are essential in the implementation of any numerical optimization routine.     

Entire functions that share a small function with their derivatives

Let f be an entire function of finite order and a an entire function of order less than f's. If f−a and f^′−a have the same zeros with the same multiplicities, then f^′−a≡c(f−a) for some non-zero constant c.     

整函数与其微分多项式IM分担两个值

设 f(Z)为非常数整函数 ,L (f )为 f (Z)的具有小函数系数的线性微分多项式 ,本文研究当 f(Z)与 L (f ) IM分担两个有穷复数时的唯一性问题 ,给出了 f (Z)与L (f )完全确定的表达式 ,改进了李平、杨重骏最近获得的一个结果 ,从而也改进了先前的一些结果 .     

The indices of Stickelberger ideals of function fields

Recently Yin [Y4] calculated the index of Stickelberger ideal in the group ring when is a (wide or narrow) ray class extension under the assumption that . In this paper, we extend Yin's result to general Received: 23 August 2001 / Published online: 27 June 2002     

Power moments of negative order for the principal spectral function of a string

We established necessary and sufficient conditions for the existence of finite power moments of all integer negative orders for the principal spectral function of a string. The necesity of this problem is explained by its relation to the so-called strong Stieltjes moment problem. Odessa Academy of Food Technologies, Odessa. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 48, No. 9, pp. 1209–1222, September, 1996.     

Entire functions that share a small function with its derivative

In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results.     

Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function

This paper continues investigations on various integral transforms of the Minkowski question mark function. In this work we finally establish the long-sought formula for the moments, which does not explicitly involve regular continued fractions, though it has a hidden nice interpretation in terms of semi-regular continued fractions. The proof is self-contained and does not rely on previous results by the author.     

On Thon's axiomatization of the Gini index

We give a short proof of Thon's characterization of the Gini index.     

Distribution of values of a real function means,moments, and symmetry

For a distribution functionD we define itsabsolute andsigned moments of orderk∈R, which generalise in a natural way the Hamburger moments of orders an even and an odd natural number. Similarly, for a real functionh we define itsabsolute andsigned asymptotic means of orderk∈R. We show that if the means exist on an infinite and bounded set of values ofk, then they exist on an intervalI and coincide onI ^o with the moments ofD=D _h, the distribution function of the values ofh, which is shown to exist (in the sense of Wintner). We also give a sufficient condition forD _h to be symmetric. These results apply to a class of functionsh that contain in particular error terms related to the Euler phi function and to the sigma divisor function. A further application on a certain class of converging trigonometrical series implies in particular classical results of A. Wintner establishing the existence for such functions of a distribution function as well as Hamburger moments of arbitrarily large orders. The remainder term of the prime number theorem belongs to this class provided the Riemann hypothesis holds, and the distribution function of its values is shown to be “almost” symmetric.     

Probability density function of the local score position

We calculate the probability density function of the local score position on complete excursions of a reflected Brownian motion. We use the trajectorial decomposition of the standard Brownian bridge to derive two different expressions of the density: the first one is based on a series and an integral while the second one is free off the series.     

An elementary characterization of the Gini index

The Gini index is one of the most used indicators of social and economic inequality. In this paper we characterize the Gini index as the unique function that satisfies the properties of scale independence, symmetry, standardization and comonotone separability. Furthermore, we propose a simpler way to compute it.     

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

We study the Epstein zeta function En(L,s) for and a random lattice L of large dimension n. For any fixed we determine the value distribution and moments of En(⋅,cn) (suitably normalized) as n→∞. We further discuss the random function c?En(⋅,cn) for c∈[A,B] with and determine its limit distribution as n→∞.