Uniform Convergence of the Non-Weighted Poverty Measures

We consider in this note the weak convergence, in the frame of the empirical processes theory, of the nonweighted poverty measures viewed as stochastic processes defined on some space of bounded functions and indexed by real numbers or monotone functions. The results include the asymptotic behavior of the Foster–Greer–Thorbecke process of poverty indices. We use them to follow up the poverty evolution in poor countries between two periods with appropriate curves.     

关于三角形内部一点的一些不等式

应用 Wolstenholme 不等式的代数形式,建立了涉及三角形内部任一点的几个二次型不等式,提出了这类不等式的几个猜想。     

On k-th record times,record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.     

The generalizations of strong law of large numbers for asymptotic even–odd Markov chains indexed by a homogeneous tree

Yang et al. (Yang et al., J. Math. Anal. Appl., 410 (2014), 179–189.) have obtained the strong law of large numbers and asymptotic equipartition property for the asymptotic even–odd Markov chains indexed by a homogeneous tree. In this article, we are going to study the strong law of large numbers and the asymptotic equipartition property for a class of non homogeneous Markov chains indexed by a homogeneous tree which are the generalizations of above results. We also provide an example showing that our generalizations are not trivial.     

A Strong Law of Large Numbers for Strongly Mixing Processes

We prove a strong law of large numbers for a class of strongly mixing processes. Our result rests on recent advances in understanding of concentration of measure. It is simple to apply and gives finite-sample (as opposed to asymptotic) bounds, with readily computable rate constants. In particular, this makes it suitable for analysis of inhomogeneous Markov processes. We demonstrate how it can be applied to establish an almost-sure convergence result for a class of models that includes as a special case a class of adaptive Markov chain Monte Carlo algorithms.     

A general function of axiomatic index numbers

Summary This paper describes an axiomatic approach to index number theory. A general bilaeral formula which generates a set of indices of prices and quantities is proposed. This formula is written as a geometric mean, weighted with logarithmic means of relative values, and includes all of those indices and related cofactors which satisfy the following axiomatic properties: strong identity, commensurability, linear homogeneity and associativity (or monotonicity). Moreover, two subsets of indices are identified: the first subset includes those which can be expressed either as geometric means or as expenditure ratios; the corresponding bounds are given by the Laspeyres' and Paasche's indices. The second subset also satisfies the desired properties of base and factor reversibility; in this case the bounds are given by Sato Vartia's and Fisher's indices. In addition it is shown that the intersection between the two subsets identifies a new bilateral ideal index which satisfies the axiomatic and reversibility properties (base and factor) and may also be written as an expenditure ratio. All other formulas, which do not belong to the family of the proposed indices, violate some of the axiomatic properties. For multilateral comparisons, a mixed system of direct and indirect indices which satisfies the transitivity condition is proposed.     

关于Fuzzy度量空间的集值映射的公共不动点

本文绘出了Fuzzy度量空间的集值映射的几个新的公共不动点定理，它们是文[1]～[4]等最近的主要结果的改进、统一和发展．     

Usefulness of Asymptotic Distributions in the Classical Occupancy Problem

Approximations to the distribution of a discrete random variable originating from the classical occupancy problem are explored. The random variable X of interest is defined to be how many of N elements selected by or assigned to K individuals when each of the N elements is equally likely to be chosen by or assigned to any of the K individuals. Assuming that N represents the number of cells and each of the K individuals is placed in exactly one of the cells, X can also be defined as the number of cells occupied by the Kindividuals. In the literature, various asymptotic results for the distributions of X and (N ? X) are given; however, no guidelines are specified with respect to their utilization. In this article, these approximations are explored for various values of K and N, and rules of thumb are given for their appropriate use.     

Fuzzy Testing for One-Sided Process Capability Indices

Process capability indices are widely used to evaluate the performance of processes in the manufacturing industry. Over the years, the issues have been investigated extensively. Some articles have studied them with fuzzy estimation. However, it seems that no article has proposed a version of triangular fuzzy numbers for critical value to test the process capability. In this article, we use Buckley's approach (2003 Buckley , J. J. ( 2003 ). Fuzzy Probabilities: New Approach and Application . Heidelberg : Physica-Verlag .[Crossref] , [Google Scholar]) to construct the triangular fuzzy numbers for C _pl and C _pu , especially, the triangular fuzzy numbers for critical values are derived to execute the fuzzy hypothesis testing for C _pl and C _pu . Some numerical examples are taken to illustrate the proposed methodology.