Strong Convergence of Empirical Distribution for a Class of Random Matrices

Let {vij}, i, j = 1, 2, …, be i.i.d, random variables with Ev11 = 0, Ev11^2 = 1 and a1 = （ai1,…, aiM） be random vectors with {aij} being i.i.d, random variables. Define XN =（x1,…, xk） and SN =XNXN^T,where xi=ai×si and si=1/√N（v1i,…, vN,i）^T. The spectral distribution of SN is proven to converge, with probability one, to a nonrandom distribution function under mild conditions.

Asymptotic Normality of the Empirical Distribution under Negatively Associated Sequences and its Applications

By the well-known large and small blocks parting method for dependent situations, we establish the asymptotic normality of the Empirical Distribution Function under Negatively Associated Sequences. As its application in reliablity problems, a natural estimate Fn（x） for the survival function F（x） = P（X 〉 x） is proposed, and the asymptotic normality of n^1/2 [Fn（x） - F（x）] is established.

Robust Variance Components Estimation in the PERG Mixed Distributions of Empirical Variances—PEROBVC Method

A mixed distribution of empirical variances, composed of two distributions the basic and contaminating ones, and referred to as PERG mixed distribution of empirical variances, is considered. In the paper a robust inverse problem solution is given, namely a (new) robust method for estimation of variances of both distributions—PEROBVC Method, as well as the estimates for the numbers of observations for both distributions and, in this way also the estimate of contamination degree.

Empirical investigation of topological and weighted properties of a bus transport network from China

Many bus transport networks （BTNs） have evolved into directed networks. A new representation model for BTNs is proposed, called directed-space P. The bus transport network of Harbin （BTN-H） is described as a directed and weighted complex network by the proposed representation model and by giving each node weights. The topological and weighted properties are revealed in detail. In-degree and out-degree distributions, in-weight and out-weight distributions are presented as an exponential law, respectively. There is a strong relation between in-weight and in-degree （also between out-weight and out-degree）, which can be fitted by a power function. Degree-degree and weight-weight correlations are investigated to reveal that BTN-H has a disassortative behavior as the nodes have relatively high degree （or weight）. The disparity distributions of out-degree and in-degree follow an approximate power-law. Besides, the node degree shows a near linear increase with the number of routes that connect to the corresponding station. These properties revealed in this paper can help public transport planners to analyze the status quo of the BTN in nature.

Red Tide Frequency Analysis Using the Extreme Value Theory

In recent years, the red tide erupted frequently, and caused a great economic loss. At present, most literatures emphasize the academic research on the growth mechanism of red tide alga. In order to find out the characters of red tide in detail and improve the precision of forecast, this paper gives some new approaches to dealing with the red tide. By the extreme values, we deal with the red tide frequency analysis and get the estimation of T-times red tide level U (T), which is the level once the consistence of red tide alga exceeds on the average in a period of T times.

Spatial and Temporal Variations of Climate in Europe

Temperature series of the original individual measurements of the minimum and maximum daily temperatures from 24 stations located in different regions of Europe are considered with the objective of studying the stability or the variability of the “climate” in time and space. The patterns of the temperature statistics, the shapes of probability density functions, in particular, at different places and times allow comparisons based on the Kolmogorov-Smirnov test, the Shannon entropy, and cluster analysis, used separately or in combination for the purposes of quantitative climate classification.

Development of EMC-based empirical model for estimating spatial distribution of pollutant loads and its application in rural areas of Korea

An integrated approach to easily calculate pollutant loads from agricultural watersheds is suggested and verified in this research. The basic concepts of this empirical tool were based on the assumption that variations in event mean concentrations（EMCs） of pollutants from a given agricultural watershed during rainstorms were only attributable to the rainfall pattern.Fifty one sets of EMC values were obtained from nine different watersheds located in the rural areas of Korea, and these data were used to develop predictive tools for the EMCs in rainfall runoff. The results of statistical tests of these formulas show that they are fairly good in predicting actual EMC values of some parameters, and useful in terms of calculating pollutant loads for any rainfall event time span such as daily, weekly, monthly, and yearly. This model was further checked in for its field applicability in a reservoir receiving stormwater after a cleanup of the sediments, covering 17 consecutive rainfall events from 1 July to 15 August in2007. Overall the predicted values matched the observed values, indicating the feasibility of this empirical tool as a simple and useful solution in evaluating timely distribution of nonpoint source pollution loads from small rural watersheds of Korea.

Empirical Analysis of Wealth Distribution Disparity of Chinese Residents and Its Changes

This article describes and analyses the disparity in wealth distribution of Chinese residents, on the basis of results of sample surveys from 1995 to 2002. It finds an obvious worsening trend, mainly due to the rapid widening of the urban-rural gap. The process of privatisation of urban housing has led to a narrowing of the wealth gap between urban residents, and has also led to a widening of the wealth gap between urban and rural residents, as well as the national wealth gap. As incomes from land decline, the relative weight of land value in rural residents＇ overall wealth possession has been decreasing and the role of land in narrowing the nation＇s urban-rural gap has been weakened. In addition, this article also holds that the financial assets of residents wiU play a greater role in enlarging the overall wealth distribution disparity.

Limiting spectral distribution of high-dimensional noncentral Fisher matrices and its analysis

Fisher matrix is one of the most important statistics in multivariate statistical analysis.Its eigenvalues are of primary importance for many applications,such as testing the equality of mean vectors,testing the equality of covariance matrices and signal detection problems.In this paper,we establish the limiting spectral distribution of high-dimensional noncentral Fisher matrices and investigate its analytic behavior.In particular,we show the determination criterion for the support of the limiting spectral distribution of the noncentral Fisher matrices,which is the base of investigating the high-dimensional problems concerned with noncentral Fisher matrices.

ASYMPTOTICS FOR THE DISTRIBUTION FUNCTION ESTIMATORS OF THE ERRORS IN SEMI-PARAMETRIC REGRESSION MODELS

This paper considers the convergence rates for nonparametric estimators of the error distribution in semi-parametric regression models. By establishing some general laws of the iterated logarithm, it shows that the rates of convergence of either the empirical distribution or a smoothed version of the empirical distribution function matches exactly the rates obtained for an independent sample from the error distribution.

Data-driven Stochastic Programming with Distributionally Robust Constraints Under Wasserstein Distance:Asymptotic Properties

Distributionally robust optimization is a dominant paradigm for decision-making problems where the distribution of random variables is unknown.We investigate a distributionally robust optimization problem with ambiguities in the objective function and countably infinite constraints.The ambiguity set is defined as a Wasserstein ball centered at the empirical distribution.Based on the concentration inequality of Wasserstein distance,we establish the asymptotic convergence property of the datadriven distributionally robust optimization problem when the sample size goes to infinity.We show that with probability 1,the optimal value and the optimal solution set of the data-driven distributionally robust problem converge to those of the stochastic optimization problem with true distribution.Finally,we provide numerical evidences for the established theoretical results.

On nonparametric change point estimator based on empirical characteristic functions

We propose a nonparametric change point estimator in the distributions of a sequence of independent observations in terms of the test statistics given by Huˇskov′a and Meintanis(2006) that are based on weighted empirical characteristic functions. The weight function ω(t; a) under consideration includes the two weight functions from Huˇskov′a and Meintanis(2006) plus the weight function used by Matteson and James(2014),where a is a tuning parameter. Under the local alternative hypothesis, we establish the consistency, convergence rate, and asymptotic distribution of this change point estimator which is the maxima of a two-side Brownian motion with a drift. Since the performance of the change point estimator depends on a in use, we thus propose an algorithm for choosing an appropriate value of a, denoted by a_s which is also justified. Our simulation study shows that the change point estimate obtained by using a_s has a satisfactory performance. We also apply our method to a real dataset.

On the Spectral Distribution of Gaussian Random Matrices

We consider the empirical spectral distribution （ESD） of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotaeh approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O（n^-1） to the Wigner distribution function uniformly on every compact intervals [u,v] within the limiting support （-1, 1）. Furthermore, the variance of the ESD for such an interval is proved to be （πn）^-2 logn asymptotically which surprisingly enough, does not depend on the details （e.g. length or location） of the interval, This property allows us to determine completely the covariance function between the values of the ESD on two intervals.

Free Probability Theory Based Event Detection for Power Grids Using IoT-enabled Measurements

Taking the advantage of Internet of Things(IoT)enabled measurements,this paper formulates the event detection problem as an information-plus-noise model,and detects events in power systems based on free probability theory(FPT).Using big data collected from phasor measurement units(PMUs),we construct the event detection matrix to reflect both spatial and temporal characteristics of power gird states.The event detection matrix is further described as an information matrix plus a noise matrix,and the essence of event detection is to extract event information from the event detection matrix.By associating the event detection problem with FPT,the empirical spectral distributions(ESDs)related moments of the sample covariance matrix of the information matrix are computed,to distinguish events from“noises”,including normal fluctuations,background noises,and measurement errors.Based on central limit theory(CLT),the alarm threshold is computed using measurements collected in normal states.Additionally,with the aid of sliding window,this paper builds an event detection architecture to reflect power grid state and detect events online.Case studies with simulated data from Anhui,China,and real PMU data from Guangdong,China,verify the effectiveness of the proposed method.Compared with other data-driven methods,the proposed method is more sensitive and has better adaptability to the normal fluctuations,background noises,and measurement errors in real PMU cases.In addition,it does not require large number of training samples as needed in the training-testing paradigm.

Discovering admissible Web services with uncertain QoS

Open and dynamic environments lead to inher- ent uncertainty of Web service QoS （Quality of Service）, and the QoS-aware service selection problem can be looked upon as a decision problem under uncertainty. We use an empiri- cal distribution function to describe the uncertainty of scores obtained from historical transactions. We then propose an approach to discovering the admissible set of services in- cluding alternative services that are not dominated by any other alternatives according to the expected utility criterion. Stochastic dominance （SD） rules are used to compare two services with uncertain scores regardless of the distribution form of their uncertain scores. By using the properties of SD rules, an algorithm is developed to reduce the number of SD tests, by which the admissible services can be reported pro- gressively. We prove that the proposed algorithm can be run on partitioned or incremental alternative services. Moreover, we achieve some useful theoretical conclusions for correct pruning of unnecessary calculations and comparisons in each SD test, by which the efficiency of the SD tests can be im- proved. We make a comprehensive experimental study using real datasets to evaluate the effectiveness, efficiency, and scal- ability of the proposed algorithm.

LARGE SAMPLE PROPERTIES OF THE SIR IN CDMA WITH MATCHED FILTER RECEIVERS

The output signal-to-interference (SIR) of conventional matched filter receiver in random environment is considered. When the number of users and the spreading factors tend to infinity with their ratio fixed, the convergence of SIR is showed to be with probability one under finite fourth.moment of the spreading sequences. The asymptotic distribution of the SIR is also obtained.

Prediction of rice productivity in Kandhamal district of Orissa using simulation modeling

Simulation technique has been employed to predict rice yield of Kandhamal plateaus in Orissa(India)using the data of previous years.Preliminary simulation model has been developed The test for uniformity and independence has been conducted using Kolmogrov–smironov test and auto-correlation test,respectively.The result obtained has been subjected to testing of hypothesis by using two sided test.Data for five years(1995 to 2000)are used for model validation and the sample size is increased to 12 years i.e.,from 1995 to 2007 for prediction up to 2012.Sensitivity analysis is conducted by changing the parameters within feasible limits to find out the effect on the model.

Imprecise Reliability Assessment for Heavy Numerical Control Machine Tools Against Small Sample Size Problem

Small sample size problem is one of the main problems that heavy numerical control(NC) machine tools encounter in their reliability assessment. In order to deal with the small sample size problem, many indirect reliability data such as reliability data of similar products, expert opinion, and engineers' experience are used in reliability assessment. However, the existing mathematical theories cannot simultaneously process the above reliability data of multiple types, and thus imprecise probability theory is introduced. Imprecise probability theory can simultaneously process multiple reliability data by quantifying multiple uncertainties(stochastic uncertainty,fuzzy uncertainty, epistemic uncertainty, etc.) together. Although imprecise probability theory has so many advantages, the existing natural extension models are complex and the computation result is imprecise. Therefore,they need some improvement for the better application of reliability engineering. This paper proposes an improved imprecise reliability assessment method by introducing empirical probability distributions to natural extension model, and the improved natural extension model is applied to the reliability assessment of heavy NC machine tool spindle to illustrate its effectiveness.

Theoretical study of fusion reactions ^(32)S + ^(94,96)Zr and ^(40)Ca + ^(94,96)Zr and quadrupole deformation of ^(94)Zr

The dynamic coupling effects on fusion cross sections for reactions^(32)S + ^(94,96)Zr and ^(40)Ca + ^(94,96)Zr are studied with the universal fusion function formalism and an empirical coupled channel(ECC) model. An examination of the reduced fusion functions shows that the total effect of couplings to inelastic excitations and neutron transfer channels on fusion in ^(32)S +^(94)Zr(^(40)Ca +^(94)Zr) is almost the same as that in ^(32)S +^(96)Zr(^(40)Ca +^(96)Zr). The enhancements of the fusion cross section at sub-barrier energies due to inelastic channel coupling and neutron transfer channel coupling are evaluated separately by using the ECC model. The results show that effect of couplings to inelastic excitations channels in the reactions with94 Zr as target should be similar as that in the reactions with ^(96) Zr as target. This implies that the quadrupole deformation parameters β_2of ^(94)Zr and^(96) Zr should be similar to each other.However, β_2 's predicted from the finite-range droplet model, which are used in the ECC model, are quite different. Experiments on^(48) Ca +^(94)Zr or^(36) S +^(94)Zr are suggested to solve the puzzling issue concerning β_2for^(94)Zr.