A class of PP l-statistics for tests

Applying projection pursuit (PP) techniques, this paper takes L-statistics as projection indices to construct a class of statistics for multivariate tests. The asymptotic distributions of the test statistics under null hypothesis are derived theoretically. Some simple applications to location and dispersion problems are given.This project is supported by the National Natural Science Foundation of China.     

Characterizations of geometric distributions based on moments

Restricted to certain classes of discrete life distributions, and based on moments, conditional binomial moments, order statistics, spacing or record values, characterizations of geometric distributions are given.This project is supported by the National Natural Science Foundation of China Grant No. 1880492.     

Edgeworth Expansion of Densities of Order Statistics with Fixed Rank

We give an Edgeworth expansion for densities of order statistics with fixed rank k.The Edgeworth expansion for densities of extreme values is then obtained as a special case k=1.     

Hypothesis testing for arbitrarily varying source

Hypothesis testing for arbitrarily varying source (AVS), which is to decide between the two hypotheses for the varying behavior of the distribution of AVS, is considered in this paper. We determine the best asymptotic exponent of the second kind of error probability when the first kind of error probability is fixed. This result generalizes the well-known lemma of Stein in statistics. As a corollary, Strassen's coding theorem for AVS is obtained.Supported by the Young Teacher Foundation of Chinese Educational Ministry and by the National Natural Science Foundation of China.     

A test of conditional heteroscedasticity in time series

A new test of conditional heteroscedasticity for time series is proposed. The new testing method is based on a goodness of fit type test statistics and a Cramer-von Mises type test statistic. The asymptotic properties of the new test statistic is establised. The results demonstrate that such a test is consistent. Project supported by the National Natural Science Foundation of China (Grant No. 19231050) and Postdoctoral Fund of China.     

PP multivariate random weighting method

According to the Projection Pursuit (PP) method and the random weighting method, we propose a PP random weighting method, and set up the asymptotic distribution theory and strong limit theorem of PP random weighting empirical process. Applying this method, we obtain two kinds of goodness-of-fit test for a multivariate distribution function, i.e., we get the random weighting approximations of PP Kolmogorov Smirnov statistics (PPKS) and PP Smirnov Cramér Von Mises statistics (PPSC), we prove that the asymptotic distribution of PPKS and PPSC are the same as those of their respective random weighting approximations.Supported by the National Natural Science Foundation of China.     

ESTIMATING OF ORDER-RESTRICTED LOCATION PARAMETERS OF TWO-EXPONENTIAL DISTRIBUTIONS UNDER MULTIPLE TYPE- Ⅱ CENSORING

In this article, Bayes estimation of location parameters under restriction is broughtforth. Since Bayes estimator is closely connected with the first value of order statistics that canbe observed, it is possible to consider “complete data” method, through which the pseudo-value of first order statistics and pseudo-right censored samples can he obtained. Thus the results under Type- Ⅱ right censoring can be used directly to get more accurate estimators by Bayes method.     

Precise asymptotics for a type of order statistics

This paper obtains some results on the precise asymptotics for the order statistics generated by the random samples of maximum domain of attraction of the Fréchet distribution, which reveal the relations among the boundary function, weight function, convergence rate and limit position in a uniform form.Research supported by National Science Foundation of China (NO. 10271087).     

Limit theorems for generalized sequential rank statistics

Summary A general class of statistics based on sequential ranks is introduced. Under suitable regularity conditions, an almost sure representation and invariance principle are established for this class. In particular, it is shown that these statistics can obey invariance principles that are radically different from those obeyed by the usual full rank statistics.Research supported by the CSIR and the University of South AfricaResearch supported by the University of Delaware Research Foundation, Grant #8325530015     

Goodness-of-fit test statistics that dominate the Kolmogorov statistics

Summary Two statistics are proposed for the simple goodness-of-fit problem. These are derived from a general principle for combining dependent test statistics that has been discussed elsewhere by the authors. It is shown that these statistics are relatively optimal in the sense of Bahadur efficiency and consequently, are more efficient than any weighted Kolmogorov statistic at every alternative. A curious pathology occurs: Under certain alternatives, the sequence of statistics has a Bahadur efficacy or exact slope only in the weak sense of convergence in law.Work supported by US National Science Foundation Grant MCS76-82618     

DISTRIBUTION FREE LAWS OF THE ITERATED LOGARITHM FOR KERNEL ESTIMATOR OF REGRESSION FUNCTION BASED ON DIRECTIONAL DATA

The authors derive laws of the iterated logarithm for kernel estimator of regression function based on directional data. The results are distribution free in the sense that they are true for all distributions of design variable. Project supported by the National Natural Science Foundation of China (Nos. 19631040; 19971085), the Doctoral Program Foundation of the Ministry of Education of China and the Special Foundation of Academia Sinica.     

Estimations on dimensions of spaces of Jacobi forms

Estimations on dimensions of spaces of Jacobi forms are given by extending the classical Siegel theorem to the theory of Jacobi forms. Project supported by the China Postdoctoral Science Foundation and the National Natural Science Foundation of China.     

Positive Periodic Solutions of a Class of Non-autonomous Single Species Population Model with Delays and Feedback Control

With the help of a continuation theorem based on Gaines and Mawhin＇s coincidence degree, several verifiable criteria are established for the global existence of positive periodic solutions of a class of non-autonomous single species population model with delays （both state-dependent delays and continuous delays） and feedback control. After that, by constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotic stable positive periodic solution of a kind of nonlinear feedback control ecosystem are obtained. Our results extend and improve the existing results, and have further applications in population dynamics.     

Algebraic maps from spheres to spheres

Some homotopy classes of the unit spheres are explicitly represented by entire rational maps. Project supported by the National Natural Science Foundation of China (Grant No. 19531050), the Special Foundation of the Chinese Academy of Sciences, Hong Kong Qiu-Shi Foundation and the Education Foundation of Tsinghua University.     

Hierarchical structure of attractors of dynamical systems

Some basic properties of the small random perturbed dynamical system of Freidlin-Wentzell type are elicited. A hierarchy structure of attractors is constructed and some further properties of this structure are confirmed. Project supported by the National Natural Science Foundation of China and the Doctoral Program Foundation of IHE 863 Program, China.     

Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems

In this paper, a gap function for a system of vector equilibrium problems is introduced and studied. Some necessary and sufficient conditions for the system of vector equilibrium problems are established. Characterizations of the solutions set for the system of vector equilibrium problems are also derived. Furthermore, some existence results of solutions for the system of vector equilibrium problems are proved. This work was supported by the National Natural Science Foundation of China, the Youth Foundation, Sichuan Education Department of China, the National Natural Science Foundation, Sichuan Education Department of China (2004C018), and a grant from the National Science Council of ROC.     

On the rate of convergence ofL- andR-statistics under alternatives

Asymptotic distributions of test statistics under alternatives are important from the point of view of their power properties. When the limiting distributions of test statistics are specified under the hypothesis in a certain sense, LeCam's third lemma ([4], Chapter 6) enables one to obtain their limiting distributions under close alternatives. In this paper we generalize LeCam's third lemma by using the rate of convergence in the case of asymptotically efficient test statistics. A general lemma is proved which is specified to linear combinations of order statistics (L-statistics) and linear rank statistics (R-statistics). Edgeworth-type asymptotic expansions for these statistics under alternatives are considered in [3]. Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Rosenthal type inequalities forB-valued strong mixing random fields and their applications

Some inequalities for moments of partial sums of aB -valued strong mixing field are established and their applications to the weak and strong laws of large numbers and the complete convergences are discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19701011) and China Postdoctoral Science Foundation.     

Non-constant stationary solutions to a prey-predator model with diffusion

In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topological degree argument and the energy method, respectively. The first author was supported by the National Natural Science Foundation of China (No. 10771032) and the Natural Science Foundation of Jiangsu province BK2006088, and the second author was supported by National Natural Science Foundation of China (No. 10601011).