Let {Xt,t ≥ 0} be a Levy process with Levy measure v on （-∞,∞）, and let τ be a nonnegative random variable independent of {Xt,t ≥ 0}. We are interested in the tail probabilities of Xτ and X（τ） = sup0≤t≤τ...Let {Xt,t ≥ 0} be a Levy process with Levy measure v on （-∞,∞）, and let τ be a nonnegative random variable independent of {Xt,t ≥ 0}. We are interested in the tail probabilities of Xτ and X（τ） = sup0≤t≤τ Xt. For various cases, under the assumption that either the Levy measure v or the random variable T has a heavy right tail we prove that both Pr（XT 〉 x） and Pr（X（τ） 〉 x） are asymptotic to ETv（（x, ∞）） ＋ Pτ（τ 〉 x/（0 V EX1）） as x → ∞, where Pr（τ 〉 x/x） = 0 by convention.展开更多
The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis.It is well known that the number of c...The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis.It is well known that the number of classes varying with sample size in the test has attached more and more attention.However,in this situation,there is not theoretical results for the asymptotic property of such chi-squared test statistic.This paper proves the consistency of chi-squared test with varying number of classes under some conditions.Meanwhile,the authors also give a convergence rate of KolmogorovSimirnov distance between the test statistic and corresponding chi-square distributed random variable.In addition,a real example and simulation results validate the reasonability of theoretical result and the superiority of chi-squared test with varying number of classes.展开更多
This paper studies the autoregression models of order one, in a general time series setting that allows for weakly dependent innovations. Let {Xt} be a linear process defined by Xt =∑k=0^∞ψ kεt-k, where {ψk, k ≥...This paper studies the autoregression models of order one, in a general time series setting that allows for weakly dependent innovations. Let {Xt} be a linear process defined by Xt =∑k=0^∞ψ kεt-k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0, ±1, ±2,...} is a sequence of random variables. Two results are proved in this paper. In the first result, assuming that {εk, k ≥ 1} is a sequence of asymptotically linear negative quadrant dependent （ALNQD） random variables, the authors find the limiting distributions of the least squares estimator and the associated regression t statistic. It is interesting that the limiting distributions are similar to the one found in earlier work under the assumption of i.i.d, innovations. In the second result the authors prove that the least squares estimator is not a strong consistency estimator of the autoregressive parameter a when {εk, k ≥ 1} is a sequence of negatively associated （NA） random variables, and ψ0 = 1, ψk = 0, k ≥ 1.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos. 10971157,11001209,70871104)the Scientific Research Foundation for the Returned Overseas Chinese Scholars
文摘Let {Xt,t ≥ 0} be a Levy process with Levy measure v on （-∞,∞）, and let τ be a nonnegative random variable independent of {Xt,t ≥ 0}. We are interested in the tail probabilities of Xτ and X（τ） = sup0≤t≤τ Xt. For various cases, under the assumption that either the Levy measure v or the random variable T has a heavy right tail we prove that both Pr（XT 〉 x） and Pr（X（τ） 〉 x） are asymptotic to ETv（（x, ∞）） ＋ Pτ（τ 〉 x/（0 V EX1）） as x → ∞, where Pr（τ 〉 x/x） = 0 by convention.
基金supported by the Natural Science Foundation of China under Grant Nos.11071022,11028103,11231010,11471223,BCMIISthe Beijing Municipal Educational Commission Foundation under Grant Nos.KZ201410028030,KM201210028005Jishou University Subject in 2014(No:14JD035)
文摘The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis.It is well known that the number of classes varying with sample size in the test has attached more and more attention.However,in this situation,there is not theoretical results for the asymptotic property of such chi-squared test statistic.This paper proves the consistency of chi-squared test with varying number of classes under some conditions.Meanwhile,the authors also give a convergence rate of KolmogorovSimirnov distance between the test statistic and corresponding chi-square distributed random variable.In addition,a real example and simulation results validate the reasonability of theoretical result and the superiority of chi-squared test with varying number of classes.
基金supported by the National Natural Science Foundation of China under Grant Nos.10971081 and 11001104985 Project of Jilin University
文摘This paper studies the autoregression models of order one, in a general time series setting that allows for weakly dependent innovations. Let {Xt} be a linear process defined by Xt =∑k=0^∞ψ kεt-k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0, ±1, ±2,...} is a sequence of random variables. Two results are proved in this paper. In the first result, assuming that {εk, k ≥ 1} is a sequence of asymptotically linear negative quadrant dependent （ALNQD） random variables, the authors find the limiting distributions of the least squares estimator and the associated regression t statistic. It is interesting that the limiting distributions are similar to the one found in earlier work under the assumption of i.i.d, innovations. In the second result the authors prove that the least squares estimator is not a strong consistency estimator of the autoregressive parameter a when {εk, k ≥ 1} is a sequence of negatively associated （NA） random variables, and ψ0 = 1, ψk = 0, k ≥ 1.
