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Huiqin Li,Zhidong Bai 한국통계학회 2015 Journal of the Korean Statistical Society Vol.44 No.1
In this paper, we study the convergence rates of empirical spectral distributions of largedimensional quaternion sample covariance matrices. Assume that the entries of Xn (p×n)are independent quaternion random variables with means zero, variances 1 and uniformlybounded sixth moments. Denote Sn = 1nXnX∗n. Using Bai’s inequality, we prove that theexpected empirical spectral distribution (ESD) converges to the limiting Marčenko–Pasturdistribution with the ratio of dimension to sample size yp = p/n at a rate of O n−1/2a−3/4n when an > n−2/5 or O n−1/5when an ≤ n−2/5, where an = (1 − √yp)2. Moreover, therates for both the convergence in probability and the almost sure convergence are alsoestablished. The weak convergence rate of the ESD is O n−2/5a−1/2n when an > n−2/5 orO n−1/5when an ≤ n−2/5. The strong convergence rate of the ESD is O n−2/5+ηa−1/2n when an > κn−2/5 or O n−1/5when an ≤ κn−2/5 for any η > 0 where κ is a positiveconstant.
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A new nonlinearity test to circumvent the limitation of Volterra expansion with application
Yongchang Hui,Wing-Keung Wong,Zhidong Bai,Zhen-Zhen Zhu 한국통계학회 2017 Journal of the Korean Statistical Society Vol.46 No.3
In this paper we study estimating the joint conditional distributions of bivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models we consider a class of time-varying transformation models and combine the two marginal models using Gaussian copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Gaussian copulas combined with time-varying transformation models may allow convenient and easy-to-interpret modeling for the joint conditional distributions for bivariate longitudinal data. We derive the asymptotic properties for the copula based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure and also investigate finite sample properties of our procedures through a simulation study.
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