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 2016 Vol.32 Issue.6,Published 2016-12-14 article
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 551 Tracy-Widom Law with Applications SU Zhonggen Tracy and Widom found a new type of probability distribution in the study of high dimensional random matrices in the 1990s, which is nowadays normally called Tracy-Widom distribution. It is used to described the limiting distribution of the extremal eigenvalues in Gaussian Unitary Ensemble. Later on, the study in the past two decades indicates that Tracy-Widom distribution is universal like normal distribution and can be well used to describe a lot of seemingly distinct random phenomena. As illustrations, the paper briefly review nine widely studied random models, each of which is more or less related to Tracy-Widom distribution. Compared to normal distribution, Tracy-Widom distribution has horribly intricate distribution function, density function and moments. people need to use deep mathematical knowledge and advanced computation technology in order to extend and to apply Tracy-Widom distribution in practice. But it is absolutely worthy further study on account of its importance. 2016 Vol. 32 (6): 551-580 [Abstract] ( 308 ) [HTML 1KB] [ PDF 986KB] ( 516 )
 581 Nonparametric Estimation of the Integrated Volatility of Jump-Diffusion Processes with Noisy High-Frequency Data YE Xuguo, LIN Jinguan This paper studies nonparametric estimation of the integrated volatility of Poisson jump-diffusion processes with noisy high-frequency data. We propose jump-robust two-scale and multi-scale estimators. The estimators are based on a combination of the multi-scale method and threshold technique, which serves to remove microstructure noise and jumps, respectively. Furthermore, asymptotic properties of the proposed estimators, such as consistency, are established. 2016 Vol. 32 (6): 581-591 [Abstract] ( 185 ) [HTML 1KB] [ PDF 639KB] ( 526 )
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 592 On the Expected Penalty Functions in a Discrete Semi-Markov Risk Model LIU Haiyan; CHEN Mi This paper considers the expected penalty functions for a discrete semi-Markov risk model, which includes several existing risk models such as the compound binomial model (with time-correlated claims) and the compound Markov binomial model (with time-correlated claims) as special cases. Recursive formulae and the initial values for the discounted free penalty functions are derived in the two-state model by an easy method. We also give some applications of our results. 2016 Vol. 32 (6): 592-602 [Abstract] ( 116 ) [HTML 1KB] [ PDF 445KB] ( 610 )
 603 The Structure and Best-Possible Bounds of Random Variables which Degree of Radial Asymmetry is $\fn_jvn \100dpi \inline 3/n$  $\fn_jvn \100dpi \inline (n\geq 3)$ $\fn_jvn \100dpi \inline (n\geq 3)$[J]. CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS, 2016,32(6): 603-616>')" href="#"> ZHANG Xiaoyu, XU Fuxia We study the random variables of radial asymmetry based on copulas. We research on the structure of random variables which radial asymmetry degree is $\fn_jvn \100dpi \inline 3/4$and get the exact best-possible bounds of random variables which radial asymmetry degree is equal to $\fn_jvn \100dpi \inline 3/4$. Then we expand to general case. We propose an essential condition of radial asymmetry degree is $\fn_jvn \100dpi \inline 3/n$ and study the structure of copula. We provide a broad bounds of copula that the radial asymmetry degree is $\fn_jvn \100dpi \inline 3/n$. 2016 Vol. 32 (6): 603-616 [Abstract] ( 144 ) [HTML 1KB] [ PDF 701KB] ( 503 )
 617 Nonparametric Bayesian Analysis of the Constant Stress Accelerated Life Test with Power Function Model LIU Bin, SHI Yimin, CAI Jing, WANG Ruibing The linear accelerated model is often used to the statistical analysis of constant stress accelerated life test, whereas it does not relate well with the facts. By adopting the power functional accelerated model, the relationship of sample quantiles among different constant stress levels is obtained, which can lead to the estimations of the parameters in accelerated model and the characteristic coefficient vectors by virtue of the least square method, then the life-time data transformation between different stress levels can be operated. For complete data and censoring data, a Dirichlet process prior is introduced to gain the posterior distribution and the nonparametric Bayesian estimation of the reliability function, meanwhile, the consistency of the posterior estimators is proved. Finally, a real life example of Metal-Oxide-Semiconductor capacitors is analyzed to illustrate the effect of our model. 2016 Vol. 32 (6): 617-631 [Abstract] ( 120 ) [HTML 1KB] [ PDF 905KB] ( 551 )
 632 backward doubly stochastic differential equations YANG Xu A class of backward doubly stochastic differential equations driven by white noises and Poisson random measures are studied in this paper. The definitions of solutions and Yamada-Watanabe type theorem to this equation are established. 2016 Vol. 32 (6): 632-642 [Abstract] ( 133 ) [HTML 1KB] [ PDF 620KB] ( 528 )
 643 GMM Estimation of Marginal Model with Missing Covariate Data HE Jie; DUAN Xiaogang; ZHANG Shumei The multivariate response is commonly seen in longitudinal and cross-sectional design. The marginal model is an important tool in discovering the average influence of the covariates on the response. A main feature of the marginal model is that even without specifying the inter-correlation among different components of the response, we still get consistent estimation of the regression parameters. This paper discusses the GMM estimation of marginal model when the covariates are missing at random. Using the inverse probability weighting and different basic working correlation matrices, we obtain a series of estimating equations. We estimate the parameters of interest by minimizing the corresponding quadratic inference function. Asymptotic normality of the proposed estimator is established. Simulation studies are conducted to investigate the finite sample performance of the new estimator. We also apply our proposal to a real data of mathematical achievement from middle school students. 2016 Vol. 32 (6): 643-654 [Abstract] ( 111 ) [HTML 1KB] [ PDF 600KB] ( 426 )

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