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.

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.

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.

We study the random variables of radial asymmetry based on
copulas. We research on the structure of random variables which radial asymmetry
degree is and get the exact best-possible bounds of random variables which
radial asymmetry degree is equal to . Then we expand to general case. We propose
an essential condition of radial asymmetry degree is  and study the structure
of copula. We provide a broad bounds of copula that the radial asymmetry degree is
.

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.

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.

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.