In this paper, we first extend the
definitions of matrix $F$ and $t$ distributions to the left
spherical distribution family, prove the density functions have no
relation with the one producing them and then show that discuss the
elliptically contoured distributions are invariant under nonsingular
matrix transformations. These distributions include the matrix Beta,
inverse Beta, Dirichlet, inverse Dirichlet, $F$ and $t$ etc. And
finally it is shown that their distribution density functions not
only have no relation with the density function generating them but
also the transformation matrix.

Semiparametric Archimedean copulas, which have a fexible
dependence structure because of the special way constructed by using
the existing archimedean generator, can describe the dependence
structure between the financial data auto-adaptively. The empirical
results on the exchange rate market suggest that the semiparametric
Archimedean copula is more flexible than the other three copulas,
and is suggestive when selecting copulas.

This paper obtains some equivelent conditions
for a type of convolution closure of local subexponential
distributions on $[0,\infty)$, which are also valid for
distributions on $(-\infty,\infty)$ under certain conditions. On the
basis of these results, the local asymptotics for the distribution
of symmetrization are given. The results above include the
corresponding, non-local results of Embrechts and Goldie
(1980)\ucite{1} and Geluk (2004)\ucite{2}. Some of our proofs are
more simple than those of Geluk (2004)\ucite{2}.

The estimation of semivarying coefficient
models are studied in this paper. The estimators of the function
coefficient and the constant coefficient are given by modifying the
profile least squares. Furthermore, the asymptotical normalities of
these estimations are investigated. A simulation study is carried
out to compare the proposed methods.

Expected Shortfall (ES) is one of the most
popular tools of risk management for financial property, and is an
ideal coherent risk measure. In this paper, we discuss the two-step
kernel estimator of ES under polynomial decay of strong mixing
coefficients of time series. The first step is the kernel estimator
of VaR (Value at Risk) and the second step is the kernel estimator
of ES. We obtain Bahadur representation of the kernel estimator of
ES. Then, we give the mean squares error and the rate of the
asymptotic normality.

Combined with semiparametric regression model
and structural change model with unknown change point, a new
regression model --- semiparametric regression model with structural
change --- has been proposed. The weighted least squares estimator
of parameter $\beta,\beta^\ast,\gamma,k$ and the kernel estimator of
$f(t)$ are given, and $\sqrt{n}$-consistency properties of
$\beta,\beta^\ast,\gamma$'s estimator is proved. Also, the
estimator's strong consistency properties and some test problem
about the model are discussed. And, at last, we give a simulation
study to verify the superiority of our new model.

This paper study the duality of heterogeneous
coagulation-fragmentation process (HCFP) which models the
coagulation, fragmentation and diffusion of clusters of particles on
lattice. The closed form of stationary distribution for HCFP is
obtained, and then the integrated form of BBGKY hierarchy of HCFP is
given.

Let $\{X_{i}\}^{\infty}_{i=1}$ be a standardized
non-stationary Gaussian sequence, and the exceedances point process
$N_{n}$ be the exceedances of level $\mu_{n}(x)$ formed by
$X_{1},X_{2},\cdots,X_{n}$, $r_{ij}=\ep X_{i}X_{j}$,
$S_{n}=\tsm_{i=1}^{n}X_{i}$. Under some conditions, the asymptotic
independence of $N_{n}$ and $S_{n}$ is obtained.

In this paper, the mean-value estimation
problem with surrogate data and validation sampling is considered. A
regression calibration kernel function is defined to incorporate the
information contained in both surrogate variates and validation
sampling. The proposed estimators are proved to be asymptotically
normal and convergent rate.

We extend the comparison method and present
a new method to derive sharp closed-form semiparametric bounds on
small value probability $\pr(X\leq t)$, where $X\in[0,M]$ is a
random variable with $\ep X=m_1$ and $\ep X^2=m_2$ fixed. The proofs
of our results are elementary.

We provide marginal coordinate tests based on
two competing Principal Hessian Directions (PHD) methods. Predictor
contributions to central mean subspace can be effectively identified
by our proposed testing procedures. PHD-based tests avoid choosing
the number of slices, which is a well-known shortcoming of similar
tests based on Sliced Inverse Regression (SIR) or Sliced Average
Variance Estimation (SAVE). The asymptotic distributions of our test
statistics under the null hypothesis are provided and the
effectiveness of the new tests is illustrated by simulations.}
\newcommand{\fundinfo}{The first and corresponding authors were supported
by National Social Science Foundation (08CTJ001).