This paper proposes a new type of random parameters AACD (RPAACD)
models, which extends the AACD model. Depending on the state of the price process, the
RPAACD models seem to be a valuable alternative to existing approaches and have the better
overall performance. We give the transition probability of the process. Moreover by
employing the transition probability, we obtain the probability properties of the RPACD
model.

The classical concentration inequalities deal with the deviations
of functions of independent and identically distributed (i.i.d.) random variables from their
expectation and these inequalities have numerous important applications in statistics and
machine learning theory. In this paper we go far beyond this classical framework by establish
two new Bernstein type concentration inequalities for -mixing sequence and uniformly
ergodic Markov chains. As the applications of the Bernstein's inequalities, we also obtain
the bounds on the rate of uniform deviations of empirical risk minimization (ERM) algorithms
based on -mixing observations.

The pricing problem of forward starting call options under a
Markov-modulated jump diffusion process is studied. Under the assumption that the dynamics
of risky asset follows a Markov-modulated jump diffusion process, the explicit analytical
formula of forward starting call options is obtained by the change of measure and no arbitrage
pricing theory. Moreover, the numerical results of option value are provided by the Monte
Carlo method, and the value of forward starting call options is compared when the risky asset
satisfies different financial models.

Kernel function method has been successfully used for the
estimation of a variety of function. By using the kernel function theory, an imputation
method based on Epanechnikov kernel and its modification were proposed to solve the
problem that missing data in compositional caused the failures of existing statistical
methods and the k-nearest imputation didn't consider the different contributions of
the k nearest samples when it used them to estimated the missing data. The experimental
results illustrate that the modified imputation method based on Epanechnikov kernel
get a more accurate estimation than k-nearest imputation for compositional data.

When the dependent variable is missing at random, the paper first
proposes the four causal effect estimation methods: propensity scores weighted method (PW),
improved propensity score weighted method (IPW), augmented propensity weighted estimator
(AIPW), regression estimator (REG) and proves the unbiasedness and consistency of the four
methods. The paper also proves that AIPW method is double robustness. The four methods are
compared when the missing ratio is in different level. It is illuminated that AIPW is more
precise and more efficient than other methods. Finally, the causal effect of the American
academy of child and adolescent welfare survey data is estimated with the four methods and
the results are reached that children accept drug intervention service show no more serious
behavior problems than the children who don't accept drug abuse services.

In this paper, we investigate the valuation of European-style call
options under an extended two-factor Markov-modulated stochastic volatility model, where the
first stochastic volatility component is driven by a mean-reversion square-root process and
the second stochastic volatility component is modulated by a continuous-time, finite-state
Markov chain. The inverse Fourier transform is adopted to obtain analytical pricing formulae.
Numerical examples are given to illustrate the discretization of the pricing formulae and
the implementation of our model.

A generalization of classical linear models is varying coefficient
models, which offer a flexible approach to modeling nonlinearity between covariates. A
method of local weighted composite quantile regression is suggested to estimate the
coefficient functions. The local Bahadur representation of the local estimator is derived
and the asymptotic normality of the resulting estimator is established. Comparing to the
local least squares estimator, the asymptotic relative efficiency is examined for the local
weighted composite quantile estimator. Both theoretical analysis and numerical simulations
reveal that the local weighted composite quantile estimator can obtain more efficient than
the local least squares estimator for various non-normal errors. In the normal error case,
the local weighted composite quantile estimator is almost as efficient as the local least
squares estimator. Monte Carlo results are consistent with our theoretical findings. An
empirical application demonstrates the potential of the proposed method.

The accelerated failure time model provides a natural formulation
of the effects of covariates on the failure time variable. The presence of censoring poses
major challenges in the semi-parametric analysis. The existing semi-parametric estimators
are computationally intractable. In this article we propose an unbiased transformation for
the potential censored response variable, thus least square estimators of regression
parameters can be gotten easily. The resulting estimators are consistent and asymptotically
normal. Based on these, we can get a strongly consistent K-M estimator for the distribution
of random error. Extensive simulation studies show that the asymptotic approximations are
accurate in practical situations.

This paper considers the optimal dividend and capital injection
strategies in the classical risk model with randomized observation periods. Assume that ruin
is prohibited. We aim to maximise the expected discounted dividend payments minus the expected
penalised discounted capital injections. We derive the associated Hamilton-Jacobi-Bellman
(HJB) equation and prove the verification theorem. The optimal control strategy and the
optimal value function are obtained under the assumption that the claim sizes are
exponentially distributed.