To study some infinite-dimensional subject (the phase transitions
in statistical physics, for instance), several mathematical tools are developed. One of
them is the speed estimation of various stabilities/instabilities. This paper collects
some unexpected, unified, nearly sharp basic estimates of various types of
stability/instability for the simplest class of Markov processes, the birth-death processes.
Some motivations and a part of extensions are also discussed. The paper is based on a talk
presented recently in several international conferences.
The local limit theorems for the minimum of a random walk with
Markovian increments is given, with using Presman's factorization theory. This result
implies the asymptotic behaviour of the survival probability for a critical branching
process in Markovian depended random environment.
The Bayes estimators of variance components are derived under
weighted square loss function for the balanced one-way classification random effects
model with the assumption that variance component has the conjugate prior distribution.
The superiorities of the Bayes estimators for variance components to traditional ANOVA
estimators are studied in terms of the mean square error (MSE) criterion. Finally, a
remark for main results is given.
In this paper, we mainly studied the limit properties for the
countable nonhomogeneous Markov chains. We established some limit properties for the
functions of the countable nonhomogeneous Markov chains with variables under the
convergence in the sense, which extended the similar conclusions for the
functions with two variables. At last, as a corollary, we given the similar result in
the homogeneous Markov stock market.
In this paper, we investigate a robust optimal portfolio and
reinsurance problem under inflation risk for an ambiguity-averse insurer (AAI), who worries
about uncertainty in model parameters. We assume that the AAI is allowed to purchase
proportional reinsurance and invest his/her wealth in a financial market which consists of
a risk-free asset and a risky asset. The objective of the AAI is to maximize the minimal
expected power utility of terminal wealth. By using techniques of stochastic control theory,
closed-form expressions for the value function and optimal strategies are obtained.
In this paper, considering of the special geometry of compositional
data, two new methods for estimating missing values in compositional data are introduced. The
first method uses the mean in the simplex space which mainly finds the -nearest neighbor
procedure based on the Aitchison distance, combining with two basic operations on the simplex,
perturbation and powering. As a second proposal the principal component regression imputation
method is introduced which initially starts from the result of the proposed the mean in the
simplex. The method uses ilr transformation to transform the compositional data set, and then
uses principal component regression in a transformed space. The proposed methods are tested
on real data and simulated data sets, the results show that the proposed methods work well.