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In this paper, we investigate optimal estimator of regression
coefficient in a general Gauss-Markov model under balanced loss function. Firstly,
necessary and sufficient conditions for linear estimators to be best linear unbiased
estimator (BLUE) are provided. Secondly, we prove the best linear unbiased estimator
is unique in the sense of almost everywhere, and also a balance between least squares
estimator and optimal estimator under quadratic loss. Thirdly, loss robustness of the
optimal estimator is discussed in terms of relative losses and relative saving losses.
Finally, we give some conditions about the robust BLUE on the mis-specification of
covariance matrix.
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In this paper, we first study the strong convergence theorem for
finite two-order nonhomogeneous Markov chains indexed by an two rooted Cayley tree, then we
obtain the strong law of large numbers for this Markov chains. Finally, we obtain the
Shannon-McMillan theorem with a.e. convergence for an two-order nonhomogeneous Markov chain
indexed by an two rooted Cayley tree.
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Under the symmetric loss functions, by means of progressively
first-failure-censored samples, the paper studies the uniformly minimum variance unbiased
estimation (UMVUE), Bayes estimation and parametric empirical Bayes estimation (PEB) based
on two-parameter Pareto distribution. According to the code of mean squared error (MSE),
the gradualism of parametric Bayes and PEB estimation is investigated by applying risk
function and by comparing the optimal property between UMVUE and PEB estimations to obtain
their convergence rate. Based on the same confidence level, parametric interval estimation
in classical and Bayes statistics is analyzed. A conclusion can be made that the precision
of interval estimation in Bayes statics is higher than that in classical statics by means
of numerical simulation.
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In this paper, we consider the partial functional linear regression
model, and construct the empirical log-likelihood ratio statistic for the unknown regression
parameter. It is shown that the proposed statistics have the asymptotic standard chi-square
distribution, and hence they can be used to construct the confidence region of the parameter.
In addition, the maximum empirical likelihood estimator of the unknown coefficient function
is constructed, and its asymptotic behavior is proved. A simulation study and real data
analysis are carried out to compare the proposed methods with the least-squares method in
terms of the confidence regions and its coverage probabilities.
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In this paper, the problem of pricing Asian options in double
stochastic jump-diffusion is researched. Firstly a double stochastic jump-diffusion model
is introduced. Secondly the inherently path dependent problem of pricing Asian options can
be eliminated by measure change. In the end the integro-differential equation that the
price of a Asian option must satisfy is given. The equation can be numerically solved and a
referred price can be got for investor.
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In this paper, we investigate the precise large deviations for
a sum of claims in compound renewal risk model with negative dependence structure, in which
we assume that is a sequence of negative dependence rv's with distribution
functions and the average of right tails of distribution functions
is equivalent to some distribution function with consistently varying tails. We try to
build a platform for the classical large deviation theory and for the compound renewal
risk model.
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In this paper, the singular stochastic partial differential
equation with an unknown parameter and a small noise is studied. The maximum likelihood
estimator of the parameter based on the continuous observation of the Fourier
coefficients is proposed. The strong convergence and asymptotic normality of the
estimator are established as the noise tends to zero.
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In this paper, we discuss some important properties of T-IPH
distribution, where T-IPH denotes the infinite phase type distribution defined on a
discrete-time birth and death process with countably many states. For the distribution,
we firstly give the probability generating function (PGF), then the obtained results
enable us to further give the recursive formula to calculating numerical results of
distributional law and factorial moment for the distribution. Finally, we also give
an application of T-IPH distribution in queueing systems.
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In this paper, we first define the concept of approximately periodic
time series, that is, the length of their periods is not any constant. For example, the sunspot
series has a period about 11 years, but the length of its periods is not just 11 years, which
is approximately periodic. Then we give a method to extract approximately periodic trend and
bring forward a generalized difference operator, which can eliminate not only time trend and
periodicity but also approximately periodicity of time series. At last, we take the sunspot
data as an example to show the application of generalized difference operator.
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The note begins with a short story on seeking for a practical
sufficiency theorem for the uniqueness of time-continuous Markov jump processes, starting
around 1977. The general result was obtained in 1985 for the processes with general state
spaces. To see the sufficient conditions are sharp, a dual criterion for non-uniqueness
was obtained in 1991. This note is restricted however to the discrete state space (then the
processes are called Q-processes or Markov chains), for which the sufficient conditions
just mentioned are showing at the end of the note to be necessary. Some examples are included
to illustrate that the sufficient conditions either for uniqueness or for non-uniqueness are
not only powerful but also sharp.
