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In this paper, we establish maximal inequalities, exponential
inequalities and Marcinkiewicz-Zygmund inequality for partial sum of random variables
which are independent on an upper expectation space. As applications, we give the complete
convergence for the partial sum of independent random variables on upper expectation space.
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The relative efficiency of the weighted mixed estimator with respect to
least squares estimator is discussed in this paper. We also give the lower and upper bounds
of those relative efficiencies. Finally, we give a numerical example to illustrate the
theoretical results.
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Nonparametric quantile regression with multivariate covariates
is a difficult estimation. To reduce the dimensionality while still retaining the
flexibility of nonparametric model, the single-index regression is often used to model
the conditional quantile of a response variable. In this paper, we focus on the variable
selection aspect of single-index quantile regression. Based on the minimized average
loss estimation (MALE), the variable selection is done by minimizing the average loss
with SCAD penalty. Under some mild conditions, we demonstrate the oracle properties
about SCAD variable section of single-index quantile regression. Furthermore, the
algorithm of the variable selection of SCAD penalized quantile regression is given.
Some simulations are done to illustrate the performance of the proposed methods.
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In this paper, let be an array of rowwise
-mixing random variables. The limiting behavior of weighted sums for arrays of
rowwise -mixing random variables is studied and some new complete convergence
results are obtained, which generalize and improve the corresponding earlier theorems.
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VaR measure has important applications in finance and insurance
practice. In this paper, the Bayesian models are established. Under some loss function,
the Bayeian estimate of VaR is derived. In addition, we prove the strongly consistency
and asymptotic normality for the Bayesian estimation of VaR under exponential-Gamma model.
Finally, the numerical simulation is done to verify the convergence rate of the estimate
of VaR with different sample sizes.
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A new class of slash distribution is studied for analyzing
nonnegative data. This distribution is defined by means of a stochastic representation
as the mixture of a half normal random variable with the power of an exponential random
variable. Density function and properties involving hazard function, moments and moment
generating function are derived. The usefulness and flexibility of the proposed
distribution are illustrated through a real application by maximum likelihood procedure.
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Regression variable subset selection is one of the most important
aspects in linear model theory. If the selected subset is consistent when the sample size
tends to infinity, and the prediction mean square error is small, then the selection method
is preferred. The BIC criterion can give consistent subset, but as the number of variables
get large, it involves too much computation. The adaptive lasso has better computational
efficiency, while keeping consistency. In this paper we propose a new approach for multiple
linear regression variable selection, which is much simpler than the other variable
selection methods, while it gives consistent subset. The new method only compute two passes
of ordinary least squares regressions, the first pass computes a complete set regression,
selects a variable subset based on the regression coefficient estimates, then the second
pass regresses on the selected variables.
Consider the following regression model:
where the indexes of the non-zero elements of are denoted by
, and
suppose the new method gives a regression variable subset indexed by , and
is the regression coefficient estimate using our new method, in which
the coefficients of the dropped out variables are defined to be zero. We proved that under
suitable conditions
where denotes the vector composed of the
elements of indexed by
,
is the error
variance, are matrix and constant relying on the limit of
.
Simulation result and application examples show that the new approach have good small
to medium sample performance, which is comparable to the other methods such as BIC, adaptive
lasso.
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This article considers Bayesian inference of the linear regression
model with one change point in observations, provided that the prior distribution of the change
point is the beta-binomial distribution or the power prior introduced by Ibrahim et al. (2003)
and the variances of the observations on two sides of the change point are the same. We get
closed forms of the posterior distributions of the change point, the regression coefficients
and the common variance. This not only generalizes the result of Ferreira (1975) from the the
discrete uniform prior distribution of the change point t to the beta-binomial distribution
which can well describe the shape of the change point distribution, but also can be further
generalized to the power prior distribution of the change point, which included the historical
information. Simulation shows higher performance or accuracy of the Bayesian method when the
change point follows the beta-binomial and power prior.
