In this paper, L_\gamma penalty method proposed by Frank and Friedman\ucite{3} is used to study the variable selection problem and its asymptotic properties based on the penalized quasi-likelihood method in generalized linear models with adaptive designs. This method can perform parameter estimation and variable selection simultaneously. For 0<\gamma<1, the existence, consistency and Oracle properties of the estimators based on L_\gamma penalty and the quasi-likelihood method in generalized linear models with adaptive designs are proved under appropriate conditions. These results generalize the related theories of generalized linear models from the case of fixed designs to the case of adaptive designs. The validity of our obtained theory is verified by numerical simulation and real data analysis inthis paper.

In this paper, we study a transformation of Dirichlet forms. We obtain the sufficient conditions ensuring the transformed bilinear forms is quasi-regular Dirichlet forms. We separately study about the Dirichlet form for a type of second order differential operator and jump measure, and obtain the relationship between the Markov processes which is corresponding to quasi-regular Dirichlet type before
and after the transformation.

In survival studies, the accelerated failure time (AFT) model is often applied to predict the event times. This article proposes a multiple-response AFT model that extends the AFT model to the multiple events case. It is assumed that the covariates are high-dimensional and the regression coefficient matrix is jointly low-rank and sparse. We also assume all the multivariate event times are subject to right-censoring by a common censoring variable. To estimate the coefficient matrix, a two-stage procedure is proposed. First weight the data with IPCW weights, and then use SESS algorithm to solve a sparse reduced-rank regression problem. The simulation results show that the proposed method performs well in many cases. The method is also applied to a real dataset of bone marrow transplant patients.

Temporal network can better describe the dynamic evolution properties of topology construction among nodes in complex network.
Taking into account the mutual influence of nodes in different time layers, and being inspired by inter-layer temporal correlation coupling relationship in multilayer network, this paper proposes a kind of temporal network with coupling based on vector autoregressive (VAR) model. How to construct temporal network is given and the empirical application in four stock markets including NDX100, S\&P 500, ZSSE100 and SSE180 is explored. Compared with the literature such as 15 and 16, the study shows that the proposed new network exhibits more obvious advantages on resolution power of nodes (stocks) importance ranking and the in-sample and out-sample performance of portfolio optimization. Meanwhile, this paper also discusses how to determine ``peripheral'' stocks on the basis of the nodes importance sequence. It is apparent that our study could further enrich temporal network theory and provide new technical tools for financial market research.

Within-subject correlation and correlation among variables are two inherent characteristics of longitudinal datasets, which contain lots of important data information. In order to use these two kinds of correlation for dimension reduction, in this paper, we propose a sufficient dimension folding method based on martingale difference divergence in the spirit of dimension folding of matrix-valued data. It can be shown that the method can find the central mean dimension folding subspace in the population level, and can reduce the dimensions of both predictors and observation times simultaneously. Further, the estimated basis directions ensures the root-n consistency. To implement the proposed method, the Kronecker product assumption is introduced, so that the process can be transformed to a constrained low-dimensional optimization problem, which can be quickly solved by exisiting nonlinear optimization algorithms. Furthermore, a consistent BIC criterion is proposed to determine the structural dimension. Simulation studies show that the proposed method is efficient and can have higher accuracy on subspace estimation and structural dimension determination. Finally, an application on primary biliary cirrhosis data is used to illustrate the effectiveness of the proposed method. 

We review some results on the recent development of statistical inference for high-dimensional linear models. We introduce three debiased LASSO estimators, which are asymptotically normal and thus we can construct statistical inference for low dimensional parameters in high-dimensional setting. In addition, we give a brief introduction to the bootstrap assistant procedures to conduct simultaneous inference based on the debiased LASSO.

Based on generalized premium principle, the paper studies the optimal reinsurance design with default risk by minimizing VaR
measure of the total risk exposure in insurance company. Suppose that the premium principle satisfies the distribution invariance, risk loading and the stop-loss ordering preserving. We obtain the general form of optimal reinsurance strategies, that is, the layer reinsurance strategy. Especially, the expression of optimal reinsurance are provided under the distortion and Dutch premium principles. Finally, two examples are given to illustrate the above methods.

This paper studies the optimal investment strategy for a defined contribution pension plan with multiple risks under the mean-variance criterion. In the pension accumulation stage, the level of interest rate, volatility level and wage level are considered to be random, in addition, it is assumed that the term structure of interest rate is driven by the stochastic affine interest rate model, and the stock price is modeled by Heston's stochastic volatility model. By using the principle of stochastic dynamic programming and Lagrangian duality theorem we obtain the explicit solutions for the efficient strategy and the efficient frontier. A numerical example is given to analyze the effects of interest rate parameters, volatility parameters, and salary parameters on efficient strategies and efficient frontiers. Research results show that the capital market line with interest rate risk, volatility risk and salary risk environments is still a straight line in a mean-standard deviation plane. That is to say, the efficient strategy only depends on both instantaneous interest rate level and instantaneous salary level, while the efficient frontier does not depend on the interest rate level, volatility level and salary level.

Partial functional linear regression model refers to a type of regression machine that contains mixed functional and numerical data at the input and numerical data at the output. In the existing partial function linear regression machine estimation algorithm, it is assumed that functional data sample follow independent and identical distribution, which is inconsistent with the dependent characteristics of functional data in the financial and other fields. Therefore, the article first proposes two data-driven functional principal components representation methods for function data, then the regression coefficient function is regularized, and finally the estimation of the partial functional linear regression machine is transformed into the estimation of the multiple linear regression machine. The Monte Carlo simulation results show that the methods proposed in this paper have smaller parameter estimation errors and higher out-of-sample prediction accuracy when dealing with dependent data, the case analysis also shows the effectiveness in stock forecasting.

In this paper we establish the existence and uniqueness of strong solution to a stochastic partial differential equation driven by Gaussian colored noise and with gradient in drift and diffusion terms and non-Lipschitz coefficients

The proposed power-power loss function which has balanced convergence rates or penalties for its argument too large and too small, has all the seven properties listed in this paper, and thus it is recommended to use for the positive restricted parameter space. We then calculate the Bayes estimator, the posterior risk, the integrated risk, and the Bayes risk of the parameter under the power-power loss function. After that, we analytically calculate these quantities under a hierarchical normal and normal-inverse-gamma model. Finally, the numerical simulations
exemplify our theoretical studies.

In this paper, we proposed a statistical framework for optimal treatment selection for a subgroup of patients, using their biomarker values based on casual inference. This new method was based on a concept, called conditional average treatment effect (CATE) curve, and CATE curve's simultaneous confidence bands (SCBs), which could be used to represent the average treatment effect for a given value of the covariate (biomarker) and to select an optimal treatment for one particular patient. We then proposed B-splines methods for estimating the CATE curves and constructing simultaneous confidence bands for the CATE curves. We derived the asymptotic properties of the proposed methods. We also conducted extensive simulation studies to evaluate finite-sample properties of the proposed simultaneous confidence bands. Finally,  we illustrated the application of the CATE curve and its simultaneous confidence bands in optimal treatment selection in a real-world data set.

With the diversification and complexity of data, functional ANOVA model has been studied by more and more scholars and applied to all walks of life. In order to reduce the sensitivity of the model to outliers, we figure out a dependent robust functional ANOVA model with t process for dependent multivariate response variables. To depict dependence between variables and ensure the positive definiteness of the covariance function, random effect variable and convolution method are introduced to construct a covariance function in this paper. In addition, the model introduces the random effect function to describe the individual characteristics of the research object. Statistical properties, such as robustness and information consistency, are also studied, and the feasibility of the model is verified by numerical simulation and example analysis.

In this paper, we study the finite-time ruin problems in the perturbed compound Poisson risk model. The finite-time Gerber-Shiu discounted penalty function and its decomposition are studied. Different from the Laplace transform method, we propose a novel method for computing the finite-time Gerber-Shiu functions by the Laguerre series expansion. When the individual claim size density function is a finite combination of exponentials,we derive the infinite series expansions for the Gerber-Shiu functions. Some numerical examples are also given to confirm the applicability of our method.

Let $\{X,X_n,n\ge1\}$ be a sequence of i.i.d. random variables taking values in a real separable Hilbert space $(\mathbb{H},\|\cdot\|)$ with covariance operator $\Sigma$, set $S_n=X_1+X_2+\cdots+X_n$, $n\ge 1$. For every $m>0$ and $a_n=O((\ln\ln n)^{-2m})$, we study the precise rates in the generalized law of the iterated logarithm for a kind of weighted infinite series of $\pr\{\|S_n\|\ge(\epsilon+a_n)\sigma\sqrt{n}(\ln\ln n)^m\}$. Let $\beta_n(\epsilon)=o(\sqrt{1/\ln\ln n})$. We also prove that, for any $r>1$ and $\alpha>-d/2$, \begin{align*} &\lim_{\epsilon\searrow\sqrt{r-1}}[\epsilon^2-(r-1)]^{\alpha+d/2} \tsm_{n=1}^{\infty}\frac{1}{n}(\ln n)^{r-2}(\ln\ln n)^{\alpha} \pr\{\|S_n\|\ge\sigma\psi(n)[\epsilon+\beta_n(\epsilon)]\}\\ =\;&\Gamma^{-1}(d/2)K(\Sigma)(r-1)^{(d-2)/2}\Gamma(\alpha+d/2) \end{align*} holds if $\ep X=0$, $\ep[\|X\|^2(\ln\|X\|)^{r-1}]<\infty$.}

In this paper, based on the basic theory and application background of stochastic differential equation and backward stochastic differential equation, and combined with stochastic optimal control theory and option price theory in financial market, we will derive the general form of fully coupled forward backward stochastic differential equations (FBSDEs in short). From the point of view of the solvability of this kind of equations, the existing methodology in the literature are analyzed and discussed, a ``unified framework'' approach is introduced to guarantee the existence and uniqueness of solutions for non-Markovian FBSDEs, and several further properties of FBSDEs are obtained. A linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs, as an important supplement and improvement of the ``unified approach'' method, which makes the application of FBSDEs more extensive.

Based on the poor interpretability and the limitation of summarizing the overall trends and local changes at the same time of the traditional neural network, it is not suitable for estimating the regression function of the partial linear model directly. In response to this problem, the semi-linear neural network structure that has both linear and non-linear parts is constructed firstly. Then, the consistency of the network estimator based on empirical risk minimization is proved under some necessary conditions, and the semi-linear network parameter estimation algorithm based on gradient descent is designed, which is called as the local back propagation algorithm. The random simulation experiments verify the large sample property, the results of the case analysis explain the necessity of introducing a linear part in the neural network. In particular, the experiment shows that the estimation effect of this method is slightly better than the N-W kernel estimation method based on the Boston House Price Dataset.

This article studies the convergence rate of the sample mean for varphi-mixing dependent random variables with finite means and infinite variances. Dividing the sample mean into sum of the average of the main parts and the average of the tailed parts, we not only obtain the convergence rate of the sample mean but also prove that the convergence rate of the average of the main parts is faster than that of the average of the tailed parts.

In this paper we  propose a semi-varying coefficient fixed effect panel data model to explore the dynamic relations between PM2.5 and the five meteorological factors: the cumulative wind speed, air pressure, dew point, temperature and hourly precipitation for the five cities: Beijing, Chengdu, Guangzhou, Shanghai and Shenyang, where PM2.5 is a key factor to determine visibility and a serious threat to public health. Then, we combine multivariate local linear fitting, transformation technique with profile likelihood, and establish semi-parametric fixed effects estimators for both parameter vector and varying-coefficient function vector. Finally, we exhibit their estimated dynamic relationships for the five cities in 2015. The proposed procedure can also be generalized to panel data analysis in other fields such as economy and finance. 

This paper studies an optimal investment and reinsurance problem in which the interest rate is driven by the Vasicek process, the surplus process is governed by a diffusion approximation model and two dependent classes of insurance business correlated through a common shock component are considered. The objective of the insurer is to minimize the variance of terminal wealth for a given terminal expected wealth. By using the stochastic linear-quadratic (LQ) control theory and the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we obtain the explicit expressions for the value function, and the optimal investment and reinsurance strategies. Furthermore, the efficient strategies and efficient frontier are derived explicitly. Finally, some examples are given to show the influence of model parameters on the optimal investment and reinsurance strategies.

The reliability of k/n system with two interrelated competitive failure mechanisms leading to component failure is considered. The competitive correlation is connected by Gumbel-Barnett (GB) Copula with Weibull marginal distributions that have different scale
parameters and shape parameters. The effects of GB Copula on the failure rate and reliability of components and k/n system are studied when the shape parameters are equal. We proved that the components have increasing, decreasing and bathtub failure rate in terms of the same shape parameter but different values. The reliability of the k/n (n>1) system is not consistently superior to that of the simple system (i.e., k/n=1/1).Based on progressively Type-II censored samples, the Bayes estimation of the parameters and reliability of the k/n system are discussed, and Monte Carlo experiments are carried out to illustrate proposed methods. Finally, a real-life data is provided to demonstrate the model and method proposed in this paper.

In this paper, confidence set estimation is considered for the parameters from a bathtub-shaped lifetime distribution when record value is available. Based on proposed pivotal quantities, exact confidence intervals and confidence regions are established firstly for the model parameters. In addition, the corresponding optimal confidence intervals and confidence regions are also constructed by using Lagrange multiplier method under given significance level. Finally, simulation studies and a real life example are provided for illustration.

By exploiting financial high frequency data, we nonparametrically estimate the jump characteristic in the presence of market microstructure noise. Our estimator is based on the realized range increments and threshold technique. Besides, the bias caused by microstructure noise can be estimated and removed, if it is modeled as ask-bid spread, which is a used frequently assumption. We further present the asymptotic properties of the proposed estimator. Simulation studies show the estimator works well under microstructure noise. Finally, the estimator is also applied to the real data.

We consider a critical Galton-Watson branching process with immigration Z_n, and study the convergence rate of the harmonic moments of this process, improving the results in previous literatures. The proof is based on the local probabilities estimations of Z_n. As applications, we obtain the large deviations of S_{Z_n}:=\tsm_{i=1}^{Z_n}X_i, where \{X_i,i\geq 1\} is a sequence of independent and identically distributed random variables, and X_1 is in the domain of attraction of an $\alpha$-stable law with \alpha\in(0,2).

In this paper, the synchronization problem of a class of nonlinear coupled neural networks with Markovian jump and variable delay is discussed. The coupling strength of the model is a random variable, the coupling structure of the network switches dynamically according to a continuous time Markov chain, and the influence of nonlinear coupling term and time-varying delay is considered. By constructing a suitable Lyapunov function and using the linear matrix inequality method, the sufficient conditions for the global mean square asymptotic synchronization of the network model are obtained. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical results.

In this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non-identical binomial exponential 2 distributed random variables. Different methods for estimating the reliability are applied. The maximum likelihood (ML), Wilson-Hilferty (WH) normal-based approximation and Bayesian methods are used in the estimation procedure. Also, we propose confidence intervals of the stress-strength reliability based on the approximate method, Bayesian method and Bootstrap methods (Boot-p and Boot-t). Different methods and the corresponding confidence intervals are
compared using Monte-Carlo simulations. Finally, analysis of a real data set is presented for illustrative purposes.

We present a weighted version of the second Borel-Cantelli lemma, which complements two earlier generalizations of this lemma.

In this paper, the Lotka-Volterra stochastic predator-prey systems in a polluted environment is considered. The existence and global attractivity of the periodic solutions are obtained. Furthermore, we obtain the sufficient conditions for the existence and global attractivity of a nontrivial positive periodic solution. Finally, simulations verify our analytical results.

With the deepening of population aging in China, the pricing method of long-term care insurance for the elderly has become a hot issue in actuarial direction. Based on the data of CLHLS from 2014 to 2018, the health statuses of the elderly are further divided into six states on the basis of the traditional three and four state Markov model. It uses the Markov model to calculate the value of each state. The transition strength matrix and transition probability matrix of health state are solved by Robinson power function, which takes gender and age into account. Then, the premium years of 65, 75 and 85 years old are estimated by using double random Lee-Carter model, random walk model and life expectancy formula, which provides a theoretical reference for the pricing of long-term care insurance in China.