The bi-directional long short-term memory neural network model is widely used in natural language processing, but hyperparameter tuning of the model is difficult in practice. In this paper, we take the semantic role recognition task as an example, consider four candidate features (word, part of speech, target word and position) and two hyperparameters (the number of layers of the network and whether CRF classifier is used) as factors in robust design, and select the optimal combination of features and hyperparameters by setting levels of each factor and performing experiments. In particular, we perform 32 cross validation on a small datasets to select the optimal configuration combination of the model based on the SNR of robust design. Then, we analyze the influence of each factor on the performance of the model by quantitatively analyze so that the model has a certain degree of interpretability. Moreover, in order to verify the superiority of our tuning method, we use the standard segmentation of natural language processing on a big dataset, adopt the traditional greedy strategy to select the optimal configuration combination, and compare with our method on the test set. The results show that our method is better than the traditional tuning method.

As the continuation and deepening of \ncite{1},this paper focuses on the center of economic equilibrium and uses mathematics as a tool to explore two themes in the economy: firstly, the ``pillar'' industry and ``bottle strength'' industry, ``top'' products and ``weak'' products in the economic system, that is, the ranking and
stability analysis of products; secondly, forecast and adjust, optimize the design and debugging of economic structure.

This paper studies an iterative method of estimating parameters in Gamma distribution based on maximum likelihood estimation and EM algorithm improved by non gradient information spectrum residual method in the case of classified data, Type-I interval censered data, Type-II interval censered data, and it proves the strong consistency of the algorithm. The simulation results show that the iterative method proposed in this paper can greatly shorten the running time while ensuring the accuracy, the estimated mean square error tends to zero with the increase of sample size.

With the deep development of the economic society,and the remarkable rise in people's living standards, customer requirements are becoming more and more intense. Meanwhile, the rapid development of the Internet technology and smart manufacturing has provided a solid industrial foundation for meeting such needs. Customized production has gradually become an important production mode. Different from traditional massive production, in a customized production mode, a product often consists of multiple customized modules, generating thousands of customized combinations, and usually a small number of products will be produced for each combination. Taking certification cost into account, the customized production makes the traditional product certification procedure unrealistic because we can not authenticate several prototypes for each customized combination. Therefore, there is an urgent requirement for developing theories and methods of customized product certification, which can accurately assess the quality of customized products at an acceptable certification cost. In this article, we propose a general framework for customized product certification based on certification big data and statistical models and illustrate it on
refrigerator safety certification. This framework transforms the safety certification of customized products into the assessment of product safety risks, establishes a quantitative statistical model to characterize the risks, and uses the principles and methods of experimental design to develop economical certification schemes. Our certification big data based simulation results indicate that this framework has the potential to achieve reliable, efficient, and intelligent customized product certification, which has important theoretical and practical significance for certification mode innovation.

This paper investigates a bidimensional risk model with interference, in which the vectors of claim process and premium process are both compound Poisson-Geometric processes. Through martingale method and stopping time technique, we get the upper bound of the ruin probability. When the marginal of claim vector and premium vector follow bivariate FGM (Farlic-Gumbel-Morgenstern) class, we have discussed some properties of the upper bound.

This paper studies the statistical inference and algorithm design problems in the mixture model with change point. For multi-classified mixture data with change point, this paper designs an improved EM algorithm based on the maximum likelihood estimation of parameters, and proves the large sample nature of the change point estimator and the mixture parameter estimator. In order to verify the effectiveness of the method, we conducted some simulation experiments.
The results show that the EM algorithm that does not consider the change point has a poor estimate on the classification result, and even loses some categories; Our improved EM algorithm can accurately locate the location of the change point, and at the same time obtain accurate estimates for the corresponding parameters of each category.

For the comprehensive ranking of urban economic level and environmental level in China, there have been some index system ranking methods, but most of them involve multivariate data. With the rapid change of data acquisition technology, data become more and more complex. The observational data produced in some fields is no longer a single type of data, but a combination of various types of data. This paper studies how to rank them when the index system involves functional data. In this paper, four ranking methods are proposed and compared. The results are as follows: when the functional data is contaminated, the
entropy weight method results are relatively stable; when the scalar data is contaminated, the multivariate modified banding depth is more stable. The research shows that the selection of ranking methods for multi-type data depends on the characteristics of the data. This research enriches the comprehensive ranking of multi-type data and has good practical significance.

When the response variables are missing randomly, in the process of statistical inference of the parameters of interest, two common working models are the regression function model and the selection probability model. In order to avoid the inference bias caused by the model setting error, the regression function model and selection probability models are necessary and meaningful for model testing. For this reason, for the first time in this paper, the feature functions are applied to the model testing problem of random missing response variables and discrete variable response variables, and a Euclidean distance between sample points is constructed based on test statistic. The proposed test avoids the selection of smoothing parameters such as bandwidth, and at the same time can detect the local alternative hypothesis at the fastest parameter speed. Further, this paper aims at the composite null hypothesis: at least one of the two working models is designed. It is correct, and a test method of the merged model is proposed. An important application scenario of this test is to determine whether the bistable estimation of the parameter is a coincident estimate. This article deeply studies the test of the merged model in the original hypothesis, the global alternative hypothesis, and the local alternative hypothesis. The asymptotic property of the following, and using the boostrap method to determine the rejection domain of the test, study the performance of the merge model test under a limited sample. Finally, this article applies the proposed merge model test method to
analyze the clinical research of AIDS research Test Data. It is worth mentioning that the combined model test mentioned in this article not only has good performance, but also the method is simple and easy to implement, and the corresponding p-value is easy to calculate.

In this summary report, we introduce some basic theory of single birth processes as the supplement of \ncite{1}, including the probabilistic meaning of some numerical characteristics, the distributions and moments of the integral-type functional of single birth processes, as well as the distributions of the staying
times, the first hitting time and the last exit times etc.

Journal impact factor is an important criterion for judging the quality of journals. In this paper, a Bayesian model is established for journal quality parameters, combined with existing journal impact factor calculation formulas, by using the theory of the credibility theory, a correction method for journal impact factors is given. We also obtained the estimation of structure parameters. The statistical properties of the estimation are discussed. Through simulations, the consistency and convergence speed of the estimators are tested. Finally, the empirical analysis of the correction formulas of the impact factors of more than twenty mathematics journals in China.

A lot of inclusive health insurance projects have been brewing in many cities in China. Among them, ``{\it WeiHuiBao}'', which adopts a multi-layer reinsurance method that includes proportion, deductibles, and limit, has attracted our attention. We define such anunderwriting method with proportion, deductibles, and limit simultaneously as generalized underwriting method, which has never appeared as the optimal contract in existing literature but very common in reality. This paper aims at finding an optimal strategy in a two-layer generalized underwriting method. We solve two models, the optimal insurance model minimizing the
premiums of policy-holders with the CTE of policy-holders' own risk less than an agreed threshold, and the optimal reinsurance model minimizing the VaR of insurer's total risk, under the variance premium principle, respectively. We find that the optimal contract forms of the two models support the clauses in ``{\it WeiHuiBao}'', and the numerical solutions under the Pareto distribution and the Gamma distribution are given. The underwriting form of ``{\it WeiHuiBao}'' is a multi-layer generalized underwriting method while this paper demonstrates the optimality of the two-layer underwriting method.

We prove a general fluctuation limit theorem for Galton-Watson branching processes with immigration. The limit is a time-inhomogeneous OU type process driven by a spectrally positive L\'{e}vy process. As applications of this result, we obtain some asymptotic estimates for the conditional least squares estimators of
the means of the offspring and immigration distributions.

The paper investigates the optimal consumption and portfolio with consistence performance under Knightian uncertainty. The agent has Knightian uncertainty for the expected return of risky asset, which is characterized by \kappa-ignorance model. Consistent performance model asks for a dynamic wealth constraint, which requires the wealth always stays at or above the weighted average of the entire historical wealth levels. In the infinite time horizon, we obtain the
optimal consumption and portfolio explicit solution for the model by the HJB equation and verification theorem.

COVID-19 has caused huge impact to financial system, among which stock market is one of the main sources of infection. This paper investigates stock market volatility based on complex networks, proposing the method of applying articulation point-targeted attack (APTA) in the model of the stock market volatility. The strategy of APTA is removing the most destructive articulation points (AP) that will result in most nodes disconnected from the giant connected component (GCC) by iterating, and eventually uncovering the residual giant bicomponent (RGB) that maintains the structural stability of the network. This paper models
stock network based on realized volatility and thresholds, separates research period into steady-developing period and risk-fluctuating period, compares and analyzes the topological properties. Network centrality indexes and APTA are used to discover the important stocks that need to be supervised specially, thus avoiding the big-scale spread of risks or concurrent risks, and helping the supervisors maintain financial stability.

A queueing-inventory system with cancellation, common life time and non-preemptive priority is considered, in which customer arrives according to a Poisson process and service time follows an exponentially distribution. By formulating the system process into a level-dependent quasi-birth-and-death process (LDQBD), the stability condition and steady-state probability vectors under Neuts-Rao truncation method are obtained. Some performance measures and expected cost function are also given. Then, the optimal maximum inventory level and minimum cost are achieved through numerical simulations. Finally, the sensitivities analysis on major parameters are performed to provide more managerial insights.

This paper discusses the optimal design problem for the hierarchical linear model with intraclass positive correlation. A-optimal designs are considered for the estimation of the fixed effects as well as the prediction of random effects. We provide two equivalence theorems to determine the $A$-optimality of a given design. Two illustrative examples of the A-optimal designs are presented.

In this paper, we present explicit sufficient and necessary conditions for the compactness of the semigroups associated with time change of non-local Dirichlet forms, whose jumping kernels have finite second moment.

In this paper, we investigate algebraic and exponential transience for continuous-time Markov chains (CTMCs). Equivalent relations of these transience are revealed between CTMCs and their jump chains and dual processes. The results are further applied to derive the criteria of these transience for general CTMCs, generalized Markov branching processes and birth-death processes.

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.

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.

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.

Based on a new lifetime testing scheme proposed in this paper, that is, double type-I hybrid censoring scheme, the maximum
likelihood estimates of the parameters are obtained for two-parameter Pareto distribution. Also, the asymptotic confidence interval of \theta is obtained from Fisher information. When \alpha is known, the Bayesian and E-Bayesian estimates of \theta, as well as the Bayesian estimates of reliability function under different loss functions are obtained using the Gamma prior distribution. When both \alpha and \theta are unknown, the joint noninformation prior distribution is taken, and the Bayesian estimates of \alpha and \theta are calculated under the squared error loss function. Using Monte Carlo method to simulate the double type-I hybrid censored samples, the estimates of the unknown parameters and reliability function are obtained, with the increase of the sample size, the relative errors and the lengths of the confidence interval decrease gradually. Finally, a numerical example is analyzed.

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.

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, 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.

In this paper, we consider the change-point problems for the functional linear model, where the explanatory variable is a random process, and the response is a scalar. Based on the projecting moment estimators of the parameters onto the truncated finite-dimensional space, we propose the detecting statistic and give the estimator of the change-point. In a theoretical investigation, we derive the asymptotic distribution for the proposed detecting statistic, and establish the consistency of the change-point estimate under some mild conditions. Some simulation studies and a real data analysis are conducted to illustrate the finite performance of the proposed testing methods.

We consider the optimal joint dividend and capital injection strategy with proportional and fixed costs. The main feature of our diffusion model is that dividend can only start at the arrival times of an exogenous uncontrolled Poisson process (signal), and the dividend rate is restricted with a bounded interval. The objective is to find the control which maximizes the expected total discounted dividends less the expected discounted costs of capital injections. We also give some numerical illustrations to optimal strategies, which indicate that the optimal threshold levels are increasing functions of exogenous parameters.

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.

As the development of the technology and material science, products become highly reliable with long lifetimes. Consequently,
the traditional life tests are gradually replaced by the degradation tests. In the degradation tests, the quality characteristic (QC) which degrades over time, can reflect the reliability status of the product. Some of the degradation paths of the QC are usually monotonically increasing and non-smooth function of time. In this paper, motivated by the degradation data from the tests of N-channel power metal oxide semiconductor field-effect transistor (MOSFET), we propose a two-phase gamma process under hierarchical Bayesian framework to model the monotonic and non-smooth degradation data. A simulation study is performed to verify the utility of the proposed model under three scenarios. An illustrative anti-radiation performance case study is analyzed to show the applicable power of the proposed model.

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

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. 

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. 

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.

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 $\ep X=0$, $\ep[\|X\|^2(\ln\|X\|)^{r-1}]<\infty$.}

Complex network is a new research field in recent years, and community discovery is one of its application directions. Clustering analysis of real data sets is an important method of data mining, but there is a situation that the effect of clustering analysis is not good. At this time, if the correlation measure is introduced to construct the data set into a complex network, the community discovery method can be used to process it. Most of the existing literatures have improved the algorithm, and few studies have
compared the results of the two methods. In this paper, the Louvain algorithm in community division and the K-means clustering algorithmin clustering algorithm are selected. First, the theories of the two algorithms are compared. Then, the complex network is constructed by using the data of patients with heart disease and kidney disease. The results of the community division of the Louvain algorithm and the clustering results of the K-means clustering algorithm are compared. Under the evaluation criteria of the correct division rate, the results of the community division of the Louvain algorithm are better than the clustering results of the K-means clustering algorithm.

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.

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.