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.

This paper considers a risk probability minimization problem for nonstationary discrete-time Markov decision processes, in which the transition probabilities and the reward functions depend on time. Different from the expected reward/cost criteria in the existing literature, the optimality performance here is to minimize the probability that the total rewards do not reach a given profit goal until the first passage time to some target set. Under mild reasonable conditions, we establish the corresponding optimality equations, verify that the sequence of the optimal risk functions is the unique solution to the optimality equations, and prove the existence of an optimal Markov policy.

We consider an optimal robust investment problem for a defined contribution DC pension plan with stochastic income and
model uncertainty. In the model, the pension account is allowed to invest into a risky asset and a risk-free asset, and the dynamic of the price of risky asset follows a Heston model. The objective of the problem is to maximize the expected utility of the terminal relative wealth by choosing admissible investment strategies. By using the stochastic control dynamic programming approach, we find the robust optimal investment strategy and the corresponding value function when the utility function has the power or the exponential form, respectively. At last, we show a numerical example to further analyze the theoretical results through the MATLAB software.

Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from 1 and ending at 0. We study the distribution of M and characterize its asymptotics, which is quite different from those of simple random walks.

For high-dimension spatial regression problems, we propose a effective sparse Bayesian model. By introducing a hierarchical Gaussian Markov random field prior, the model can obtain sparse spatial varying parameters, and meanwhile, it can obtain homogeneous parameters estimation for adjacent spatial regions. We use a fast-converging variational EM algorithm for posterior inference, rather than the traditional sampling-based methods. In the M-step of the algorithm, the optimization can be transformed into a classic adaptive lasso problem by simple deformation. The simulation result demonstrate the better performance of our model both in parameters estimation and variable selection. Finally, the proposed model is used to analyze the impact of the socio-demographic factors on the death rate of the COVID-19 in countries of Europe.

In this paper, for a widely applicable semi-parametric model, partially linear additive model, we study the estimation of its coefficients and nonparametric functions when responses are censored. For this, a composite quantile regression estimation method based on data augmentation is proposed. This method utilizes the relationship between quantile regression and distribution function to construct the imputation dataset, and the final estimators are obtained by composite quantile regression through iterations. The proposed method relaxes the assumptions of the model, not only has low requirements for initial values of iterations but also allows the case when different types of censoring are present in the same dataset. Numerical simulations
show that the proposed method can accurately estimate the coefficients and nonparametric functions of the censored partially linear additive model. In real data analysis, this paper studies the air quality in Beijing, and measures the effects of PM10 concentration, CO concentration, temperature, air pressure, and dew point on PM2.5 concentration. The results show that the composite quantile regression of the partially linear additive model can describe well the influence of these factors on PM2.5 from the perspective of linear and nonlinear relationships, and the proposed method performs well in the processing of censored data.

Longitudinal data is an important type of data that is widely used in sociology, economics, biomedicine, epidemiology and other fields. However, in practical problems, people often encounter the situation that the variable dimension is very high and the variable concerned cannot be directly observed, that is, there is a measurement error. In order to solve such problems, this paper studies the estimation of the longitudinal data order index model with measurement error. Based on local linear method and simulation extrapolation (SIMEX) method, this paper constructs a new method for estimating single-index parameters and nonparametric link functions. The effectiveness of the proposed estimation method is verified by Monte Carlo numerical simulation. Compared with the Naive estimation which ignores the measurement error and the estimation which ignores the intra-individual correlation, the estimation constructed in this paper has a smaller mean square error. Finally, we apply the method in this paper to the actual data analysis of the investment demand of listed companies, and the results show that the measurement error has a significant impact on the parameter estimation in practical problems.

Currently, spectroscopy technology is widely used in traditional Chinese medicine analysis. In this paper, from the functional
data analysis perspective, we study outlier detection methods for spectral data, detect outliers, and propose the ``Oja depth detection method''. Simulation studies demonstrate the advantages of the Oja depth detection method. We compare the Oja depth detection method with three existing methods on a Chinese medicine spectral data of 73-dose six-mixture liquid. The results show the proposed Oja depth method is able to detect all six unqualified samples and has the highest accuracy.

In this paper, we consider the nonparametric estimation of the ruin probability in a compound Poisson risk model perturbed by diffusion. We approximate the ruin probability based on the complex Fourier series expansion (CFS) method, and use a random sample on claim number and individual claim sizes to construct a nonparametric estimator of the ruin probability. We also perform an error analysis of the estimator under a large sample size, and provide simulation results to verify the effectiveness of this estimation method under a finite sample size.

We study the numerical solutions for a class of coupled mean-field forward-backward stochastic differential equations. Under suitable regularity assumptions, a posteriori estimate of forward-backward stochastic differential equation is provided. This posterior estimate indicates that the error of the solution for the forward-backward equation can be controlled by the error of the terminal term. Furthermore, we propose a numerical algorithm based on deep neural network and conduct convergence analysis on the discretization scheme.

With the advance of the era of big data, high-dimensional data are frequently collected in many research fields such as economics, finance, and biomedicine. One of the characteristics of high-dimensional data is that the variable dimension p increases with the increase of the sample size n and usually exceeds the sample size. At the same time, outliers are also prone to appear in high-dimensional data. Therefore, how to overcome the influence of outliers on high-dimensional statistical inference, so as to obtain a
more accurate model, is one of the hot issues in current statistical research. This paper is an overview of robust variable selection methods under high-dimensional linear models. Specifically, first of all, we introduce three indicators to evaluate robustness: influence function, breakdown point and maximum deviation. Secondly, it focuses on the selection methods of robust variables, including response variables with outliers, response variables and covariates with outliers, high breakdown point and efficient variable selection methods. Then, the related algorithms are introduced, and different variable selection methods are compared through simulation and examples. Finally, the problems of high-dimensional robust effective variable selection methods and the possible development direction in the future are briefly discussed.

By utilizing the robust loss function, B-spline approximation and adaptive group Lasso, a nonparametric additive model
is investigated to identify insignificant covariates for the ``large p small $n$'' setting. Compared with the ordinary least-square adaptive group Lasso, the proposed method is resistant to heavy-tailed errors or outlines in the responses. To prove facilitate presentation, a more general weighted robust group Lasso estimator is considered. Moreover, the weight vectors play a pivotal role for the suggested estimators to enjoy the model selection oracle property and asymptotic normality. The robust group Lasso and adaptive robust group Lasso can be seen as special circumstances of different weight vectors. In practice, we use the robust group Lasso to obtain an initial estimator to reduce the dimension of the problem, and then apply the iterative adaptive robust group Lasso to select nonzero components. The results of simulation studies show that the proposed methods work well with samples of moderate size.\! A high-dimensional gene TRIM32 data is used to illustrate the application of the proposed method.

This study considers the reliability of a multicomponent stress-strength model involving one stress and multiple strengths from a series system. We derive the Jeffreys prior when the stress and strength variables follow Weibull distribution with a common shape parameter. The necessary and sufficient conditions of the propriety of the posterior distribution based on the Jeffreys prior are obtained. Lindley's approximation and Markov chain Monte Carlo method are presented to obtain the estimates of the system reliability. The performance of the proposed methods is evaluated by Monte Carlo simulation. The simulation results show the Bayesian method outperforms maximum likelihood method, especially in the case of a small sample size. Finally, a real dataset is analyzed for illustration.

This paper concerns the exponential utility maximization problem for semi-Markov decision process with Borel state and action spaces, and nonnegative rewards. The optimal criterion is maximize the expectation of exponential utility of the total rewards
in infinite horizon. Under the regular and compactness-continuity conditions, we establish the corresponding optimality equation, and prove the existence of an exponential utility optimal stationary policy by an invariant embedding technique. Moreover, we provide an iterative algorithm for calculating the value function as well as the optimal policies. Finally, we illustrate the computational aspects of an optimal policy with an example. 

This paper mainly proposes Bayesian methods to analyze the accelerated failure time models. In this model, the distribution of the error terms is unknown and approximated with a P\'{o}lya tree distribution. This paper employs the Bayesian Lasso and Markov chain Monte Carlo methods for parameter estimation and variable selection. Simulation studies demonstrate that the proposed methods can identify the important factors and provide accurate estimates. Finally, the proposed model is applied to identify the risk factors of survival times of the Type II diabetic patients.

In this paper, we study a class of one-dimensional time-inhomogeneous stochastic differential equations with mean field. We
show that the unique solution is ergodic under certain conditions. We further show that, as the strength of the mean field tends to 0, the solution and stationary distribution of such equation respectively converge a.e. \!\!uniformly and in Wasserstein distance to those of corresponding equation without mean field.

The probabilistic combination method is a very useful method to analyze the reliability of wireless sensor network (WSN), and it can effectively deal with the isolation effect and competition failure in the network. However, there are some shortcomings in the reliability modeling and calculation of WSN based on the probabilistic combination method in the existing literature, which leads to a limited application of the research results and even an incorrect evaluation for the reliability of WSN. This paper studies the reliability modeling and calculation of a typical WSN which has only one relay node and the probabilistic functional dependence mechanism. Under the general assumption that WSN faces local failures of components and global failures caused by external attacks, a reliability model that is more realistic and more rigorous is established for WSN. Based on rigorous probability combination analysis, it shows systematic method and compact formulas for the reliability of WSN. The current study enhances the research on such issues in the literature efficiently. Finally, as applications, we calculate the reliability of two special WSNs ---body sensor network and air monitoring system.

The pricing problem of correlation options with exchange rate risk under the regime-switching jump-diffusion model is studied. Under the risk neutral measure, it is assumed that the exchange rate follows the regime-switching mean reversion model and the asset prices follow the regime-switching jump-diffusion models. The pricing formula of the correlation options with exchange rate risk is derived by using the measure transform and Fourier transform method. Moreover, the numerical results of option value are provided by the fast Fourier transform algorithm, and the effects of different models and some important parameters on the value of correlation options with exchange rate risk are analyzed.

In financial market, VaR and ES are applied to measure the risk of asset, portfolio management and margin calculation, which are the international unified standards for bank capital and risk supervision. However, VaR has some certain limitations, ES, as an important risk measurement method, has attracted the attention of financial institutions in recent years. Based on the sublinear expectation theory and G-VaR, this study proposes a new calculation method for ES, and denoted as G-ES. This calculation method can be naturally combined with the back testing of G-VaR. Based on the data of S\&P 500 index and CSI300 index, comparing with other commonly used models, such as historical simulation, AR-GARCH model and POT model based on extreme value theory, it is found that this G-ES method has a good performance within different historical data windows.

The outbreak of COVID-19 in Shanghai in the spring of 2022 had a serious impact on the society, economy, and daily life of residents. The spread of COVID-19 often exhibits complex non-linear dynamics influenced by environment, demographics, medical conditions, frequency of nucleic acid or antigen testing, epidemic control strategies, etc. Long-short term memory (LSTM) models with complex network structures and extensive training are widely adopted to learn and predict the spreading of epidemic. However, such a model neither explains the uncertainty in data, nor takes the influence of various covariates and heterogeneities into account. Therefore, a two-stage LSTM nested generalized Poisson regression (LNGPR) model is proposed in this paper to analyze COVID-19 infectious data in Shanghai outbroke in the Spring of 2022. In the first stage, a multi-layer LSTM network is trained to learn district-specific infectious data, then the trained LSTM is used to fit and predict the number of symptomatic COVID-19 infections. In the second stage, the predicted number of cases is modeled by a generalized Poisson regression model under a hierarchical Bayesian framework, in which the logarithm of the relative risks is modeled as a linear function of covariates and random effects with spatio-temporal heterogeneities. Facilitated by a deep learning approach, the spatio-temporal generalized Poisson regression model can forecast and quantifies uncertainty of the number of daily new symptomatic infections. Furthermore, the predictions based on the proposed Bayesian deep learning approach performs better than those based on LSTM method in virtue of borrowing strength from covariates, and spatial and temporal heterogeneity.

In this paper we study a discrete-time Geo/T-IPH/1 queue model, where T-IPH denotes the discrete-time phase type distribution defined on a birth and death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countably many phases. Using operator-geometric solution method, we first give the expression of the joint stationary distribution. Then we obtain the explicit stationary queue length distribution of the queue we considered. Finally, a numerical example is also presented to illustrate the computational procedure.

The Laplacian distribution is one of the most important distributions used to characterize the peak and thick-tailed data. This paper proposes a linear approximation Bayesian estimation with explicit solutions for the two parameters of the Laplace distribution. The superiority of linear approximate Bayesian estimation over other estimators is verified by theoretical derivation and numerical simulations, and the asymptotic behavior of the linear estimation with the increase of sample size is investigated.

With the increasing complexity of production processes, there has been a growing focus on online algorithms within the domain of multivariate statistical process control (SPC). Nonetheless, conventional methods, based on the assumption of complete data obtained at uniform time intervals, exhibit suboptimal performance in the presence of missing data. In our pursuit of maximizing available information, we propose an adaptive exponentially weighted moving average (EWMA) control chart employing a weighted imputation approach that leverages the relationships between complete and incomplete data. Specifically, we introduce two recovery methods: an improved K-Nearest Neighbors imputing value and the conventional univariate EWMA statistic. We then formulate an adaptive weighting function to amalgamate these methods, assigning a diminished weight to the EWMA statistic when the sample information suggests an increased likelihood of the process being out of control, and vice versa. The robustness and sensitivity of the proposed scheme are shown through simulation results and an illustrative example.

Let \{X,X_n,n\geq1\} be a sequence of identically distributed NA random variables and set S_n=\sum_{i=1}^nX_i, n\geq 1. Let h(\cdot) be a positive nondecreasing function on (0,\infty) such that \int_1^\infty[th(t)]^{-1}\md t=\infty. Denote Lt=\ln\max\{e,t\}, S_n=\sum_{i=1}^nX_i, \psi(t)=\int_1^t[sh(s)]^{-1}\md s, t\geq 1. In this paper, we prove that \sum_{n=1}^\infty[nh(n)]^{-1}\pr(\max_{1\leq j\leq n}|S_j|\geq (1+\varepsilon)\sigma\sqrt{2nL\psi(n)})<\infty, \forall\,\varepsilon>0 if and if \ep(X)=0 and \ep(X^2)=\sigma^2\in(0,\infty). The result partially extends and improves the theorems of \ncite{7}.

The time series in various kinds of fields not only exist a period but also easily affected by external variables of which the effect may vary with time. Sometimes, the period of some time series may be unknown. For the time series with unknown period and affected by external variables, we use a periodic varying-coefficient model to model it. We write the classical decomposition time series model as a partial linear varying-coefficient model with an unknown parameter. Then we approximate the varying-coefficient functions with B-spline and obtain the estimators of the period as well as the periodic sequence and the varying-coefficient functions in the decomposition model. The asymptotic behaviors of the estimators are given in our paper, including the consistency of period estimation and the asymptotic behavior of estimated periodic sequence and the varying-coefficient functions. We illustrate the superiority of our method through simulation studies in Section 4 and the applications to three real data examples including the number of the tourists in Hongkong and Macao and the crude oil price data in Section 5 show the utility of our method.

In this paper, an adaptive Lasso mode regression model is proposed to solve the problem of variable selection in mode regression model. Compared with the traditional mean regression and median regression, mode regression is robust when the distribution is heavy-tail or asymmetric. Kernel estimation is widely used in mode regression. When coping with high dimension, the method will result in calculation errors that hard to ignore. In this paper, the adaptive Lasso penalty is added to estimated parameters based on the mode regression model of the kernel estimation method, and the independent variables with low contribution rate are effectively eliminated. Therefore, the method can improve the accuracy of the calculation. The calculation method is described in detail in this paper, and this paper proposes the estimation methods of the parameters of the model and the asymptotic normality of the estimated values under some regular conditions. The simulation experiment and empirical analysis are performed to investigate the properties of the proposed method in finite sample. Compared with the traditional mode regression model and the mean regression model, the mode regression model with adaptive Lasso penalty greatly improves the accuracy of parameter estimation.

The actuarial calculation of annuities is closely related to the interest rate model. In standard annuities, the interest rate for each period is a fixed constant. In practice, the interest rate for each period can be a variable or even a random variable. These random
variables constitute a stochastic process of interest rate. In many cases, the stochastic process of interest rate is a Markov process. This article studies the actuarial calculation of annuities under the Markov stochastic interest rate model. It is proved that if the interest rate process is a time-homogeneous Markov chain, then the discounting process is also a time-homogeneous Markov chain, and they have the `same' initial distribution and the `same' one-step transition probability matrix. With the help of the interest rate discounting process, the expectation and variance of the present value of annuities under the Markov stochastic interest rate model are calculated. This article introduced annuity polynomials, operators, and annuity operator polynomials. It makes the expressions of expectation and variance for annuities very concise, and easy to program and calculate.

Cox-Ingersoll-Ross (CIR) process is an important tool to study stochastic interest rate and stochastic volatility in financial market. The statistical behavior of fractional CIR process is mainly simulated and discussed in this paper. Since there is no analytical expression of the CIR process, two different functions wfbm and fbmld are used to simulate the fractional Brownian motion, and the Euler-Maruyama (EM) method is used to investigate the expectation and variance of the fractional CIR process. Because the distribution of fractional CIR process can not be expressed by the solution of Fokker-Planck equation, the empirical distribution of fractional CIR process is simulated, and the change of empirical distribution with time is obtained. In order to further verify the algorithm and compare the advantages of the two different algorithms, a backward Euler type scheme of the CIR model and the fractional Ornstein-Uhlenbeck (OU) model with analytical solution is carried out. By comparing figure and table, it is found that simulation by the function fbmld have a very high fitting precision with the theoretical analytical solution with expectation and variance.

Simple random sampling, both with and without replacement, are fundamental in traditional survey sampling. Based on an idea of ``imaginary census'', we provide in this note a new way for calculating the sample variance of simple sample average, as well as understanding the intrinsic relationship between simple random sampling with and without replacement. The key concept is an ``imaginary census'' matrix, which records the exact sampling trajectory of each draw without replacement, until all population units were sampled out. The random matrix possesses a nice probabilistic symmetry, and each of its column summed to be a fixed number. The new framework, in a sense, is a fusion of two existing classic techniques in traditional survey sampling. One depends on the random vector of 0\,--\,1 valued random variables indicating which population units were sampled, and the other is the symmetrization technique. Our method appears valuable for understanding several important sampling strategies, even to the branch of survey sampling itself. For illustration, we present two examples pertain to unequal probability sampling with replacement and adaptive cluster sampling, with a focus on understanding these sampling strategies from the perspective of simple random sampling.

Precision medicine emphasizes the importance of correctly identifying heterogeneous subgroups to develop individualized treatment strategies that can be prescribed for each subgroup. Despite the recent methodological advances in subgroup analysis, how to effectively identify subgroups for censored data remains largely unexplored. In this paper, we propose a new methodology for subgroup analysis based on the accelerated failure time model, in which the heterogeneity caused by latent factors can be represented by subject-specific intercepts. We consider the most common right censoring situation, and processes the censored data by the mean imputation method. The hard thresholding penalty function is applied to pairwise differences of the intercepts, thus automatically dividing the observations into different subgroups. We also establish the theoretical properties of our proposed estimator. Our proposed method is further illustrated by simulation studies and analysis of a wisconsin breast cancer dataset.

We consider quadratic weighted branching processes with immigration and instantaneous resurrection. Under the condition that
the resurrection rate can be summed, we show that the target process does not exist. The existence and uniqueness criterion for the process are obtained under the assumption that the sum of the resurrection is infinite. Also, the equivalent conditions for the existence criterion are given for easy verification. It is proved that there exist infinitely many of Q processes. Among them, there exists a unique honest process and the corresponding construction method is then investigated. We prove that this honest process is always ergodic and the second-order differential equation of the equilibrium distribution is established.

Semi-supervised data contains a labeled data set with both responses and covariates and an unlabeled data set with covariates only. The inference based on semi-supervised data is gaining more and more interests in statistics. When the response in the labeled data is binary, case-control sampling is commonly used to alleviate the imbalanced data structure. When the response and the covariates satisfy the logistic model, the slope parameter of the model can be consistently estimated even for the case-control sampling. However, when the logistic model is incorrectly specified for the data, the case-control samples can not estimate the population risk minimizer consistently. With the help of the unlabeled data, we derive a consistent estimator for the case population proportion. Then, an inverse probability weighted loss function is developed to obtain a consistent estimator for the population risk minimizer. The proposed estimators are shown to be asymptotically normal and the limiting variance-covariance matrix can be consistently estimated. Simulation results show that the proposed method gives out reasonable finite sample performances. A real data example is also analyzed for illustration.

In this article, we propose a nonconcave penalized M-estimation of least product relative error (penalized M-LPRE) method for
multiplicative regression models whose dimension of parameters is sparse and can increase with the sample size. Under some mild conditions, consistency and asymptotic normality of the penalized M-LPRE estimator are established. Numerical simulations and a real data analysis on the body fat are carried out to assess the performance of the proposed method.

Biased sampling is a pervasive issue that transcends various disciplines, impacting fields such as econometrics, epidemiology, medicine, survey research, and more recently, machine learning and artificial intelligence (AI). This ubiquitous challenge arises when the selection of data points for analysis or research introduces systematic biases, potentially compromising the accuracy

and reliability of research outcomes. In this paper, our objective is to provide a comprehensive overview of the foundational concepts related to biased sampling problems and the methods of inference. Furthermore, we aim to establish a connection between biased sampling issues and the more recent discussions in machine learning regarding distribution shift problems. Additionally, we will delve into the latest advancements in biased sampling, particularly within the context of transfer learning and conformal inference for predictive confidence intervals. Our ultimate goal is to present this material in a manner that is accessible to graduate students, enabling them to identify applications of biased sampling problems within their own research endeavors.

It is with deep respect and gratitude that we dedicate this paper to the memory of the late Professor Shisong Mao, whose guidance and wisdom have been invaluable throughout the years.

We investigate the equilibrium reinsurance strategy in an infinite reinsurance space for an ambiguity-averse insurer (AAI) under a continuous-time framework. We assume that the surplus process of the AAI follows the Cram\'{e}r-Lundberg (C-L) model perturbed by standard Brownian motion, and the insurer invests his surplus in a risk-free asset. We present the equilibrium reinsurance strategy and its corresponding value function by solving extended Hamilton-Jacobi-Bellman (HJB) system equations, and we find that the AAI's equilibrium reinsurance strategy to maximize the time-inconsistent penalty-dependent mean-variance
reward function is a combination of quota-share with excess of loss reinsurance or its dual form. Detailed numerical analyses are presented to illustrate the various effects of insurer aversion to various uncertainties and other parameters on the equilibrium reinsurance strategy and its corresponding value function.

In this paper, we consider a risk model where the aggregate premium process is a compound Poisson process. Moreover, there are a constant interest and a constant dividend barrier strategy in this model. The integro-differential equations for the expectation and the $n$th moment and the moment generating function of the cumulative discounted dividend payments until ruin are obtained. Meanwhile, the explicit expressions for the expectation and the $n$th moment and the moment generating function of the cumulative dividend payments until ruin are given when the individual stochastic premium amount and claim amount are exponentially distributed. Finally, numerical example is also given to illustrate the effect of the related parameters on the expected value of the cumulative discounted dividend payments until ruin.

In this paper we study a continuous-time optimal portfolio selection game problems between two risk-averse institutional investors when each investor takes into account his relative return by comparison to her competitor. Both institutional investors can invest freely in the risk-free asset and only one of the two correlated risky stocks is available to each investor, reflecting asset specialization. Each investor chooses a dynamic portfolio strategy in order to maximize her expected terminal utility of the weight sum of absolute log-return and relative log-return. We first characterize explicitly the unique Nash equilibrium portfolio strategies. Secondly, the Nash equilibrium portfolio strategy and the value function of each investor are obtained in closed forms for the case of each investor with an exponential utility function. The effects of the relative return on the Nash equilibrium portfolio strategies and the value function are also analyzed. Numerical examples are also provided to illustrate how the Nash equilibrium portfolio strategies change when some model parameters vary. The investment performance between the Nash equilibrium portfolio strategy and the traditional strategy is also compared by using the certainty equivalent. The results reveal that the relative return can change the investment performance of the institutional investors.