Constructing the variance estimator of the total population estimator is an important part of the data analysis in China’s 1% population sample survey. However, due to the stratified, two-stage, probability proportion, cluster sampling method used in China’s 1% population sampling survey, and in principle, only one secondary unit is selected from each selected primary unit, the traditional variance estimation method is no longer applicable. In order to improve the accuracy of variance estimation in population sampling survey, this paper proposes the unequal probability rescaled bootstrap. This method introduces unequal probability sampling into the re-sampling process, and redesigns the sampling probability for the case that only one secondary unit is selected from most primary units. Theoretical derivation and Monte Carlo simulation show that the new method can reduce the deviation of the variance estimator. An example analysis verifies the practical effectiveness of the method in China’s 1% population sampling survey.

Suppose that a fire starts at some vertex of a graph $G$. At every step, a firefighter can protect at most $k$ vertices, and then the fire spreads to all unprotected neighbors. The $k$-surviving rate $\rho_{k}(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. A graph is NIC-planar if it can be drawn in the plane such that each pairs of crossing edges share at most one vertex. In this paper, we prove that every NIC-planar graph $G$ satisfies $\rho_5(G)>\frac 1{73}$.

Finite-time stability problem of nonlinear fractional singular system with time-varying delay is studied in this paper. First, the sufficient conditions for the systems to be regular and impulse-free are obtained by using singular system theory. On this basis, by constructing Lyapunov function, the finite-time stability criterions for fractional singular differential system with time-varying delay factor are derived via the generalized Gronwall inequality and linear matrix inequality method. Finally, a numerical examples is given to verify the validity of the theorem conditions.

In this paper, we are concerned with the existence problems of periodic solutions for the Liénard equation with impulses. By analyzing the variation characteristics of Poincaré mapping at impulse points and using Poincaré-Bohl fixed point theorem, we prove that if a series of impulse points have periodic distribution characteristics on the time axis and under appropriate impulse conditions, the potential function satisfies the Lipschitz condition and the forcing term is a periodic function, then the existence of periodic solutions for Liénard equation $x''+f(x)x'+g(x)= p(t)$ is still maintained. A concrete example of Liénard equation is provided to show the effectiveness of the main results of present paper.

Second-order derivative information plays an important role in full waveform inversion. However, its application in full waveform inversion is hindered by its huge computation and memory requirements. In this study, an efficient truncated Newton full waveform inversion method based on MINRES-QLP method is proposed. This full waveform inversion method can make full use of the second-order derivative information of the objective functional to improve the inversion accuracy. In addition, this MINRES-QLP truncated Newton method can also use the negative eigenvalue information of Hessian matrix to improve the inversion resolution and computational efficiency. For the computational challenge of the Hessian matrix, we present a fast matrix-vector product algorithm. Numerical experiments based on 2004 BP model and Sigsbee model are presented to show the performance of the MINRES-QLP truncated Newton method. Numerical results demonstrate that MINRES-QLP truncated Newton method can take advantage of the second-order derivative information and the negative eigenvalue information of Hessian matrix, accelerating the convergence speed of algorithm and improving the imaging accuracy.

By introducing the harmonic averaging point, we propose a new cell-centered finite volume scheme for evolutionary diffusion equation in this paper. On a mesh edge, we choose its two vertices and a harmonic averaging point as the auxiliary interpolation points. To make the finite volume scheme a cell-centered one, we replace the unknowns on these auxiliary points by the unknowns on the central point of the corresponding grid elements. Our scheme is linearity-preserving and local conservative, it can be applied on an arbitrary polygonal mesh. Considering the cases that the diffusion coefficient is continuous, discontinuous and nonlinear, respectively, four numerical experiments are implemented on six different polygonal meshes. The numerical results show that our scheme is second-order convergent in $L^2$ norm, it maintains good robustness for different types of diffusion coefficients, besides, it is easy to extend to 3D cases in programming.

The data from different populations are highly heterogeneous, and has more “zero values” and large dispersion. The zero-inflated Poisson (ZIP) mixture regression model can be used for modeling, but the mixed model may have many irrelevant independent variables. In order to find out the important independent variables, this paper uses adaptive LASSO to select variables for the ZIP mixture regression model, that is, adding a penalty term to the likelihood function, and then using the EM algorithm to estimate the parameters. The effectiveness of parameter estimation and variable selection is verified by simulation. In order to verify the feasibility of the proposed method, the ZIP mixture regression model is applied to the analysis of the failure times of online lending, and variables with significant impact on the failure of lending are screened out; Through the test set, the fitting effect of the model is better than Poisson, Negative-Binomial(NB) and ZIP regression models.

In this paper, the existence result for fixed points of fuzzy mappings is investigated, and the stability result for a special class of fuzzy fixed points problems is studied by utilizing a technique of bounded rationality. That is, under the framework of bounded rationality, we prove that most of the fuzzy fixed points problems are stable on the meaning of Baire category with the perturbation of fuzzy mappings and feasible sets. Moreover, an approximation theorem for fuzzy fixed points problems is proved under appropriate conditions, which provides a theoretical support for the solving algorithm on the fuzzy fixed point problems.

An edge-partition of a graph $G$ \ is a decomposition of $G$ into subgraphs $G_1, G_2$, $\cdots,G_m$ such that $E(G)=E(G_1)\cup\cdots\cup E(G_m)$ and $E(G_i)\cap E(G_j)=\emptyset$ for $i\neq j$. A linear forest is a forest in which each connected component is a path. The linear arboricity $la(G)$ is the least integer $m$ such that $G$ can be edge-partitioned into $m$ linear forests. In this paper, we study the linear arboricity $la(G)$ of IC-planar graphs, and prove that $la(G)=\lceil\frac{\Delta(G)}{2}\rceil$ for each IC-planar graph $G$ with $\Delta(G)\ge15$, where $\Delta(G)$ is the maximum degree of $G$.

In this paper, a class of neutral fuzzy BAM neural networks with timevarying delays in the leakage term on time scales is considered. By using the exponential dichotomy of linear dynamic equations on time scales, Banach’s fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of Fuzzy BAM neural networks on time scales. The results of this paper are completely new and are complementary to the previously known results.

Due to the huge amount of data stored in different machines, common statistical methods cannot be directly used in big data analysis. Hence, it is necessary to develop the distributed algorithms. Both divide-and-conquer and multi-center methods require the data interaction, in which data privacy and efficient communication are two keys. Furthermore, too many transmissions not only affect the efficiency of computing, but also challenge data privacy protection. Inspired by this, the paper propose two types of privacy-preserving estimation in communication-efficient distributed cases based on the differential privacy. Meanwhile, we strictly prove that the scheme can not only effectively protect data security, but also does not affect the validity of parameter estimation. Finally, both simulation results and a real example illustrate the loss of privacy protection in estimation under the linear model assumption.

Nonparametric spatial dynamic models have been widely used in researching multiple types of problems. It is a kind of modeling technology that simultaneously deals with spatial nonstationarity and spatial dependence in regression analysis. This paper focuses on the spatial varying coefficient models when hierarchical structure exists in the dependent variables. The profile maximum likelihood method is employed for parameter estimation. The AIC criterion is considered for identifying the parametric and nonparametric components of the model. Asymptotic properties of the proposed estimator are also established. Monte Carlo simulations show that the model and method are effective in finite samples. The illustration shows that the spatial spillover effect of economic development between cities in the province is more significant than that between cities at the province’s border. Both these two spatial spillover effects can not be ignored. Besides, the impact of foreign trade and urbanization rate on economic development does not change with spatial location,however,other influencing factors,Including: population density, human capital, savings level, industrial structure, government intervention, social demand and road density have significant spatial heterogeneity on the impact of economic development.

In this paper, we study the well-posedness of solutions of a class of nonlinear stochastic differential equations with multi-term fractional Caputo derivatives. Specifically, firstly, the multi-term fractional stochastic differential equation is equivalently transformed into stochastic Volterra integral equation. Then, the Euler-Maruyama (EM) scheme of the stochastic integral equation is given. Finally, by using the EM scheme, the well-posedness of the solution of the multi-term fractional stochastic differential equation is proved.

In this paper, we investigate the dynamical behavior of a multigroup SIRI epidemic model with stochastic perturbation. We firstly formulate a class of multi-group SIRI epidemic model with stochastic white noises and give the global existence and uniqueness of the solution of the stochastic model. Then, according to the obtained basic reproduction number and the properties of irreducible matrix, we obtain the oscillating properties of the solution of stochastic model around the disease-free equilibrium or endemic equilibrium of the deterministic model and the estimate for the oscillation amplitude is given by using some novel Lyapunov functions. Furthermore, the existence of a stationary distribution and the ergodicity of solutions are obtained. Our results extend and improve some existing ones in previous publications.

In this paper, a new quantization structure for networked control system under DoS attacks is proposed and an H_∞ output feedback controller is designed. In the network environment, assuming that there are DoS attacks in the controller-to-actuator transmission channel, a new quantization structure is used to quantify the sensor signal to reduce the quantization error. DoS attacks are described as the probability model subject to Bernoulli distribution, and the networked control system is modeled as a linear discrete system with random parameters. Unlike the ordinary quadratic Lyapunov function which is used in analyzing networked control system stability, this paper selects a Lyapunov function based on quantization error and attack probability. Based on the stability theory and linear matrix inequality method, the design method of output feedback controller which makes the closed-loop system satisfies exponentially stable in the mean-square sense and the given H_∞ performance index. A simulation example illustrates the effectiveness of the proposed method.

Considering the impact of age in the transmission of some infectious diseases, an age-structured model with general incidence function is presented in this paper. By lemmas and theorems related with Hille-Yosida operator, dynamical properties of the model are examined. Specifically, stabilities of equilibria and the condition for Hopf bifurcation due to the destabilization of endemic equilibrium are investigated. By linearizing the model at the endemic equilibrium, the roots of characteristic equation are analyzed when the basic reproductive number R₀> 1, which reveal that the equilibrium may be destabilized by the perturbation of immune age and Hopf bifurcation can occur. Meanwhile, when critical immune age is taken as the bifurcation parameter, numerical simulations are conducted to illustrate the influence of immune age on dynamical bahaviors of the model.

In this paper, the existence and multiplicity of positive solutions to a class of p-Laplacian fractional differential equations whose nonlinearity contains the derivative explicitly is considered. First, the Green function of the boundary value problem is given by using fractional differential equations and boundary value conditions, and then the existence of one or three positive solutions to the boundary value problem are obtained by the Guo-Krasnosel'skii's fixed point theorem and Leggett-Williams fixed point theorem. As applications, two examples are given to verify the applicability of the conclusion, in particular, the solution of the graphics are given by using the iterative method and simulation approximate approach. It is worth mentioned that the nonlinear term of the differential equation which is studied in this paper involves the Riemann-Liouville fractional differentiation.

Tungiasis is a common zoonotic disease in economically disadvantaged regions such as the Caribbean, Latin America and sub-Saharan Africa. It is of great concern due to its high incidence and rapid transmission. Timely public health education is a means to control the disease in these economically poor areas. Based on this, in this paper, the dynamic behavior of a stochastic model of tungiasis with public health education and saturation treatment rates was studied. By using Itô formula and constructing Lyapunov function, the existence and uniqueness of global positive solutions for stochastic systems are firstly proved; then, the asymptotic behavior of the positive solution of the stochastic system around the disease-free equilibrium and endemic equilibrium of the deterministic system is analyzed. Finally, the correctness of the theoretical results is verified by numerical simulation. The conclusion of theoretical research and numerical simulation shows that increasing the intensity of random interference is a means to accelerate the extinction of diseases.

We investigate the optimal investment problem of a DC pension fund manager under loss aversion and a limited expectation losses (LEL) constraint. We apply the concavification technique to solve the LEL-constrained problem and derive the closedform representations of the optimal wealth and portfolio processes. Furthermore, we compare the effects of a VaR and a LEL constraint on the optimal investment behavior under prospect theory. Although a LEL constraint can provide a better protection for the investors’ benefits than a VaR constraint in a concave optimization problem since the VaR-based risk management will incur heavier losses than the LEL-based risk management in the worst financial states under a concave utility, in our non-concave optimization problem, theoretical and numerical results show that for a relatively low protection level, a VaR and a LEL constraint induce the same optimal terminal wealth and the same investment behavior, that is to say, there is an equivalence between a LEL and a VaR constraint. Therefore, under a non-concave utility, the LEL-based risk management has lost its advantage over the VaR-based risk management. It needs to design a more efficient risk measure for loss averse investors to improve the risk management for a DC pension plan.

This paper studies the jackknife model averaging method of the accelerated failure time model with current status data. Firstly, through the unbiased transformation, we can obtain the LSE of regression parameters. Then the delete-one cross validation criterion is introduced to select the weights of candidate models, and under some regularity conditions, the asymptotic optimality of the model averaging estimator is established. Numerous simulation results show that the proposed method is superior to other existing model averaging and model selection methods in terms of prediction performance. Finally, we applied the proposed method to the NDHS data, and the real data also verify the excellent properties of the proposed method.

Based on Tobin’s $q$-theoretical model, this paper studies the dynamic optimization problem of family business in incomplete market. Under the assumption that enterprise capital and productivity are random, we study the optimization problem to maximize the expected utility of CRRA for entrepreneurs. Besides, the problem refers to the optimal decision making of production, consumption and portfolio during dynamic operation. By the principle of dynamic programming, we derive the HJB equation satisfied by the enterprise optimal decision. In addition, we solve the HJB equation through the applicable production and consumption equilibrium theory as well as risk hedging demand. Then we get the optimal production, optimal consumption and optimal investment strategy of the family business. Finally, we select appropriate parameters for a numerical analysis of the optimal strategy for family businesses, and illustrate the changing relationship between the optimal strategy and its liquidity.

In this paper, we consider semi-Fredholmness for upper triangular operator matrices. We obtain the characterization of semi-Fredholmness by using the relationship between the single-valued extension property and ascent, descent, nullity, deficiency of operators and give conditions for diagonal operators to characterize the semi-Fredholmness property. Besides, we study the perturbation of semi-Fredholmness for upper triangular operator matrices. Further, we study the spectrum, the essential spectrum and the Browder spectrum of operator matrices, give conditions for operator matrices to satisfy Browder theorem, a-Browder theorem, Weyl theorem, and a-Weyl theorem, show connection between these theorems and illustrate with examples, which generalizes the results of Fredholm theory and further reveals the deep connection between Fredholm theory and local spectral theory.

In the investment decision-making problem on precautionary effort with random income, based on the perspective of restricted Ross more risk aversion, this paper obtains some comparative static results consistent with individual’s risk preferences: Whether the temporal or not the intertemporal decision-making problems on precautionary effort, under corresponding sufficient conditions, a restricted Ross more risk averse individual always invests more in precautionary effort, thus this paper generalizes the existing results on precautionary effort under certain income.

This paper studies the Kirchhoff-Schrödinger-Poisson system. Unlike most studies, this paper allows the potential function to be indefinite, that is, the potential function has a finite dimensional negative space. With the help of Morse theory, the author obtains the existence of nontrivial solutions for the Kirchhoff-Schrödinger-Poisson system.

In this paper, by using the time map, we are considered with the shape of bifurcation curves as well as the existence and multiplicity of positive solutions for a class of Minkowski-curvature Dirichlet problem $$\left\{\begin{array}{l} -\left(\frac{u^{\prime}}{\sqrt{1-u^{\prime 2}}}\right)^{\prime}=\lambda f(u), \quad x \in(-1,1), \\ u(-1)=u(1)=0, \end{array}\right.$$ in the case of the nonlinear term does not satisfy the sign condition, where $\lambda>0$ is a parameter, $f\in C[0, \infty)\cap C^2(0, \infty)$. The results revealed the relationship between the number of the nonlinear term $f$ and the number of the positive solutions. Our results improve and extend the existing results.

The potential outcome framework and structural causal model are two main frameworks for causal modeling, and there are efforts to combine the merits of each framework, such as the single world intervention graph (SWIG) and its potential outcome calculus. In this paper, we propose the info intervention inspired by understanding the causality as information transfer, and provide the corresponding causal calculus. On one hand, we explain the connection between info calculus and do calculus. On the other hand, we show that the info calculus is as convenient as the SWIG to check the conditional independence, and most importantly, it owns an operator σ(·) for formalizing causal queries.

In the formulation of reinsurance contract, the relationship between an insurance company and a reinsurance company is competitive. By using the relative performance, this paper quantifies this kind of competition. Then, we suppose that the insurance company is engaged in two kinds of dependent insurance business, and derive the relative wealth process of the insurance company under the competition. The aim of the insurance company is to find an optimal time-consistent reinsurance strategy so as to maximize the expected terminal wealth and minimize the variance of the terminal wealth. By using stochastic calculus and stochastic control theory, the explicit solutions for the optimal time-consistent reinsurance strategy and the optimal value function are obtained, and the economic significance of the optimal strategy is discussed theoretically. Finally, the influence of model parameters on the optimal timeconsistent reinsurance strategy is analyzed through numerical experiments, and the relationship between the optimal reinsurance strategy in two special cases and a general case is compared. Through the research of this paper, some new findings are obtained, which can guide the reinsurance decision of insurance companies more reasonably.

In this work, we introduce a new iterative algorithm for solving equilibrium problems involving pseudomontone and Lipschitz-type bifunctions in real Hilbert space. The algorithm use inertial method and a non-monotonic step size, strong convergence of the algorithm is established without the knowledge of the Lipschitz-type constants of bifunction. Some numerical experiments are reported to show the advantage of the proposed algorithm.

This paper concerns the decay rates of weak solutions to the 3D generalized Navier-Stokes equations with fractional Laplacian dissipation $\Lambda^{2\alpha}u$. It is proved that if the weak solution $u(x,t)$ to the 3D generalized Navier-Stokes equations lies in the regular class $$\nabla u\in L^p(0,\infty;{B}^0_{q,\infty}( \ { R}^{3})),\ \frac{2\alpha}{p}+\frac{3}{q}=2\alpha, \ \frac{3}{2\alpha}<q <\infty,\ \frac{1}{2}\leq\alpha\leq 1,$$ the large initial perturbation $w_{0}\in L^2(\mathbb{R}^{3})$ satisfies \begin{equation}\nonumber \int_{\mathcal{S}^2}|\widehat{w_{0}}(r\omega)|^2\text{d}\omega =Cr^{2\alpha\gamma-3}+o(r^{2\alpha\gamma-3})(r \rightarrow 0),\ \ \frac{10}{\alpha}-8\leq\gamma\leq\frac{25}{2\alpha}-10, \end{equation} then every weak solution $v(x,t)$ of the perturbed system converges algebraically to $u(x,t)$ with the optimal upper and lower bounds $$C_1(1+t)^{-\frac{\gamma}{2}}\leq \|v(t)-u(t)\|_{L^2}\leq C_2 (1+t)^{-\frac{\gamma}{2}},$$ for large $t>1$.

In this paper, a time-delayed vector-borne plant disease model with state-structure is presented. Firstly, the analytical formula for the basic reproductive number $R_0$ is given by using the next generation matrix method. Theoretical results show that the basic reproductive number serves as a threshold parameter provided that the invasion intensity is not strong: the disease dies out if $R_0<1$, and breaks out if $R_0>1$. Moreover, by means of fluctuation method, sufficient conditions for the global attractivity of the endemic equilibrium are obtained if removing infected trees is ignored. Finally, numerical simulations are given to verify the analytical results, and illustrate that spraying insecticides is a very effective control measure.

In this paper, we study blow-up phenomena of solutions to a nonlocal porous medium parabolic equation with space-dependent coefficients and inner absorption terms under nonlinear boundary conditions. By using a differential technique, we obtain the sufficient conditions for the global existence for the parabolic equation with space-dependent coefficients and inner absorption terms under nonlinear boundary conditions in high dimensional spaces. Moreover, an upper bound and a lower bound estimates of blow up time are derived by formulating energy expressions and using Sobolev inequalities and other differential methods.

Let $G$ be a connected graph with $n\ge 2$ vertices. Suppose that a fire breaks out at a vertex $v$ of $G$, a firefighter starts to protects the vertices of $G$ by choosing a vertex not yet fire. Then the fire spreads to all the unprotected vertices that have a neighbor on fire. Let sn$(v)$ denote the maximum number of vertices in $G$ that the firefighter can save when a fire breaks out at vertex $v$. The surviving rate $\rho(G)$ of $G$ is defined to be $\sum\limits_{v\in V(G)} \frac {{\rm sn}(v)} {n^2}$. It is easy to see that $0<\rho(G)<1$. In this paper we show that if $G$ is a planar graph without all cycles of length from 4 to 11, then $\rho(G)>\frac 1{481}$.

Let $G$ be a connected graph with $m$ edges. $\mathcal{H}=\{H_1,H_2,\cdots,H_m\}$ is a set of disjoint graphs. $H_i$ is a connected graph with two distinct vertices $u_i$ and $v_i$, $i=1,2,\cdots,m$. The tensor product of $G$ and $\mathcal{H}$, denoted by $G[\mathcal{H}]$, is the graph obtained from $G$ by replacing edge $e_i$ of $G$ with $H_i$ for $1\leq i\leq m$. In this paper, only using classical graph theory, we obtain a formula for the Tutte polynomial of the tensor product $G[\mathcal{H}]$. Firstly, we divide the set of spanning subgraphs of $H_i$ into two disjoint subsets according to whether the two distinct vertices $u_i$ and $v_i$ are contained in the same connected component or not. In this way, we split the Tutte polynomial of $H$ into two parts, $T_1(H;x,y)$ and $T_2(H;x,y)$. By using the connection between the subgraphs of $H$ and that of $H/uv$, we derive expressions for $T_1(H;x,y)$ and $T_2(H;x,y)$ in terms of $T(H;x,y)$ and $T(H/uv;x,y)$. Secondly, we partition the set of spanning subgraphs of $G[\mathcal{H}]$ into $2^{|E(G)|}$ disjoint subsets according to the construction method of $G[\mathcal{H}]$. Applying an indirect way, we obtain a formula for the contribution to $T(G[\mathcal{H}];x,y)$ of each subset. Thirdly, we sum over the contribution of each subset and obtain an expression for $T(G[\mathcal{H}];x,y)$. As applications, we show that the Tutte polynomials of several operation graphs can be deduced from our result directly. Finally, we discuss why our method can work well and in which situation our technique can be applied to study graph polynomials.

In this work, we come up with the concept of fuzzy rate of an operator in linear spaces for the very first time. Some of its properties are studied. Fuzzy rate of an operator $B$ in a plane is discussed. As its application, a new fixed point existence theorem is obtained. Like the classical fixed point theory applied to differential equations, we believe that the Fixed Point Existence Theorem with respect to fuzzy theory might be applied to fuzzy equations or fuzzy differential equations for future study.

In this paper, a class of nonsmooth semi-infinite multiobjective programming problems is considered by using tangential subdifferentials, and its dual and saddle theorems are discussed. Firstly, Mond-Weir type dual of semi-infinite multiobjective programming problems is established, under the assumptions of the generalized convexity, the weak duality, strong duality and inverse duality theorems of approximate solutions for semi-infinite multiobjective optimization problems are given. Secondly, the $\varepsilon$-quasi-saddle of vector Langrange function is defined, then some necessary and sufficient conditions of the $\varepsilon$-quasi-saddle point are derived. Our results extend and improve some known results, some examples are given to illustrate the main results of this paper.

In this paper, a heterogeneous meta-population SEIAR model is constructed against the background of inoculation of susceptible population, and the basic reproduction number and control reproduction number of the model are calculated by using the next generation matrix method. The effects of preferential mixing mode and heterogeneity on reproduction number are studied. The results show that the heterogeneity of activity, subpopulation size and vaccine coverage have important effects on the number of regenerated cells, and the effects are amplified by preferential mixing. Finally, the optimal control problem with vaccination rate as control variable is studied. The work in this paper can provide reference for the formulation of infectious disease vaccination strategy.

In this paper, we study the initial value problem and the existence of statistical solutions for impulsive discrete nonlinear Schrödinger-Boussinesq equations. The authors first prove the global well-posedness of the impulsive problem, then prove that the process generated by the solutions operator possesses pullback attractors and a family of invariant Borel probability measures. Then we give the definition of statistical solutions for the impulsive problem and prove its existence. The results show that the statistical solution of the impulsive problem satisfies satisfies merely the Liouville type theorem piecewise.

This work is devoted to study the evolution properties of the solutions with compactly supported initial data of a time-periodic S-I reaction-diffusion epidemic model with distributed delays by using the theory of asymptotic spreading speed, by which we can explain the geographic spreading phenomena of newly introduced diseases. Firstly, by applying the uniform persistence idea and comparison skill, and taking three steps, we prove the uniform persistence of the model system in the region where the disease has invaded. Along the way, the main difficulty caused by delay and the periodicity of the coefficients is solved by constructing initial boundary value problems posed on truncated intervals. Secondly, we analyze the evolution properties of the host population in the disease-free region by constructing monotone equation and further employing the spreading speed theory of monotone systems combining with the comparison principle.

Existing research on regression models are limited to directly observed explanatory variables, which increases the error of estimation. The existing studies on models with measurement error mainly focus on the normality assumption of regression error. However, it is not reasonable to use the assumption to study asymmetric data. The performance of the mode is better than that of the mean and median for skewed data. Considering measurement error data, this paper will introducethe skew-normal mode regression model, and extend an EM algorithm which is developed to estimate the parameters based on the traditional method. The results of simulation study indicate that the performance of mode regression is better than mean regression when covariates with measurement error. A real example is further provided to investigate the performance of the proposed methodologies. Emulation experiments and instance analysis show that the model and the proposed parameter estimation method have strong practicability and effectiveness.

The combine estimation function(CEF) method is applied to the statistical study of the generalized random coefficient autoregressive model(GRCA) in this paper. The parameter estimators of the generalized random coefficient autoregressive model are obtained by using the combine estimation function method. The consistency and asymptotic normality of the proposed parameter estimators are proved. The comparison is made by using numerical simulation. The results show that the combine estimation method is better than the estimators based on estimation function method, pseudo-maximum likelihood method and least squares method. The empirical study also shows that CEF method has a good effect.