In this paper, we explain the solution of the non-autonomous stochastic delay lattice equation driven by white noise generating a continuous cocycle. Uniform estimation and tail estimation are performed, and it is proved that the system has a $ \mathcal{D}$-pullback random attractor. We prove the double upper semicontinuity of random attractors for delay system when the delay and a certain parameter converge simultaneously.

The aim of this paper is to investigate the well-posedness and stability in set optimization. For the first time, by embedding the original set optimization problem into a family of perturbed problems with the same structure, the notions of generalized $m_{1}$-well-posedness and $m_{1}$-well-setness under perturbation for set optimization problems are introduced. The relationships between them are obtained. Sufficient condition, necessary conditon and many characterizations are given for generalized $m_{1}$-well-posedness under perturbation, respectively. Two new definitions of monotonicity of set-valued mapping are introduced, and from this, the semi-continuity and closedness of $m_{1}$-efficient solution mappings for parametric set optimization problems are studied.

In this paper, we consider the dynamics of a second-order rational difference equation. Through the linear stability analysis, the parameter conditions are gained to guarantee the existence and stability of the equilibrium. By applying the center manifold theorem, the Normal Form theory and the bifurcation theory, we derive the critical parameter values of the fold bifurcation, transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation respectively. In order to identify chaos among regular behaviors, we calculate the maximum Lyapunov exponents and fractal dimensions. These results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamic behaviors, such as period doubling cascade, periodic windows, limit cycles, chaotic behaviors and so on. Interestingly, with the selection of parameters, the dynamic behaviors of the system are completely symmetric.

This work is concerned with a stochastic evolutionary equation with a multiplicative noise. Verifying the convergence from the solution of the stochastic evolutionary equation to one of its Wong-Zakai approximation and applying the exponential martingale argument, the Kallianpur-Striebel formula and Itô formula, we prove that the nonlinear filter generated by the stochastic evolutionary equation converges to one generated by its Wong-Zakai approximation in the observation system with colored noise.

Malaria is a vector-borne disease caused by a pathogen. In order to study the multiple effects of spatial heterogeneity, vector-bias effect, and seasonality on disease transmission, this study proposes a temporal periodic reaction-diffusion model for malaria transmission. Firstly, the basic reproduction number ($R_{0}$) of the model is introduced. Then it is proved that if $R_{0}\leq1$, the disease-free periodic solution is globally asymptotically stable; while if $R_{0}>1$, the system possesses a globally asymptotically stable positive periodic solution. These proofs utilize the monotone dynamical system theorem, the theory of periodic semiflows, and the chain transitivity theory. Numerical studies on malaria transmission in Maputo Province, Mozambique are carried out to validate the theoretical analysis results. The impact of key parameters in the model is discussed, and it is concluded that neglecting the diffusion of human and mosquito populations and the vector-bias effect underestimates the risk of disease transmission. In addition, the impact of medical resources on disease transmission is analyzed from two aspects: quantity and distribution. It is found that increasing medical resources would reduce the risk of disease transmission. If medical resources are fixed, reducing the variability in the distribution of medical resources would also decrease the risk of disease transmission.

This article proposes a hypothesis testing method to detect serial correlation and ARCH effect in high-dimensional time series based on $L_{2}$ norm and Spearman's correlation. In this article, We study the asymptotic behavior of our test statistic and provide a bootstrap-based approach to generate critical values, we prove our test can control Type-I errors. Our test is dimensional-free, which means it is independent of the dimension of the data, hence our test can be used for high dimensional time series data. Our test does not require tail properties of data, hence it can be used for heavy-tailed time series. The simulation results indicate that our new test performs well in both empirical sizes and powers and outperforms other tests. The practical usefulness of our test is illustrated via simulation and a real data analysis.

The paper is concerned with the traveling wave solutions of a nonlocal dispersal delayed SEIR model with saturated recovery rate. First, we proved the existence of traveling wave solutions for R₀>1,c> $c^{*}$ by applying the Schauder's fixed point theorem and a limiting argument, and the asymptotic boundary of the traveling wave solutions was obtained by Lyapunov functional. Secondly, we proved the nonexistence of the traveling wave solutions for R₀>1, 0<c<c^* or R₀<1 by two-side Laplace transform. Finally, we discussed how the latent period and the movement of the infective individuals affect the minimal wave speed. Generally speaking, the longer the latent period, the slower the spread of the disease.

The discounted Hamilton-Jacobi equation is a special form of the contact Hamilton-Jacobi equation. Hence,the study of the discounted Hamilton-Jacobi equation is more intuitionistic.In this paper,we mainly study the convergence of the viscosity solutions of time-periodic discounted Hamilton-Jacobi equation by variational method. For the evolutionary Hamilton-Jacobi equation $u_t+\lambda u+H\left(x, D_x u, t\right)=0$, under the assumptions that $H(x, p, t)$ is a Tonelli Hamiltonian,and $H(x, p, t)$ is 1-period in the variable $t$. we can get the 1-period solution $\bar{u}_\lambda(x, t)$ and the 1-periodic solution is unique under the given conditions. Furthermore,we prove $\lim _{n \rightarrow \infty} u_\lambda(x, t+n)=\bar{u}_\lambda(x, t)$, where $u_\lambda(x, t+n)$ is the viscosity solution of $u_t+\lambda u+H\left(x, D_x u, t\right)=0$. At last,we explain the conclusion with a specific example of Hamilton-Jacobi equation.

Based on the spatial heterogeneity, the asymptomatic hosts and the multiplicity of pathogen transmission routes, a model of reactive diffusion host-pathogen with asymptomatic hosts and multiple infection routes is proposed, which is discussed the existence and uniqueness of the global positive solution of the model by using semigroup theory. Further, according to the spectral radius method of the next generation operator, the basic reproduction number $\mathcal{R}_{0}$ of the model is given, and the extinction and persistence of the disease are described. That is, if $\mathcal{R}_0<1$, the disease-free steady state is globally asymptotically stable; while if $\mathcal{R}_0>1$, the disease is uniformly persistent and the model admits at least one endemic steady state. In addition, the global asymptotic stability of the disease-free and endemic equilibrium states of the model in a spatially homogeneous environment is obtained by constructing suitable Lyapunov functions. Finally, some numerical simulations are conducted to explain the main theoretical results and to explore the influence of diffusion rates on the distribution of infected hosts.

In this paper, we extend the network oligopoly with finitely many markets to the model with infinitely many markets, and propose the network oligopoly model with continuous-variable markets. Under the fully noncooperative hypothesis, we first prove the existence of Cournot-Nash equilibria. Furthermore, we assume that there exists a partition of the set of firms. By defining the coalitional cost function and coalitional profit function of every coalition, we establish a noncooperative game basing the coalition structure, and prove the existence of Nash equilibria. New network oligopoly models and Nash equilibrium existence theorems are our main contributions.

In this paper, we study a new (2+1) dimensional modified Camassa-Holm equation, which is integrable since it admits bi-Hamiltonian structure. The aim of this paper is to study the existence of peakons for such equation. The suitable definitions of weak solutions both on real line and circle are defined. The single-peakon, two-peakon, and periodic peakon of such equation are derived. A new type of two-peakon is obtained, and the reason why such phenomena may not occur in (1+1) dimension case is also illustrated.

This paper studies the equilibrium strategy of a fluid queue with breakdowns and working vacations in the fully observable case, where the buffer switches alternately in the busy period, working vacation period, and fault maintenance period. Based on the renewal process and the standard theory of linear ordinary differential equations, the Nash equilibrium behavior of the fluid and the steady-state probability distribution are obtained, then the expected buffer content can be derived by using the classical Laplace-Stieltjes Transform (LST) method. Based on the economics and utility theory, the expected social benefit function is constructed reasonably and the global balking thresholds that maximize the social welfare can be obtained by some numerical examples. The admission fee revenue model can be constructed with the globally optimal thresholds and admission fees as joint decision variables, and the effect of the global optimal thresholds on the maximal admission fee revenue can be illustrated. The parameter optimization strategies and social benefit problems are crucial for improving the secure transmission performance of cognitive wireless networks, where the nodes fail and become semi-dormant. Simulation results can provide a theoretical basis for the optimal allocation of limited wireless network resources.

In this paper, we consider the non-collapsing property on a class of planar convex curves deformed by non-linear curvature flow.Firstly, we give the definition of the inner non-collapsing property and the outer non-collapsing property on planar curves, then by defining a function $Z$, we prove that the inner non-collapsing property on planar curves is equivalent to the non-negativity of $Z$ and the outer non-collapsing property on planar curves is equivalent to the non-positivity of $Z$. Finally, we use maximum principle to prove the non-collapsing property on the planar convex curve shortening flow is maintained under some conditions.

Ecological researches show that there exist dominance ranks of individuals in many species. Moreover, natural populations are actually subject to seasonal fluctuations which makes their habitats often undergoes some periodic changes. Motivated by these considerations, in this paper, we investigate the optimal harvesting problem for a hierarchical age-structured population system in a periodic environment. Here the objective functional represents the net economic benefit yielded from harvesting. Firstly, by means of frozen coefficients and fixed point theory we show that the state system is well posed if the reproducing number is less than one. Meanwhile, it is shown that the population density depends continuously on control parameters. Similarly, we show that the adjoint system is also well posed. Then, the optimality conditions given by the feedback forms of state variable and adjoint variable are obtained by using the adjoint system and tangent-normal cone techniques. The existence of optimal harvesting policy is verified via Ekeland's variational principle and fixed point reasoning. Finally, we use numerical simulations to verify the main results and find other dynamic properties of the system. The results in this paper generalize and improve the previous related results.

In the realm of financial data analysis, where data often exhibits volatility clustering, heteroscedasticity, and asymmetry, capturing these inherent features authentically requires the adoption of conditional heteroscedasticity models within a skewed distribution framework. This study addresses the Bayesian statistical diagnostic challenges associated with skew-normal GARCH models. The research begins by employing the Griddy-Gibbs algorithm for effective parameter estimation within the skew-normal distribution embedded in the GARCH model. The investigation considers three distinct sources of perturbations: disruptions from prior assumptions, anomalies within the data, and variations in the model itself. To facilitate comprehensive statistical diagnosis, the study leverages three objective functions: Bayes factor, Kullback-Leibler divergence, and posterior mean, enhancing the precision of the diagnostic process. Empirical validation is achieved through rigorous numerical simulations, conclusively establishing the method's efficacy and resilience. This is further supported by an empirical application involving GARCH modeling for Chevron Stock. By utilizing the skew-normal distribution to encapsulate weekly logarithmic returns, the study empirically underscores the distinct advantages of Bayesian local influence analysis, demonstrating superior results.

In this paper, a multi-fidelity Monte Carlo (Multi-fidelity Monte Carlo, MFMC) estimation method based on data-driven low-fidelity models is proposed for the multi-parameter uncertainty quantitative analysis of the numerical solution of the Advection-Diffusion-Reaction equation. In our method, a high-fidelity model of numerical solution is first obtained according to the finite element method. Then, two types of low-fidelity models of DEIM and POD-DEIM are given respectively based on the POD reducing dimension method of the finite element discrete linear equations and the DEIM interpolation method for parameter space. Finally, through numerical experiments, the mean value and sensitivity of the multi-parameter uncertainty are analyzed for the ADR equation. The results show that, compared with the standard Monte Carlo method, the MFMC estimation method based on the data-driven low-fidelity model can effectively reduce the computational cost and the relative mean square error.

In this paper, we undertake a comprehensive study the orbital stability of standing waves for the inhomogeneous nonlinear Schrödinger equation $$ \begin{cases} i\partial_t\psi+\Delta \psi+|\psi|^{p}\psi+|x|^{-b}|\psi|^q\psi=0, &\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \\ \psi(0,x) = \psi_0 (x), &\quad x\in\mathbb{R}^N, \end{cases} $$ where $N\geq3$, $\psi:\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{C}$, $\frac{4-2b}{N}<q<\frac{4-2b}{N-2}$, $0<b<2$. In the case of $0<p<\frac{4}{N}$, the energy functional corresponding to this equation has a local minimizing structure. Therefore, we introduce a local minimizing problem. By studying the compactness of the minimizing sequence of this minimizing problem, we prove the existence of the minimizer of this minimizing problem, and finally obtain that the set of minimizer is orbitally stable.

Delay differential equation is a very important problem in the study of infinite-dimensional dynamical system and application. Recently, delay differential equation has attracted the attention and exploration of many scholars. In this paper we study the existence of global attractors and generalized exponential attractors in two-dimensional beam equations with state delay. By applying the Banach fixed-point theorem and the operator semigroup theory, we prove the existence and uniqueness of the mild solutions and the continuous dependence on the initial data. Further combining with the quasi-stability property we show the existence of generalized exponential attractors and global attractors with finite fractal dimensions.

Quantum Bernoulli noises are the family of annihilation and creation operators on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. Stochastic Schrödinger equation is a classical stochastic differential equation describing the evolution of open quantum systems in continuous quantum measurement. In this paper, the diffusion of linear stochastic Schrödinger equation with quantum Bernoulli noise method is studied. This model is expected to play a role in describing the evolution of open quantum systems interacting with QBN. Firstly, we prove some technical conclusions in Bernoulli functional space. In particular, we get spectral decomposition of counting operators. And then, with these theorems as main tools, inspired by the ideas of Pellegrini and Mora, we establish a theorem about the existence and uniqueness of a regular solution to the associated stochastic Schrödinger equation, which gives particular versions of previous results via QBN. Finally, we prove that the diffusion case can be obtained by the limit of the discrete process obtained by the measurement of non-diagonal observables, and some further results are obtained.

During the process of androgen deprivation therapy for prostate cancer (PCa), stochastic effects in the tumor microenvironment can cause treatment to fall short of the desired effect on tumor clearance. In this paper, the Lévy noise is introduced to describe the random changes of tumor cell number, and a system of stochastic differential equations (SDEs) driven by Lévy noise is established and analyzed. Firstly, through appropriate Lyapunov functions and the solution formula of SDEs driven by Lévy noise, the existence and uniqueness of the global positive solution are proved. Then, employing the theories and methods of stochastic dynamical systems, such as Lévy-Itô's formula, the comparison theorem of SDEs driven by Lévy noise, the exponential martingale inequality and the Borel-Cantelli lemma, the stochastic dynamics of extinction, non-persistence in the mean, weak persistence in the mean and stochastic permanence of PCa cells are studied. Finally, numerical simulation further verifies the theoretical results. Combined with theoretical analysis and numerical simulation, it can be found that the higher the intensity of Lévy noise, the easier the PCa cells are removed.

This paper is concerned with the time decay rates of strong solutions for the one-dimensional isentropic compressible micropolar fluid model with density-dependent viscosity. The pressure $p(\rho)=\rho^\gamma$ and the viscosity coefficient $\mu(\rho)=\rho^\alpha$ for some parameters $\alpha,\gamma\in\mathbb{R}$ are considered. By using a priori assumption and some refined energy estimates, we show that the global existence and large-time behavior of strong solutions with large initial data for the Cauchy problem under the perturbation of the constant state. Furthermore, by using the anti-derivative and time weighted energy method, the algebraic time decay rates for the specific volume $v(t,x)$ and the velocity $u(t,x)$ are also established.

In this paper, we investigate a partially observation insider trading model with different market structures and different risk preferences agents. With the help of conditional expectation, game theory and projection theorem, the related market characteristics are given, and then we explicate the corresponding economic significance under equilibrium conditions. It shows that: (1) The higher the observation precision by market makers, the market is basically strong efficient; on the contrary, it is counterproductive, and even the insider can get higher expected profits when she/he releases less private information. (2) Under Cournot game, the risk averse doesn't willing to trade, so the private information releases relatively slow. In Stackelberg game, the leader is conservative, while the follower is impulsive. In short, it is beneficial for risk neutral trader under Cournot game, and it is useful to follower under Stackelberg game. (3) When both insiders adopt a constant strategy, with the decreasing of market makers' observation accuracy, both the market liquidity and the residual information are increasing, however the price pressure of partially observation decrease slowly.

The adapted power generalized Weibull (APGW) model, which is generalized from the Weibull distribution, is a flexible parametric model and includes many popular survival models as special cases. In this paper, we extend the APGW model to the modeling of recurrent event data. A maximum likelihood estimating approach is proposed and the asymptotic properties are established. A model selection method is provided via variable selection through the SCAD penalty likelihood approach. The performance of both parameter estimation and model selection under finite samples is examined by simulation studies. To the end, the proposed method is applied to a bladder cancer recurrent event data for illustration.

The metric dimension problem (MDP) for graphs is a class of combinatorial optimization problems widely used in machine navigation, chemistry, network discovery and other fields. In order to solve this problem, a general optimization model with combinatorial properties is established in this paper, the resolving relationships between vertices and vertex pairs are described theoretically, the resolving graph and resolving probability distribution of graphs are proposed, and then the resolving probability distributions of some special graph classes are determined. An approximate metric dimension computation framework based on reinforcement learning and graph convolutional neural network is established by using machine learning sampling method. Numerical experiments show the effectiveness of graph machine learning algorithm to solve the graph metric dimension problem.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this article, we consider maximum likelihood estimator(MLE) of the parameter of Topp-Leone distribution and its properties under ranked set sampling(RSS), maximum ranked set sampling with unequal samples(MaxRSSU) and minimum ranked set sampling with unequal samples(MinRSSU). Theoretical results of asymptotic efficiencies of the above MLE show that the MaxRSSU estimator and the simple random sampling estimator have the same efficiency, the RSS estimator and the MinRSSU estimator are more efficient than the simple random sampling estimator.

In this paper, a nonlocal dispersal dynamic model of HIV latent infection with spatial heterogeneity is studied. We overcome the difficulty of non compactness caused by the nonlocal dispersal operator, and obtain the functional expression of the next generation operator $\mathcal{R}$ by using the renewal equation. Then, the basic reproduction number $R_0$ of the model is obtained, which is defined by the spectral radius of the next generation regeneration operator $\mathcal{R}$. Finally, the threshold dynamics of the system is analyzed. Specifically, by constructing appropriate Lyapunov functional, it is proved that the uninfected steady state is globally asymptotically stable when $R_0<1 $; Applying the consistent persistence theory of point dissipative systems, we prove that the system is uniformly persistent and has at least one positive steady state when $R_0>1$.

In this paper, under the assumption that the regularity domain $\Pi(T_{0})$ of the minimal operator $T_{0}$ on two intervals is non-empty, we give an analytic characterization of the $J$-self-adjoint extension domains for singular higher-order $J$-symmetric differential operators on two intervals by employing the direct sum theory and the solutions to the equation $\tau y=\lambda y$. This result includes the case when the endpoints are regular or singular and the deficiency index is arbitrary.

This paper deals with the uniqueness of a finite-order meormophic solution $f(z)$ of some delay differential equation $$ f(z+1)-f(z-1)+a(z)\frac{f'(z)}{f(z)}=b(z) $$ sharing $0, 1, \infty$ CM with a meromorphic function $g(z)$, where $a(z), b(z)$ are nonzero rational functions, then either $f(z)\equiv g(z)$ holds, either $f(z)=Ce^{ik\pi z}$ and $f(z)g(z)\equiv1$ holds, where $C$ is a nonzero constant, $k$ is a nonzero integer.

In a general unbounded domain, we study the large time behavior for the initial boundary value problem of a three-dimensional incompressible viscous magneto-hydrodynamic system. Using the theory of polishing operators, we first establish an approximate solution sequence; secondly, using spectral decomposition method and analytic semigroup theory, we give a new unified estimate for all the nonlinear terms in the equation system. Combining the energy estimation method and weak convergence theory, the existence of the global weak solution is ultimately proved, and long time decay rate is also give. In addition, it reveals that the algebraic decay property of the weak solution is generally dominated by its linear part (i.e., the semigroup solution of the Stokes equation).

Uniform designs are a kind of space-filling designs that are widely used in scientific experiments and industrial production due to their runs flexibility. Discrepancies are used to measure the overall uniformity of designs, but in practical problems, it is often necessary to consider the uniformity of the low-dimensional projection designs. In this paper, the conclusions of Wang et al. (2021) are generalized. Firstly, the uniformity pattern of any $q$-level design based on the reproducing kernel function is defined, and the minimum projection uniformity criterion is provided to select the designs with minimum low-dimensional projection uniformity. Secondly, the analytical relationship between the uniformity pattern and the generalized word length pattern of $q$-level designs is built, and an improved lower bound of the uniformity pattern is obtained, which can be a basis to measure the uniformity of projection designs. Finally, some numerical examples are given to verify the theoretical results.

In this paper, we study a water-borne epidemic model with multiple transmission ways. Firstly, we show the existence and uniqueness of global positive solution of the stochastic model by using suitable Lyapunov function. Moreover, by applying the Has'minskii theory, we obtain the existence of a ergodic stationary distribution of the positive solution of the model system under certain sufficient conditions. At last, we carry out numerical simulations to verify the analytical results. The results show that Random noise has a great influence on the spread of infectious diseases, and larger noise is beneficial to control the outbreak and spread of infectious diseases.

In this paper, we consider a one-dimensional Schrödinger equation with Dirichlet boundary condition, where Neumann control is suffered from bounded disturbance. On the one hand, we design sliding mode control and analyze stability property of the closed-loop system. Firstly, we transform the system by an invertible transformation. And then we prove the system is well-posed. Secondly, we design the sliding mode surface. And we show the system on the sliding mode surface is exponentially stable. Thirdly, we construct sliding mode control, and prove the system will reach the sliding mode surface in a finite time interval which shows stability of the closed-loop system. On the other hand, we construct high gain estimators and design active disturbance rejection control, and then analyze asymptotical stability of the closed-loop system. Finally, we simulate states of the closed-loop systems under both sliding mode control and active disturbance rejection control respectively. Simulation results show the two controllers are effective to make the original system stable.

We mainly consider the existence of random uniform exponential attractors for the Boussinesq lattice system with quasi-periodic forces and multiplicative white noise. Firstly, by using the Ornstein-Uhlenbeck process, we transfer the stochastic Boussinesq lattice system (SDE) with multiplicative white noise into a random Boussinesq lattice system (RDE) without white noise. Then, we verify that this RDE system's solutions can define a jointly continuous random dynamical system. Next, we testify the existence of a uniform absorbing set for this system and construct a tempered bounded and closed absorbing random set. And we verify the Lipschitz continuity and the random squeezing property on this absorbing random set, which can be solved by estimating the “tail” of the solutions and decomposing the difference between two solutions of the system appropriately. Finally, according to the criterion for the existence of a random uniform exponential attractor for the jointly continuous random dynamical system, we obtain the existence of random uniform exponential attractors for the considered system of this paper.

Within the framework of statistical learning theory, this paper studies the learning rate of the maximum correntropy regression model under a mixture of symmetric triangular noise, the efficiency and robustness of estimates under the limited samples, and the application on real data. The results show that the maximum correntropy regression model has the asymptotically optimal convergence rate, a good estimation effect under limited samples, and is better than the Huber regression model and least square regression model in robustness.

This paper establishes an existence and uniquness result and a comparison theorem for solutions to backward stochastic differential equation driven by $G$-Brownian motion, where $p>1$, the generators $f$ and $g$ satisfy the $p$-order weak monotonicity condition in $y$, and Lipschitz condition in $z$, the terminal condition $\xi$ satisfies the $p'$-order integrable condition, and $p'>p$.

We study dynamical systems which have bounded complexity with respect to the Bowen metric $d_{n}$. It is shown that any topological dynamical system $G\curvearrowright X$ for actions of countable group $G$ is equicontinuous if and only if $X$ has bounded topological complexity with respect to $\{d_{n}\}_{n=1}^{\infty}$. Meanwhile, it is shown that for any topological dynamical system $G\curvearrowright X$ for actions of countable group $G$ and a Borel probability measure $\mu$ on $X$, $\mu$ has bounded measure-theoretic complexity with respect to $\{d_{n}\}_{n=1}^{\infty}$ if and only if $G\curvearrowright X$ is $\mu$-equicontinuous. These generalize some results of Huang, Li, Thouvenot, Xu and Ye.

At present, research on functional regression models is mainly based on the estimation of mean regression. However, mean regression only studies the influence of covariates on the mean position of response variables in the conditional distribution, and cannot reflect the relationship between the two at the tail of the conditional distribution, which can lead to information leakage and be easily affected by outliers. At the same time, there is currently no relevant research on partially functional linear additive models in the spatial dimension. In fact, economic relationships between variables exhibit more nonlinear characteristics in space, and ignoring this nonlinear relationship in spatial lag models can easily lead to model setting errors. To overcome the above shortcomings, this paper combines parametric models and semi parametric models with functional data to propose a new partially functional linear additive spatial lag quantile. Regression model. Furthermore, A tool variable estimation method for the model was constructed based on functional principal component analysis and B-spline approximation. Under some regular conditions, the consistency and asymptotic normality of the model parameter estimates were given, and the optimal convergence speed of the function estimates was obtained. The large sample nature of these estimates was also proved. The model can reflect spatial dependence and the influence of functional data, as well as capture multiple nonlinear effects caused by covariates, reducing the risk of model error, solving the curse of dimensionality, and having high robustness. Finally, numerical simulations and practical applications show that the proposed model and method are effective.

In specific practical problems, structures with ordered and inequality constraints find extensive applications in multiple fields. When estimating parameters in such cases, one often encounters restrictions imposed by ordered constraints. For instance, in pharmaceutical testing, researchers may impose constraints on the dosage of a particular drug, necessitating consideration of ordered constraints. Therefore, ordered constraint mean estimation holds significant practical value in these applications. In previous literature, the focus on mean estimation under ordered constraints has been mainly on 2 and 3 multivariate normal populations. This paper extends this focus to the estimation problem of $k$ multivariate normal population means under simple partial order constraints. It introduces a new estimate $\tilde{\mu}$, based on the PAVA algorithm when the population covariance matrix $\Sigma_{i}$ is known, and proves its consistent superiority over the unordered maximum likelihood estimate $\bar{X}$. Finally, the effectiveness of the proposed estimation method is validated through simulation experiments and compared with the maximum likelihood estimation method.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this paper, a maximum likelihood estimator (MLE) of the parameter of the SBB distribution and its properties are respectively studied under simple random sampling (SRS) and ranked set sampling(RSS). Both theoretical and numerical results demonstrate that the MLE under RSS is asymptotically more effective than the MLE under SRS. Additionally, we investigate the asymptotic efficiency of the MLE under imperfect ranked set sampling (IRSS), taking into account the potential presence of ranked errors. Numerical results show that the asymptotic efficiency is influenced by the ranked judgement, but the MLE under IRSS is at least as effective as the MLE under SRS.