JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Piecewise linear Tobit regression model estimation based on kernel function method

WANG Xiao-gang, LI Bing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 1-9. doi:10.6040/j.issn.1671-9352.0.2019.865

Abstract ( 1488 )

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In the change point estimation model, the common grid search method has the disadvantages of cumbersome calculation and unreal meaning, and the linearization technique cannot prove the large sample properties. In order to overcome these shortcomings, the kernel function method is proposed in the piecewise linear Tobit regression model to solve the problem of non-differentiable of the objective function, the change point and parameter estimation are obtained, the asymptotic properties are proved, and the validity and robustness of the estimation are verified by Monte Carlo simulation. The empirical analysis of the family annual financial asset data shows that family financial assets would be positively affected by the level of education, and the educational level would change before and after the undergraduate level.

Parameter estimation of linear exponential distribution based on truncated and censored data

SUI Yun-yun, HU Jiang-shan, MA Shu-cai, FU Yun-peng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 10-16. doi:10.6040/j.issn.1671-9352.0.2019.752

Abstract ( 1170 )

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The parameter estimation of linear exponential distribution based on truncated and censored data model is studied. The maximum likelihood estimation method and EM algorithm are respectively used to estimate the unknown parameters, and stochastic simulations are carried out. It is found that the parameter estimation obtained by maximum likelihood estimation is similar to that obtained by EM algorithm, but the convergence speed of EM algorithm is faster.

Hypothesis testing of maximum likelihood estimation for stochastic PDE

WANG Xiao-wen, LYU Yan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 17-22. doi:10.6040/j.issn.1671-9352.0.2019.026

Abstract ( 1183 )

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The hypothesis testing problem for the parameter estimation for a class of stochastic partial differential equation driven by additive noise is studied. The explicit expressions for rejection domain of the unknown parameter in two asymptotic cases: T→∞ with N fixed and N→∞ with T fixed are shown and their asymptotic properties are further proved.

Bochner-Wick integral for S ^*(M)space

SHI Jia, WANG Cai-shi, ZHANG Li-xia, ZHANG Yin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 23-31. doi:10.6040/j.issn.1671-9352.0.2019.655

Abstract ( 1273 )

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Let S ^*(M)be a generalized functional space of discrete-time normal martingale M. This paper introduces and discusses the integral operation of S ^*(M)-valued measure and S ^*(M)-valued function in the framework of Bernoulli noise analysis. First of all, the concept of S ^*(M)-valued measure is defined. On this basis, this paper deeply investigates the properties of S ^*(M)-valued measure by Fock transform, and obtains the appropriate conditions for the countably additive of this kind of measure in the sense of norm. Next, Bochner-Wick integral for S ^*(M)-valued function with respect to S ^*(M)-valued measure is defined and the corresponding control convergence theorem is established.

Precise asymptotics of complete moment convergence for ρ-mixing sequence

DENG Xiao-qin, WU Qun-ying

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 32-40. doi:10.6040/j.issn.1671-9352.0.2019.703

Abstract ( 1070 )

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There is a ρ-mixing sequence with mean zero and same distribution. Under some suitable moment conditions, by using the central limit theorem for ρ-mixing sequence on moving average process and its moment inequality, and the method of multiple censoring, the paper obtain the relevant conclusion that precise asymptotics of complete moment convergence of the ρ-mixing random variables on the moving average process, which extends the result of independent random variables.

Existence of solutions to initial value problems of fractional differential equations of variable-order

AN Jia-hui, CHEN Peng-yu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 41-47. doi:10.6040/j.issn.1671-9352.0.2019.757

Abstract ( 1110 )

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By using Schauder fixed point theorem, this paper obtains the existence of solution for the following initial value problem of fractional differential equations of variable-order{D^q(t)0+ x(t)=f(t,x), 0<t≤T,x(0)=0,where 0<q(t)<1, 0<T<+∞, D^q(t)₀₊ is the fractional derivative of Riemann-Liouvile of variable-order q(t).

Existence of solutions for fractional Kirchhoff-type equation with variable exponent

ZHANG Shen-gui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 48-55. doi:10.6040/j.issn.1671-9352.0.2019.785

Abstract ( 1251 )

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This paper studies a class of Dirichlet boundary value problem for fractional Kirchhoff-type equation with variable exponent. When the nonlinear term is p⁺-superlinear at infinity, some sufficient condition for the existence of infinitely many solutions is established by employing the critical point theory, variational methods and the theory of fractional variable exponent space.

Local Gevrey regularity and analyticity of the solutions to the initial value problem associated with the two-component Novikov system

WANG Hai-quan, CHONG Ge-zi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 56-63. doi:10.6040/j.issn.1671-9352.0.2019.921

Abstract ( 1166 )

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Considered herein is the initial value problem for the two-component Novikov system. At first, the Gevrey regularity and analyticity of the solutions to this problem in Sobolev-Gevrey spaces G1_r,s(R)×G¹_r,s-1(R)with s>3/2, r≥1 are investigated by making use of the generalized Ovsyannikov theorem. Next, the continuity of the solution map z0→z(t) is discussed. The results can be directly applied to Novikov equation.

Blow up of solutions to wave equations in the de Sitter spacetime

MENG Xi-wang, WANG Juan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 64-75. doi:10.6040/j.issn.1671-9352.0.2019.459

Abstract ( 1046 )

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This paper mainly studies the lifespan of a solution to Klein-Gordon wave equations in the de Sitter spacetime by considering a special form of the Klein-Gordon equation and studying its Cauchy problem. By a transformation of unknowns and by using the method of heat kernel, as well as the properties of semigroup, the blow up of this Cauchy problem is proved, and the upper bound of its life span is also obtained.

Local gradient estimates for weak solutions of obstacle problems to a class of A-harmonic equations

ZHANG Ya-nan, YANG Ya-qi, TONG Yu-xia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 76-83. doi:10.6040/j.issn.1671-9352.0.2019.871

Abstract ( 1025 )

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This paper deals with the local gradient estimates for weak solutions of obstacle problems to a class of A-harmonic equations. L^p-type estimates to weak solutions of obstacle problems are derived. The L^φ-type estimates in Orlicz space are also obtained by a new normalization method and a new iteration-covering approach.

Existence of positive solutions for a class of semi-positive nonlinear elastic beam equation boundary value problems

WANG Jing-jing, LU Yan-qiong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 84-92. doi:10.6040/j.issn.1671-9352.0.2019.781

Abstract ( 1110 )

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By using the fixed point theorem of cone mapping, it obtains the existence and multiplicity of positive solutions for the semipositive nonlinear elastic beam equation boundary value problems{y⁽⁴⁾(x)+(k₁+k₂)y″(x)+k₁k₂y(x)=λf(x,y(x)), x∈［0,1］,y'(0)=y'(1)=y(0)=y(1)=0with Neumann boundary conditions, where 0<k1<k2≤(π2)/4, λ>0 is a parameter and f∈C(［0,1］×［0,∞),(-∞,∞)), with f(x,y)≥-X for some positive constant X.

Existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem

LI Zhao-qian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 93-100. doi:10.6040/j.issn.1671-9352.0.2020.003

Abstract ( 1272 )

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This paper is devoted to investigate the existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem{u⁽⁴⁾(t)=f(t,u(t),u'(t),u″(t),u(t)), t∈(0,1),u(0)=u'(0)=u'(1)=u″(1)=0,where f:［0,1］×R⁴→R is a Carathéodory function. The existence of the solution is proved in the case f is utmost linear growth. Furthermore the existence and uniqueness of the solution is also obtained when f is Lipschitz type. The proof of the main results is based on Leray-Schauder fixed point theorem.

Existence and uniqueness of solutions for a class of boundary value problems of nonlinear fourth-order ordinary differential equations

YANG Li-juan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 101-108. doi:10.6040/j.issn.1671-9352.0.2020.062

Abstract ( 968 )

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This article considers the existence and uniqueness of solutions of boundary value problems of nonlinear fourth-order ordinary differential equations{u⁽⁴⁾(t)=f(t,u(t),u'(t),u″(t)), a.e. t∈(0,1),u(0)=u″(0)=u(1)=u″(1)=0,where nonlinearity f:［0,1］×R³→R is a Carathéodory function. The existence of solutions is obtained when f satisfies the condition of proper utmost linear growth by using the Leray-Schauder principle. Furthermore, the uniqueness of solutions is proved when f satisfies the Lipschitz condition.

Existence of positive periodic solutions for a class of non-autonomous fourth-order ordinary differential equations

YANG Hu-jun, HAN Xiao-ling

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 109-114. doi:10.6040/j.issn.1671-9352.0.2019.762

Abstract ( 1214 )

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This paper studies the existence of positive periodic solutions for a class of non-autonomous fourth-order ordinary differential equations u^(iv)+pu″+a(x)uⁿ-b(x)uⁿ⁺¹-c(x)uⁿ⁺²=0,where p≥-1, n is a finite positive integer, a(x), b(x), c(x) are continuous T-periodic functions. By using Mawhins continuation theorem, the existence of positive periodic solutions for this kind of equations is proved.

Crank-Nicolson finite difference method for parabolic optimal control problem

YANG Cai-jie, SUN Tong-jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 115-121. doi:10.6040/j.issn.1671-9352.0.2020.101

Abstract ( 2390 )

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A one dimensional parabolic optimal control problem with Neumann boundary condition is considered. The co-state equations and optimality conditions are presented and the optimality system is obtained. By applying a ghost-point based central difference approximation to the boundary condition, Crank-Nicolson finite difference discrete schemes are established for the optimality system. The maximum norm error estimates are proved to be of second-order convergence in both time and space for the state, co-state and control variables. Finally, numerical example is presented. In order to avoid solving large coupled algebraic equations, the iterative method is used. The numerical results validate the theoretical conclusion.

A modified Jacobi iterative method for large-size linear systems

WANG Yan-rong, CHEN Yun-lan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(6): 122-126. doi:10.6040/j.issn.1671-9352.0.2019.003

Abstract ( 1322 )

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An improved Jacobi method is proposed for solving large-size linear system. The improvement of the proposed method is to use the combination of the current point obtained by Jacobi method and the previous point to get the new point, to get the optimal factor of linear combination by solving the least square optimization. Under the same conditions as the classic Jacobi method, the global convergence of this method is proved, the computational results show the effectiveness of this method.

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