26 December 2022, Volume 38 Issue 6

  

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  Variable Selection and Its Asymptotic Theory Based on L_\gamma Penalty in Generalized Linear Models with Adaptive Designs

  GAO Qibing, GUO Zihan, ZHU Guimei, SHI Qianqian

  2022, 38(6): 791. https://doi.org/10.3969/j.issn.1001-4268.2022.06.001

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  In this paper, L_\gamma penalty method proposed by Frank and Friedman\ucite{3} is used to study the variable selection problem and its asymptotic properties based on the penalized quasi-likelihood method in generalized linear models with adaptive designs. This method can perform parameter estimation and variable selection simultaneously. For 0<\gamma<1, the existence, consistency and Oracle properties of the estimators based on L_\gamma penalty and the quasi-likelihood method in generalized linear models with adaptive designs are proved under appropriate conditions. These results generalize the related theories of generalized linear models from the case of fixed designs to the case of adaptive designs. The validity of our obtained theory is verified by numerical simulation and real data analysis inthis paper.
  
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  Precise Rates in the Generalized Law of the Iterated Logarithm in the Hilbert Space#br#

  XU Mingzhou, CHENG Kun

  2022, 38(6): 807. https://doi.org/10.3969/j.issn.1001-4268.2022.06.002

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  Let $\{X,X_n,n\ge1\}$ be a sequence of i.i.d. random variables taking values in a real separable Hilbert space $(\mathbb{H},\|\cdot\|)$ with covariance operator $\Sigma$, set $S_n=X_1+X_2+\cdots+X_n$, $n\ge 1$. For every $m>0$ and $a_n=O((\ln\ln n)^{-2m})$, we study the precise rates in the generalized law of the iterated logarithm for a kind of weighted infinite series of $\pr\{\|S_n\|\ge(\epsilon+a_n)\sigma\sqrt{n}(\ln\ln n)^m\}$. Let $\beta_n(\epsilon)=o(\sqrt{1/\ln\ln n})$. We also prove that, for any $r>1$ and $\alpha>-d/2$, \begin{align*} &\lim_{\epsilon\searrow\sqrt{r-1}}[\epsilon^2-(r-1)]^{\alpha+d/2} \tsm_{n=1}^{\infty}\frac{1}{n}(\ln n)^{r-2}(\ln\ln n)^{\alpha} \pr\{\|S_n\|\ge\sigma\psi(n)[\epsilon+\beta_n(\epsilon)]\}\\ =\;&\Gamma^{-1}(d/2)K(\Sigma)(r-1)^{(d-2)/2}\Gamma(\alpha+d/2) \end{align*} holds if $\ep X=0$, $\ep[\|X\|^2(\ln\|X\|)^{r-1}]<\infty$.}
  
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  Optimal Estimation of Confidence Set for a Bathtub-Shaped Lifetime Distribution under Records#br#

  WANG Liang, ZUO Xuanjia

  2022, 38(6): 825. https://doi.org/10.3969/j.issn.1001-4268.2022.06.003

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  In this paper, confidence set estimation is considered for the parameters from a bathtub-shaped lifetime distribution when record value is available. Based on proposed pivotal quantities, exact confidence intervals and confidence regions are established firstly for the model parameters. In addition, the corresponding optimal confidence intervals and confidence regions are also constructed by using Lagrange multiplier method under given significance level. Finally, simulation studies and a real life example are provided for illustration.
  
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  Synchronization of Nonlinearly-Coupled Neural Networks with Markov Jump and Time-Varying Delay#br#

  DONG Hailing, TANG Juan, XIAO Dichang

  2022, 38(6): 836. https://doi.org/10.3969/j.issn.1001-4268.2022.06.004

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  In this paper, the synchronization problem of a class of nonlinear coupled neural networks with Markovian jump and variable delay is discussed. The coupling strength of the model is a random variable, the coupling structure of the network switches dynamically according to a continuous time Markov chain, and the influence of nonlinear coupling term and time-varying delay is considered. By constructing a suitable Lyapunov function and using the linear matrix inequality method, the sufficient conditions for the global mean square asymptotic synchronization of the network model are obtained. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical results.
  
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  A Defined Contribution Pension Plan with Multiple Risks under the Mean-Variance Criterion#br#

  ZHAO Leilei, CHANG Hao, LI Jiaao

  2022, 38(6): 847. https://doi.org/10.3969/j.issn.1001-4268.2022.06.005

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  This paper studies the optimal investment strategy for a defined contribution pension plan with multiple risks under the mean-variance criterion. In the pension accumulation stage, the level of interest rate, volatility level and wage level are considered to be random, in addition, it is assumed that the term structure of interest rate is driven by the stochastic affine interest rate model, and the stock price is modeled by Heston's stochastic volatility model. By using the principle of stochastic dynamic programming and Lagrangian duality theorem we obtain the explicit solutions for the efficient strategy and the efficient frontier. A numerical example is given to analyze the effects of interest rate parameters, volatility parameters, and salary parameters on efficient strategies and efficient frontiers. Research results show that the capital market line with interest rate risk, volatility risk and salary risk environments is still a straight line in a mean-standard deviation plane. That is to say, the efficient strategy only depends on both instantaneous interest rate level and instantaneous salary level, while the efficient frontier does not depend on the interest rate level, volatility level and salary level.
  
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  Solving the Finite-Time Ruin Problems by Laguerre Series Expansion#br#

  XIE Jiayi, ZHANG Zhimin, YU Wenguang

  2022, 38(6): 867. https://doi.org/10.3969/j.issn.1001-4268.2022.06.006

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  In this paper, we study the finite-time ruin problems in the perturbed compound Poisson risk model. The finite-time Gerber-Shiu discounted penalty function and its decomposition are studied. Different from the Laplace transform method, we propose a novel method for computing the finite-time Gerber-Shiu functions by the Laguerre series expansion. When the individual claim size density function is a finite combination of exponentials,we derive the infinite series expansions for the Gerber-Shiu functions. Some numerical examples are also given to confirm the applicability of our method.
  
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  Nonparametric Estimation of Jump Characteristics under Market Microstructure Noise#br#

  FU Jinyu, LIN Jinguan

  2022, 38(6): 887. https://doi.org/10.3969/j.issn.1001-4268.2022.06.007

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  By exploiting financial high frequency data, we nonparametrically estimate the jump characteristic in the presence of market microstructure noise. Our estimator is based on the realized range increments and threshold technique. Besides, the bias caused by microstructure noise can be estimated and removed, if it is modeled as ask-bid spread, which is a used frequently assumption. We further present the asymptotic properties of the proposed estimator. Simulation studies show the estimator works well under microstructure noise. Finally, the estimator is also applied to the real data.
  
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  Estimation of Partial Functional Linear Regression Model under Dependent Condition#br#

  LI Qifang, SU Zhifang

  2022, 38(6): 904. https://doi.org/10.3969/j.issn.1001-4268.2022.06.008

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  Partial functional linear regression model refers to a type of regression machine that contains mixed functional and numerical data at the input and numerical data at the output. In the existing partial function linear regression machine estimation algorithm, it is assumed that functional data sample follow independent and identical distribution, which is inconsistent with the dependent characteristics of functional data in the financial and other fields. Therefore, the article first proposes two data-driven functional principal components representation methods for function data, then the regression coefficient function is regularized, and finally the estimation of the partial functional linear regression machine is transformed into the estimation of the multiple linear regression machine. The Monte Carlo simulation results show that the methods proposed in this paper have smaller parameter estimation errors and higher out-of-sample prediction accuracy when dealing with dependent data, the case analysis also shows the effectiveness in stock forecasting.
  
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  Limit Behaviors for a Critical Galton-Watson Process with Immigration#br#

  SHI Wanlin, LI Doudou

  2022, 38(6): 919. https://doi.org/10.3969/j.issn.1001-4268.2022.06.009

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  We consider a critical Galton-Watson branching process with immigration Z_n, and study the convergence rate of the harmonic moments of this process, improving the results in previous literatures. The proof is based on the local probabilities estimations of Z_n. As applications, we obtain the large deviations of S_{Z_n}:=\tsm_{i=1}^{Z_n}X_i, where \{X_i,i\geq 1\} is a sequence of independent and identically distributed random variables, and X_1 is in the domain of attraction of an $\alpha$-stable law with \alpha\in(0,2).
  
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  Existence and Uniqueness of Strong Solution to a Stochastic Partial Differential Equation with Gradient#br#

  YANG Xu, TANG Mengting

  2022, 38(6): 931. https://doi.org/10.3969/j.issn.1001-4268.2022.06.010

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  In this paper we establish the existence and uniqueness of strong solution to a stochastic partial differential equation driven by Gaussian colored noise and with gradient in drift and diffusion terms and non-Lipschitz coefficients