Table of Content

  

20 September 2026
Volume 61 Issue 9

Kernel-based estimation for threshold modal regression model

WANG Xiaogang, ZHU Qingjun, SHI Yating

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  1-13.  doi:10.6040/j.issn.1671-9352.0.2025.255

Abstract ( 60 )   PDF (1183KB) ( 41 )   Save

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For the threshold modal regression model, effective estimate of the threshold location is constructed using the kernel-based method, and the large-sample properties of the threshold location estimation are established, thereby overcoming the limitations of grid search and linearization techniques. The effectiveness and robustness of the estimation are verified by numerical simulations. An empirical analysis based on the income of urban households shows that income is positively correlated with the education level, and there is a threshold around the undergraduate education level.

Local sparse estimation of functional complementary log-log regression model

GAO Haiyan, ZHAO Jingxian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  14-22.  doi:10.6040/j.issn.1671-9352.0.2024.354

Abstract ( 62 )   PDF (2008KB) ( 15 )   Save

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A sparse functional complementary log-log regression model(SFCLR)is proposed to effectively identify the zero sub-regions of the coefficient function in scalar response variables and functional covariates. The model leverages the compact support property of B-spline basis functions, integrates smoothing techniques for functional data with sparse regularization strategies, and utilizes the Newton-Raphson algorithm to optimize the penalized likelihood function, thereby achieving local sparse estimation. Numerical simulations based on common functions and Raman spectroscopy data demonstrate that the SFCLR model not only successfully identifies zero regions of coefficient functions but also produces continuous and smooth estimates in non-zero coefficient regions. Additionally, an empirical analysis using the Tecator fat dataset reveals that the SFCLR model can identify wavelength intervals that significantly contribute to determining obesity characteristics, further validating the models effectiveness.

Sparse and robust estimation of longitudinal data regression models based on the exponential squared function

CHENG Yuhe, ZHANG Bin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  23-34.  doi:10.6040/j.issn.1671-9352.0.2024.309

Abstract ( 43 )   PDF (3048KB) ( 7 )   Save

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First, the exponential squared function is used to construct a new score function to achieve a robust estimation of the regression parameters. Second, the empirical likelihood function is penalized by the smoothly clipped absolute deviation, so that the mean and covariance matrix estimates simultaneously possess good statistical properties of sparsity and robustness. Additionally, a Newton-Raphson iterative algorithm for solving the model is provided. Finally, the asymptotic normality of the parameter estimators is proved, and the proposed method is compared with existing methods through numerical simulations, demonstrating superior performance in terms of robustness and sparsity across different covariance structures and error distributions.

Variable selection of SCAD-L₂ penalty for the partially linear additive higher-order spatial autoregressive model

HUANG Shi, CHENG Suli, GUO Shengliang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  35-48.  doi:10.6040/j.issn.1671-9352.0.2025.200

Abstract ( 35 )   PDF (2531KB) ( 13 )   Save

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A SCAD-L₂ penalized least squares method for partially linear additive higher-order spatial autoregressive models is proposed, which can simultaneously perform group variable selection and parameter estimation for higher-order spatial lag terms and linear explanatory variables. It is proved that the proposed variable selection method possesses sparseness and asymptotic normality. The Monte Carlo numerical simulation further confirmed the excellent properties of the SCAD-L₂ method in small samples. Finally, the proposed approach is applied to the analysis of the Boston housing price dataset, and the results show that the average number of rooms and the proportion of lower-status population have a significant impact on Boston housing price.

Robustness comparison of ROBPCA methods

FANG Hongyan, ZHANG Qiaoqiao, YANG Xinyu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  49-55.  doi:10.6040/j.issn.1671-9352.0.2024.120

Abstract ( 34 )   PDF (2431KB) ( 6 )   Save

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An adaptive robust PCA method is developed by employing the CUSUM chart method in change point detection. The outlier detection performances of robust PCA methods are compared through numerous simulation studies. The robust PCA based on the CUSUM chart method obtains the closest sample size to the true value, has the highest screening accuracy, and has a superior performance even when the variable dimension becomes larger. The analysis and application of the actual data of national economic indicators in counties and cities in Zhejiang province also verifies the accuracy and reliability of this method.

Prediction of HIV infection rates with missing data: based on a functional data analysis approach

PENG Qianqian, LING Nengxiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  56-63.  doi:10.6040/j.issn.1671-9352.0.2025.053

Abstract ( 48 )   PDF (3357KB) ( 14 )   Save

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A partial functional linear quantile regression model is constructed to address the prediction of human immunodeficiency virus(HIV)infection rates. The model overcomes the limitations of traditional mean regression and captures the influence characteristics of different indicators on HIV infection rates. For the complex scenario where both functional explanatory variable and scalar response variable are missing in real-world data, a joint processing method integrating functional data reconstruction and regression imputation is proposed. Additionally, the proposed method is compared with other commonly used curve reconstruction and data imputation approaches. The results demonstrate that the proposed method exhibits advantages in handling missing data and improving prediction accuracy.

Existence and smoothness for multiple local time of fractional Brownian motion

ZHANG Cuiyun, GUO Jingjun, HU Feng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  64-73.  doi:10.6040/j.issn.1671-9352.0.2024.345

Abstract ( 33 )   PDF (800KB) ( 10 )   Save

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The Wiener chaos decomposition for the multiple local time of one-dimensional fractional Brownian motion is established. Its existence and smoothness are proved. The relevant results are extended to the d-dimensional fractional Brownian motion for d≥2. When x=0, the Wiener chaos decomposition and smoothness for the multiple local time of d-dimensional fractional Brownian motion are obtained.

European option pricing with double jump driven by sentiment

DUAN Lina, FAN Xiaoming

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  74-83.  doi:10.6040/j.issn.1671-9352.0.2024.338

Abstract ( 44 )   PDF (5218KB) ( 12 )   Save

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On the basis of log mean-reverting(LMR)model, double exponential jump(DEJ)and investor sentiment are taken as influencing factors to study their impact on European option pricing, and the log mean-reverting double exponential jump diffusion model with investor sentiment(LMR-DEJ-S)is constructed. The Shanghai Stock Exchange 50 ETF option price is predicted using the LMR-DEJ-S model, BS model, Merton model, Heston model and log mean-reverting double exponential jump diffusion model(LMR-DEJ)without investor sentiment. The mean square error, mean absolute error and mean absolute percentage error are selected as evaluation indicators. The results show that the LMR-DEJ-S model has a smaller prediction error and better fitting accuracy.

Gap option pricing under the mixed sub-fractional Vasicek stochastic rate model

LI Li, SONG Ruili

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  84-95.  doi:10.6040/j.issn.1671-9352.0.2024.405

Abstract ( 45 )   PDF (2769KB) ( 8 )   Save

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A mixed sub-fractional Vasicek stochastic interest rate model is established based on the risky asset price process driven by mixed sub-fractional Brownian motion combined with the Vasicek stochastic interest rate model. With the help of Duffies idea of affine structure solution, the pricing formula of zero-coupon bond is derived. The partial differential equation satisfied by the option price is obtained using the Δ-hedging principle, and finally, the pricing formula for gap options is solved through variable substitution. Numerical simulations analyze the effects of Hurst index H, initial underlying asset price S, gap G, maturity date T, stock volatility σ₂, and interest rate volatility σ_r2 on option prices. The results show that the increase in H, G, and σ_r2 will reduce the option price, while the increase in the S, T, and σ₂ will increase the option price, which extends the existing research results on gap options.

Volatility forecasting of stock market based on deep learning model with mixed-frequency data

ZHANG Yufeng, ZHANG Tonghui, WANG Yuanyue

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  96-107.  doi:10.6040/j.issn.1671-9352.0.2025.119

Abstract ( 52 )   PDF (1718KB) ( 15 )   Save

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To address challenges in stock market volatility prediction, such as multiple influencing factors, mixed data frequencies, and nonlinear relationships, the Shanghai Composite Index is selected as the modeling target, and a hybrid framework is constructed by combining the GARCH-MIDAS model with the CNN-BiLSTM-Attention deep learning model. Monthly macroeconomic indicators, daily economic policy uncertainty, and technical trading indicators are introduced as predictive variables. Empirical results show that the CNN-BiLSTM-Attention model outperforms benchmark models, achieving the lowest out-of-sample prediction loss. In particular, its mean squared error is reduced by 47 percent compared to the LSTM model. The inclusion of macroeconomic and other predictors also provides valuable information, enhancing forecasting performance.

Robust detection of persistence change point in heteroscedastic sequences

SONG Jiale, JIN Hao, YANG Yunfeng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  108-122.  doi:10.6040/j.issn.1671-9352.0.2025.133

Abstract ( 50 )   PDF (1759KB) ( 12 )   Save

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In persistence change-point testing for strongly mixing heteroscedastic sequences, a ratio-based test statistic using least absolute deviation(LAD)estimation is proposed to address the bias in least squares estimation caused by heteroscedasticity. Theoretically, this approach not only avoids the necessity for consistent estimation of long-term variance but also eliminates the need for assumptions or estimation of the variance function. Under some certain conditions, the limit distribution of the statistic under the null hypothesis is derived, and its consistency is proven under the alternative. Numerical simulation indicates that the Ratio-type test based on LAD estimation exhibits good empirical size, and the empirical power also demonstrates satisfactory robustness compared with the least squares estimation. Finally, a set of stock price data are used to further validate the effectiveness and feasibility of the proposed method.

Ratio test for the mean gradual model of an independent identically distributed sequence

ZHENG Dongxue, WEI Yuesong, XU Tianming

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(9):  123-130.  doi:10.6040/j.issn.1671-9352.0.2024.099

Abstract ( 30 )   PDF (2689KB) ( 18 )   Save

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In view of the possible change point problem in the random error term from an independent identically distributed sequence, an improved Ratio test statistic is proposed to test the existence of change points, and the limit distributions of the Ratio test statistic under the null hypothesis and alternative assumptions are also obtained. Finally, the effectiveness of the Ratio method is illustrated by numerical simulation and empirical analysis of instances.