The difference-based partially linear model is an appropriate regression model when both linear and nonlinear predictors are present in the data. However, when we want to optimize the weights using the difference-based method, the problem of variable selection can be difficult since low-variance predictors present a challenge. Therefore, this study aims to establish a novel methodology based on marginal theory to tackle such mixed relationships successfully, emphasizing variable selection in low dimensions. We suggest using a marginalized LASSO estimator with a penalty term that is not as severe and related to the difference order. As part of our numerical analysis of small sample performance, we undertake comprehensive simulation experiments to numerically demonstrate the strength of our proposed technique in estimation and prediction compared to the LASSO under a low-dimensional setup. This is done so that we can numerically demonstrate the strength of our proposed method in estimation and prediction. The bootstrapped method is utilized to evaluate how well our proposed prediction method performs when examining the King House dataset.

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