Robust Variable Selection for High-dimensional Regression with Missing Data and Measurement Errors (2410.16722v3)

Abstract: In our paper, we focus on robust variable selection for missing data and measurement error. Missing data and measurement errors can lead to confusing data distribution. We propose an exponential loss function with a tuning parameter to apply to Missing and measurement errors data. By adjusting the parameter, the loss function can be better and more robust under various data distributions. We use inverse probability weighting and additive error models to address missing data and measurement errors. Also, we find that the Atan punishment method works better. We used Monte Carlo simulations to assess the validity of robust variable selection and validated our findings with the breast cancer dataset.

• The paper introduces an exponential squared loss function enhanced by a tuning parameter h to robustly address missing data and measurement errors.
• It integrates inverse probability weighting and an additive error model to correct biases in high-dimensional regression.
• The use of the Atan penalty method demonstrates superior variable selection accuracy via Monte Carlo simulations and real-world data analysis.

Robust Variable Selection for High-Dimensional Regression with Missing Data and Measurement Errors: A Summary

Zhang presents a robust methodology for variable selection in high-dimensional regression settings, particularly where datasets are challenged by missing data and measurement errors. The paper introduces an exponential loss function with a tuning parameter, h, and integrates inverse probability weighting (IPW) alongside an additive error model. The Atan penalty method is also applied to enhance robustness and accuracy.

Methodology

The traditional squared loss function, which assumes a normal data distribution, often struggles with datasets that harbor missing values and measurement errors. These issues can lead to substantial estimation biases if not addressed effectively. To mitigate this, the paper proposes an exponential squared loss function. By adjusting the tuning parameter $h$, the function is adaptable to various data distributions, ensuring robustness for any configuration within the interval $h \in (0, +\infty)$.

Zhang explores the linear regression model where the relationship between the response variable $Y$ and covariates $X$ is expressed in a standard linear form. The exponential squared loss function modifies this by transforming the typical squared error using the parameter $h$. For large values of $h$, the behavior approaches traditional least squares, while small $h$ enhances robustness against outliers.

Managing Missing Data and Measurement Errors

To handle missing data, the methodology applies inverse probability weighting (IPW). IPW compensates for selection bias that arises in non-randomized treatment assignments commonly observed in observational studies. For measurement errors, the paper employs an additive error model, where the errors are presumed independent and normally distributed.

Variable Selection and Penalty Methods

Given the high dimensionality of the data, it is critical to select sparse, relevant variables. The paper employs the Atan penalty method, which is noted for its unbiasedness and sparsity properties compared to conventional penalties like Lasso, SCAD, and MCP.

The Atan penalty function enhances model selection accuracy by effectively identifying and retaining significant variables. This is crucial in high-dimensional settings where information matrices are often singular, complicating traditional minimization approaches.

Numerical Results and Experimentation

Through Monte Carlo simulations, Zhang's approach shows superior performance across various conditions, validating its robustness and adaptability. The experiments compare several penalty methods and demonstrate the efficacy of the Atan penalty over traditional methods. The results indicate that the proposed method reliably selects variables even in the presence of missing data and measurement errors.

Additionally, the methodology's application on a real-world breast cancer dataset further corroborates its viability in producing less biased, more robust models.

Conclusion and Implications

Zhang's paper advances the field by offering a method that overcomes the limitations of traditional squared loss functions in the presence of data irregularities. The innovative use of an exponential squared loss function combined with IPW and a non-convex penalty enhances the robustness and reliability of variable selection in complex datasets.

This work opens avenues for further research into more adaptive and resilient model selection methods, particularly as data complexities continue to evolve in modern datasets. Future explorations might extend these concepts to other forms of data irregularities and incorporate additional machine learning techniques to refine model performance further.