Iterative Least Squares Algorithm for M-Estimation

The iterative least squares algorithm for M-estimation is a robust regression method designed to reduce the influence of outliers on model fitting. Compared to traditional ordinary least squares (OLS), M-estimation introduces specific weight functions to adjust the contribution of outliers, resulting in more stable parameter estimates.

### Core Concepts Implementation of Weight Functions: M-estimation dynamically adjusts data point weights by analyzing residual magnitudes. Points with larger residuals (potential outliers) receive lower weights, while points with smaller residuals retain higher weights. In code implementation, this is typically achieved through a weight function w(r) where r represents residuals, with common implementations including Huber, Tukey, or Hampel weighting schemes. Iterative Optimization: The algorithm employs an Iteratively Reweighted Least Squares (IRLS) strategy. Each iteration recalculates weights based on current residuals and updates parameter estimates until convergence. The IRLS algorithm can be implemented with a while-loop structure that continues until parameter changes fall below a specified tolerance threshold, typically using a convergence criterion like ||β_new - β_old|| < ε.

### Common Estimators Huber Estimator: Features a smooth transition weight function that provides robust handling of moderate outliers. In practice, the Huber loss function transitions from quadratic to linear at a specified threshold k, implemented as: w(r) = 1 if |r| ≤ k, else w(r) = k/|r|. Andrews Estimator: Utilizes a sine function as the weight function, suitable for symmetric data distributions. The Andrews wave weight function is defined as w(r) = sin(r/c)/(r/c) for |r| ≤ πc, zero otherwise. Hampel Estimator: Employs a piecewise weight function that more strictly suppresses extreme outliers. This three-part weight function uses different thresholds (a, b, c) to gradually reduce influence: w(r) = 1 for |r| ≤ a, then linear decay to zero. Ramsay Estimator: Uses an exponential decay weight function, ideal for handling long-tailed distribution data. The Ramsay Ea weight function follows w(r) = exp(-a|r|) where parameter a controls the decay rate.

### Application Scenarios This algorithm finds widespread application in financial modeling, engineering signal processing, and biostatistics. Particularly when data contains noise or outliers, its robustness significantly outperforms traditional least squares methods.

### Advantages and Limitations Advantages: Reduces outlier interference and improves model generalization capability. Algorithm implementation typically shows better performance on contaminated datasets compared to OLS. Limitations: Higher computational complexity, and weight function selection must consider data characteristics. The IRLS algorithm requires multiple matrix inversions, making it computationally intensive for large datasets. Weight function parameters often need calibration through cross-validation.