L Statistic

TL;DR

• Quantifies a linear regression model’s fit by summing the squared differences between observed and predicted values.
• Lower L-statistic values indicate a better-performing model.
• Can be used to compare models and guide improvements to predictive performance.

Definition

The L-statistic is a statistical measure used to evaluate the performance of a linear regression model. It is calculated by taking the sum of the squared residuals of the model, where each residual is the difference between an observed value and the model’s predicted value.

$$L = \sum_i (y_i - \hat{y}_i)^2$$

Explanation

The L-statistic provides a single numeric summary of how closely a linear regression model’s predictions match observed data. By squaring residuals before summing, larger prediction errors contribute more to the statistic. Comparing L-statistic values across models or iterations allows assessment of which model predicts observed values more accurately; a lower L-statistic corresponds to better predictive performance according to this measure.

Examples

Stock market example

A linear regression model is used to predict a stock’s future value from its past performance. The L-statistic is calculated for this model by summing the squared residuals (differences between observed and predicted stock values). A lower L-statistic indicates the model predicts the stock’s future value more accurately.

Weather example

A linear regression model is used to predict future temperature at a location from past temperature data. The L-statistic for this model is the sum of the squared residuals between observed and predicted temperatures. A lower L-statistic indicates the model predicts future temperature more accurately.

Use cases

• Evaluating the predictive performance of a linear regression model.
• Using the L-statistic value to inform model improvement and enhance performance.