Among the modeling methods selected, which one does not hold true for k-predictor models?

The assertion that all three models are the same for k=1 is correct because when k=1, the models are essentially reduced to single predictor models. In this scenario, regardless of the method used (forward stepwise, backward stepwise, or best subset selection), the outcome will consist of one predictor variable being selected. This means that each modeling approach will end with the same single predictor chosen from the set of available variables, leading to identical model outcomes. Therefore, for k=1, there is no distinction between the methodologies; they will yield the same predictions and coefficients.

The other statements do not hold true universally for k-predictor models. For instance, while forward stepwise may yield the model with the smallest training residual sum of squares, this does not guarantee it is optimal or generalizable to new data when evaluated on a test set. Similarly, best subset selection could lead to a model with the smallest test residual sum of squares, but it is dependent on the specific features being modeled and the robustness of the chosen subset. Lastly, backward stepwise does involve considering multiple models as it starts with all predictors and sequentially removes them, which adds to the complexity. Thus, the unique characteristic of k=1 leading to equivalent model outcomes