Part CCSIR NET December 2024shrinkage-wins-on-variance-and-can-lose-on-mse

Shrinkage wins on variance and can lose on mse

Consider a linear regression model $Y = X β + ε$ , with r regressors and an intercept. Random error $ε$ ~ $N_{n} (0, σ^{2} I_{n})$ and X has full column rank. Here $I_{n}$ denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let $σ$ ̂ $^{2}$ and $σ$ ̂ $^{2} ($ MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of $σ^{2}$ . Then, which of the following statements are true?

A.MSE $(σ$ ̂ $^{2} ($ MLE)) < MSE $(σ$ ̂ $^{2})$ if r = 3, n = 12✓
B. $V a r (σ$ ̂ $^{2} ($ MLE $)) < V a r (σ$ ̂ $^{2})$ if 1 ≤ r ≤ n − 2, n ≥ 3✓
C. $V a r (σ$ ̂ $^{2} ($ MLE $)) > V a r (σ$ ̂ $^{2})$ if 1 ≤ r ≤ 6, n ≥ 12
D.MSE $(σ$ ̂ $^{2} ($ MLE)) > MSE $(σ$ ̂ $^{2})$ if r = 6, n = 12✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: OLS is the BLUE in every linear model

Shrinkage wins on variance and can lose on mse

More on this topic