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 $Y = X\beta + \varepsilon$ , with r regressors and an intercept. Random error $\varepsilon$ ~ $N_{n}(0, \sigma^{2}I_{n})$ 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 $\sigma$ ̂ $^{2}$ and $\sigma$ ̂ $^{2}($ MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of $\sigma^{2}$ . Then, which of the following statements are true?

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

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, and why this one tests what the inference means.

See pricing

50 are analysed free — try those first.

The trap it tests

What the inference means

p-values, size, power and error types say specific things. This was not one of them.

Drill statements like this

Related counterexample: OLS is the BLUE in every linear model

Shrinkage wins on variance and can lose on mse

More on this topic