Part CCSIR NET December 2024at-those-numbers-the-two-errors-coincide

At those numbers the two errors coincide

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 = 2, n = 12✓ Correct
B. $Var(\sigma$ ̂ $^{2}($ MLE $)) > Var(\sigma$ ̂ $^{2})$ if 2 ≤ r ≤ n − 2, n ≥ 12
C. $Var(\sigma$ ̂ $^{2}($ MLE $)) < Var(\sigma$ ̂ $^{2})$ if 1 ≤ r ≤ n − 2, n ≥ 3✓ Correct
D.MSE $(\sigma$ ̂ $^{2}($ MLE)) > MSE $(\sigma$ ̂ $^{2})$ if r = 7, 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, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: OLS is the BLUE in every linear model

At those numbers the two errors coincide

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