Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 18H / d

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- Solution d

Solution

The reverse-regression estimate is

\hat{b} = \sum X_{i} Y_{i} / \sum Y_{i}^{2}

. Therefore

\hat{b} \hat{β} = \frac{( \sum X _{i} Y _{i} ) ^{2}}{( \sum X _{i}^{2} ) ( \sum Y _{i}^{2} )} \leq 1

by Cauchy–Schwarz. Equality means the two data vectors are proportional, which is exactly when both fitted residual sums, and hence both variance MLEs, vanish.

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