Let Y1,Y2,Y3 and Y4 be uncorrelated observations with common unknown variance sigma^2. Let E(Y1)=beta1+beta2+beta3=E(Y2),qquad E(Y3)=beta1-beta2=E(Y4) where beta1,beta2 and beta3 are unknown parameters. Define e1=dfrac1sqrt2(Y1-Y2) and e2=dfrac1sqrt2(Y3-Y4). An unbiased estimate of sigma^2 is:

Q54Section I (Objective (Q1–Q80))54 of 80

Unbiased estimator of $\sigma^2$ from contrasts

Let $Y_1,Y_2,Y_3$ and $Y_4$ be uncorrelated observations with common unknown variance $\sigma^2$ . Let

$\mathbb{E}(Y_1)=\beta_1+\beta_2+\beta_3=\mathbb{E}(Y_2),\qquad \mathbb{E}(Y_3)=\beta_1-\beta_2=\mathbb{E}(Y_4)$

where $\beta_1,\beta_2$ and $\beta_3$ are unknown parameters. Define $e_1=\dfrac1{\sqrt2}(Y_1-Y_2)$ and $e_2=\dfrac1{\sqrt2}(Y_3-Y_4)$ . An unbiased estimate of $\sigma^2$ is:

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