Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / O / 2 / b

Put $z_i=e^{-t_i}-c$. Assume the measurement errors are independent and centered, so $Y_i=\alpha +\beta z_i+{ϵ}_i$ with independent ${ϵ}_i\sim N(0,{\sigma }^2)$. Treat the observation times as fixed. The design matrix has rows $(1,z_i)$ and parameter vector $(\alpha ,\beta {)}^{\mathsf{T}}$. Require at least two distinct times so its columns are independent, equivalently ${\sum }_i{z}_i^2>0$. This gives the required Gaussian linear regression model.

Centering gives ${\sum }_i{z}_i=0$, and henceThe least-squares covariance formula from part (a) therefore givesExplicitly $\hat{\alpha }=\overline{Y}$ and $\hat{\beta }={\sum }_i{z}_i{Y}_i/{\sum }_i{z}_i^2$. Their zero covariance follows directly from the centered predictor, and their joint normality additionally makes them independent.

With independent Gaussian errors, maximizing the likelihood over $\alpha ,\beta$ minimizes the residual sum of squares, so the maximum-likelihood estimators are the same as least squares. The invariance property of maximum likelihood estimation givesThe two estimates have zero covariance by part (ii). ThereforeThis is the uncertainty in the estimated expected response. Estimating an additional noisy observation would add its own error variance, but the quantity requested is the mean response $E_t$.