I have been attempting to work through a best linear predictor example within the textbook Semiparametric Regression by Ruppert, Wand, and Carrol (pdf). The specific example concerns a linear mixed model on pig weight over time:

$$ weight_{i,j} = \beta_0 + U_i + \beta_1week_j + \epsilon_{i,j}$$

Where j indexes the weeks and i indexes the pigs. U is a random variable distributed according to $U \sim N(0,\sigma_U^2)$, epsilon is the error term distributed according to $\epsilon \sim N(0,\sigma_\epsilon^2)$. We are also told that $cov(weight_{i,j},weight_{i,j'}) = \sigma_U^2$. The purpose of the example is to predict U using the equation:

$$BP(U) = E(U) + CV^-1(y-E(y)) \\ C = E[(U - E(U))(y - E(y))] \\ V = Cov(y)$$

My work is as follows:

$$ BP(U) = 0 + E[(U - 0)(y - \beta_0 - \beta_i\bar{x})]\frac{1}{\sigma_U^2}(y - \beta_0 - \beta_i\bar{x}) \\ BP(U) = E[Uy]\frac{1}{\sigma_U^2}(y - \beta_0 - \beta_i\bar{x}) \\ $$

The answer as provided in the textbook is:

$$ BP(U) = \frac{n_i\sigma_U^2}{\sigma^2_\epsilon + n_i\sigma_U^2}(y - \beta_0 - \beta_i\bar{x})$$

Any steps that connect the starting point to the final answer in the book would be unbelievably appreciated.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.