Today

• Nonparametric regression (ELMR 14).

Announcement

• I’ll post final (take home) by this Thursday 6/4. Due next Tue 6/9 @ 11:59pm.

• HW4 Q2 summary. Balanced one-way ANOVA random effects model.

  • ANOVA and REML estimate. Unbiased. $\hat{\sigma}_\epsilon^2 = \frac{\text{SSE}}{a(n-1)}$ $\hat{\sigma}_\alpha^2 = \frac{\text{SSA}}{(a - 1)n} - \frac{\text{SSE}}{an(n - 1)}$

  • MLE. $\hat{\sigma}_\alpha^2$ is biased but might have a smaller MSE than ANOVA/REML estimate. $\hat{\sigma}_\epsilon^2 = \frac{\text{SSE}}{a(n-1)}$ $\hat{\sigma}_\alpha^2 = \frac{\text{SSA}}{an} - \frac{\text{SSE}}{an(n - 1)}$

  • Analog to the linear regression model $Y \sim N(X \beta, \sigma^2 I)$. Unbiased estimate of $\sigma^2$ is $\text{SSE} / (n - p)$. MLE is $\text{SSE} / n$. Best quadratic invariant estimator is $\text{SSE} / (n - p + 2)$.

• Course evaluation: https://my.ucla.edu