Question

3. Consider three models for a response and two predictors xil and xi2, with additive errors єї і.hd. N(0. ơ2). (a) For each of the above models, explain why it is not a linear model in the sense of PSTAT 126. (b) Propose a change to model (M1) so that it becomes a linear model in the sense of PSTAT 126. (c) Write down the likelihood for n data points (xn, Yi) from model (M2)

PSTAT 126 meaning a Linear Regression course.

Answer 1

$(a)~Since~all~models~can't~be~expressed~as~linear~combinations~of~parameters~\beta_i's\\ so~these~are~not~linear~model.\\ (b)~Y_i=\beta_0+\beta_1^* x_{i1}+\beta_2x_{i2}+\epsilon _i~where~\beta^*_1=\frac{1}{\beta_1}\\ Hence~M1~is~linear~model~in~terms~of~parameters~\beta_0,\beta_1^*,\beta_2.\\ (c)~Likelihood~function~of~\beta_0,\beta_1=L(\beta_0,\beta_1)\\ =\prod_{i=1}^n\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(y_i-\exp(\beta_0+\beta_1x_{i1}))^2}{2\sigma^2} \right )=\frac{1}{(2\pi)^{\frac{n}{2}}\sigma^n}\exp\left(-\sum_{i=1}^n\frac{(y_i-\exp(\beta_0+\beta_1x_{i1}))^2}{2\sigma^2} \right )$