Is this a linear or non linear model, and why?

Question

A model $Y=(\beta_0+\beta_1x)^{-1}+\epsilon$, where $\epsilon \sim N(0,\sigma^2)$ is to be fitted to the data $(x_1,Y_1), (x_2,Y_2), \dots (x_n,Y_n)$.

Is this a linear or non linear model, and why?

Show how it can be made into a GLM and state the link function?

Any help would be great!

Answer 1

This is just a somewhat unusual generalised linear model with a Gaussian response $Y\sim N(\mu,\sigma^2)$ where the mean of the response $\mu$ is linked to the linear predictor $\eta$ via $$ g(\mu)=\underbrace{\beta_0 + \beta_1 x}_\eta. $$ where $g$ is the inverse link function $g(\mu)=1/\mu$,

The linear predictor $\eta$ is clearly linear in $x$ and, as implied by its name, in the regression coefficients $\beta_0$ and $\beta_1$. However, because of the non-linear link function, $\mu=EY$ is non-linear in the model parameters and in $x$.

To fit this in R do glm(y ~ x, gaussian(link = "inverse")).