Suppose that the random variables $Y_1,...,Y_n$ satisfy $$Y_i=\beta x_i + \epsilon_i, i=1,...,n$$ where $x_1,...,x_n$ are fixed constants, and $\epsilon_1,...,\epsilon_n$ are i.i.d. $N(0,\sigma^2), \sigma^2$ unknown.
(a) Find a two-dimensional sufficient statistic for $(\beta, \sigma^2)$.
I have no idea where to even start with this one. Prof didn't do any examples like this in class. Any help would be appreciated.
Let's compute the likelihood function for this model. Since $Y_i=\beta x_i+\epsilon_i\sim\mathcal{N}(\beta x_i,\sigma^2)$: $$\mathcal{L}(\beta,\sigma^2)=\prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma²}}\exp\left(-\frac{(y_i-\beta x_i)^2}{2\sigma^2}\right)\\ =\frac{1}{({2\pi\sigma^2})^\frac{n}{2}}\exp\left(\frac{1}{2\sigma^2}\sum_{i=1}^n (y_i-\beta x_i)^2\right)\\ =\frac{1}{({2\pi\sigma^2})^\frac{n}{2}}\exp\left(\frac{1}{2\sigma^2}\sum_{i=1}^n y_i^2-\frac{1}{2\sigma^2}\sum_{i=1}^n2\beta x_iy_i+\frac{1}{2\sigma^2}\sum_{i=1}^n\beta^2x_i^2\right)\\ =\frac{1}{({2\pi\sigma^2})^\frac{n}{2}}\exp\left(\frac{1}{2\sigma^2}\sum_{i=1}^n y_i^2\right)\exp\left(-\frac{\beta}{\sigma^2}\sum_{i=1}^nx_iy_i\right)\exp\left(\frac{\beta^2}{2\sigma^2}\sum_{i=1}^nx_i^2\right)$$ Now, since the likelihood function can be factorized in terms that depend on the sample only through $\sum_{i=1}^ny^2$ and $\sum_{i=1}^nx_iy_i$, we can affirm that $T(Y)=\left(\sum_{i=1}^nx_iY_i,\sum_{i=1}^nY_i^2\right)$ is a sufficient statistic.
