Consider the simple linear correlation model $Y_i=\alpha+ \beta X_i+\epsilon_i \ ; \ \ i=1,..n$. Here $\alpha,\beta \in \mathbb{R}$ Here $X_1,X_2,...X_n \sim N(0,\tau^2)$(i.i.d) and $\epsilon_i \sim N(0,\sigma^2)$(i.i.d). Also, $X_i's$ are independent of $\epsilon_i's$ Find a minimal sufficient statistic for this model.
Can it be said that $Y_i \sim N(\alpha,\beta^2\tau^2+\sigma^2)$ here? I can't guarantee that it is in the exponential family. Any approach?
Assuming that you are interested in the MSS for all the unknown parameters $ \theta = ( \alpha, \theta, \beta, \sigma^2 )$. The distribution of $Y$ is indeed normal with mean $\alpha$ and variance $\beta^2 \tau ^ 2 + \sigma^2$.Intuitively you need a vector of $4$ statistics, which should be $T(X) = (\sum X_i, \sum Y_i, \sum X_i^2, \sum Y_i^2)$. For the proof you can use the factorization criteria where you have a sample of $n$ i.i.d observations (realizations) from a bivariate normal distribution of $(X_i, Y_i)$ where the mean vector is $(\mu_Y, \mu_X) = (0, \alpha)$ and the covariance matrix is
\begin{align} \Sigma = \begin{pmatrix} \tau ^ 2 && \beta \tau^2 \newline \beta \tau ^2 && \beta^2 \tau ^ 2 + \sigma^2 \end{pmatrix} \end{align}