# Sufficiency for AR(1) model

Consider the following AR(1) model: $$X_1=\epsilon_0\,,\,X_t=\rho X_{t-1} + \epsilon_t\,,$$ where $$t=2,3\ldots,n$$ and $$\epsilon_t \sim N(0 , \sigma^2)$$ independently. It is given that $$|\rho| <1$$.

Let $$X_1,X_2,\ldots,X_n$$ be a random sample drawn from this model. Find the minimal sufficient statistic for this model.

The thing is I cannot write the joint distribution of $$X_i$$'s for this model explicitly as it is necessary to determine whether it belongs to the exponential family. Please help.

First, $$X_t=\rho X_{t-1}+\epsilon_t=\rho^2 X_{t-2}+\rho\epsilon_{t-1}+\epsilon_t=\ldots=\sum_{s=0}^{t-1}\rho^s\epsilon_{t-s},$$ and $$\begin{bmatrix} X_1 \\ X_2 \\ \vdots \\ X_t \end{bmatrix}= \begin{bmatrix} 1 & 0 & \cdots & 0 \\ \rho & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ \rho^{t-1} & \rho^{t-2} & \cdots &1 \end{bmatrix} \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_t \end{bmatrix}.$$ Therefore, $$[X_1,\cdots, X_t]^{\top}\sim N(0, \sigma^2 R_{\rho}R_{\rho}^{\top})$$, where $$R_{\rho}$$ is the $$t\times t$$ matrix in the preceding equation.
Let $$\vec{x}\equiv [x_1,\ldots,x_t]^{\top}$$. Then the likelihood function for $$(\sigma,\rho)$$ is given by $$\mathcal{L}(\sigma,\rho;\vec{x})=(2\pi\sigma^2)^{-k/2}e^{-\frac{1}{2\sigma^2}\vec{x}^{\top}\left(R_{\rho}R_{\rho}^{\top}\right)^{-1}\vec{x}}$$ because $$\operatorname{det}(R_{\rho}R_{\rho}^{\top})=1$$. The quadratic form appearing in $$\exp(\,\cdot\,)$$ can be written as $$\vec{x}^{\top}\left(R_{\rho}R_{\rho}^{\top}\right)^{-1}\vec{x}=(1+\rho^2)\sum_{s=1}^{t-1}x_s^2+x_t^2-2\rho\sum_{s=1}^{t-1}x_sx_{s+1}.$$ Now use the Neyman-Pearson factorisation criterion.
• Okay.. what is $r$ here? Do you mean $\rho$?