Sufficiency for AR(1) model

Question

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.

Answer 1

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.

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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.