Finding a sufficient statistic for $\theta$ that takes values in a subset of $\mathbb R^3$

Question

This is the problem that I'm working on, for reference:

For some $n\ge 3$, let $\varepsilon_1, \ldots , \varepsilon_n$ be i.i.d $N(0, 1)$. Set $X_1 = \varepsilon_1$ and $X_i=\theta X_{i-1}+(1-\theta^2)^{1/2}\varepsilon_i$ for $i = 2, \ldots , n$ and some $\theta \in (−1, 1)$. Find a sufficient statistic for $\theta$ that takes values in a subset of $\mathbb R^3$.

So, the way I went about trying to tackle this question was by trying to figure out the probability that $X_i$ equals a certain value given that we know the value of $X_{i-1}$: in other words

$$P(X_i = x_2 \mid X_{i-1} = x_1)$$

The way I did this was by assuming that $X_{i-1}$ was a constant ($x_1$), and then figuring out the probability distribution of $X_i$ with this constant included (which is just $N(\theta x_1, 1-\theta^2)$). Is this method of determining the conditional probability correct/allowed?

Answer 1

Hint:

What you can do instead is solve the recurrence to get

$$X_i=\theta^{i-1}\varepsilon_1+(1-\theta^2)^{1/2}\sum_{j=0}^{i-2}\theta^j \varepsilon_{i-j}\quad,\,i=2,3,\ldots,n$$

Then express the vector $X=(X_1,\ldots,X_n)^T$ as a linear transformation of $\varepsilon=(\varepsilon_1,\ldots,\varepsilon_n)^T$, i.e.

$$X=R_{\theta}\,\varepsilon$$

for some nonsingular matrix $R_{\theta}$. Since $\varepsilon\sim N_n(0,I_n)$, that would imply $X\sim N_n(0,R_{\theta}R_{\theta}^T)$.

So just write down the density of $X$ and apply Factorization theorem.