# Sufficient statistic for class of distributions

For the class $$\{F_{\theta_1}, F_{\theta_2}\}$$ of two DFs where $$F_{\theta_1}$$ is $$N(0,1)$$ and $$F_{\theta_2}$$ is $$C(0,1)$$, find a sufficient statistic.

Let, $$X_1, X_2, \dots, X_n$$ is a random sample from the class $$\{F_{\theta_1}, F_{\theta_2}\}$$.

Let, $$Y_i$$ is $$1$$ if $$X_i$$ is from $$F_{\theta_1}$$ and is $$0$$ if $$X_i$$ is from $$F_{\theta_2}$$.

Then the joint pdf of $$x = (x_1,x_2,\dots,x_n)$$ given $$y = (y_1,y_2,\dots, y_n)$$ is:

$$f(x|y) = \prod_{i=1}^{n}(f_{\theta_1}(x_i))^{y_i}(f_{\theta_2}(x_i))^{1-y_i}$$

where $$f_{\theta_1}$$ is pdf of $$N0,1)$$ and $$f_{\theta_2}$$ is pdf of $$C(0,1)$$.

(Note that $$N(0,1)$$ is standard normal distribution and $$C(0,1)$$ is standard Cauchy distribution.)

I don't know how to find sufficient statistic in this case (I don't even know if I am doing in right way). Thanks for any help.

Define $$I(\theta)=\begin{cases}1&,\text{ if }\theta=\theta_1 \\ 0&,\text{ if }\theta=\theta_2\end{cases}$$
Then pdf of $$X\sim F_{\theta}$$ can be written as
\begin{align} G_{\theta}(x)&=[f_{\theta_1}(x)]^{I(\theta)}\,[f_{\theta_2}(x)]^{1-I(\theta)} \\\\&=\underbrace{\left[\frac{f_{\theta_1}(x)}{f_{\theta_2}(x)}\right]^{I(\theta)}}_{g(\theta,T(x))}\,\underbrace{f_{\theta_2}(x)}_{h(x)}\qquad,\,\,\theta\in\{\theta_1,\theta_2\} \end{align}
By Factorisation theorem, a sufficient statistic for $$F_{\theta}$$ is $$T(X)=\frac{f_{\theta_1}(X)}{f_{\theta_2}(X)}$$