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.
Best Answer
This might work :
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 $$T(X)=\frac{f_{\theta_1}(X)}{f_{\theta_2}(X)}$$