In the textbook "Theory of Point Estimation" 2nd Ed. by Lehmann and Casella, Theorem 6.18 states:
Suppose $X_1, ..., X_n$ are real-valued IID according to a distribution with density $f_\theta(x_i)$ with respect to Lebesgue measure, which is continuous in $x_i$ and whose support for all $\theta$ is an interval I. Suppose that for the joing density of $X = (X_1, ..., X_n)$ $p_\theta(x) = f_\theta(x_1)...f_\theta(x_n)$ there exists a continuous $k$-dimensional sufficient statistic. Then
(i) if $k=1$, there exists function $\eta_1, B$ and $h$ such that (5.1) holds;
(ii) $k > 1$, and if ......
where (5.1) is: $p_\theta(x) = h(x) \exp[\sum_{i=1}^s \eta_i(\theta) T_i(x) - B(\theta)]$.
I seem to have a simple counter-example to this.
Consider the case of $n=1$ and distribution is $N(\mu, \sigma^2)$. Then we have a "continuous 1-dimensional sufficient statistic", the data point itself. So, in this case $k = 1$. Per the Theorem, the PDF can be written in the exponential family with $s = 1$. But we know that cannot be done.
What am I understanding wrong here?
Note: I want to point out one thing (even though it is irrelevant for the problem at hand). Parameters here are well-identified even with just a single data point. Maximum likelihood would not exist but that really is another problem.
