Is This A Counter-Example To The Theorem by Barndorff-Nielsen-Pedersen (1968)?

Question

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.

Answer 1

OK. I glanced through the original paper, which explicitly assumed that the number of data observations is strictly bigger than the dimension of the sufficient statistic.

So the paper is OK. The Theorem in the text is poorly stated.

Also, regarding Cat's comment, the original paper indeed stated that "if k = 1, then s = 1".