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

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.

• Does the theorem explicitly say that $s = 1$? In my copy it merely says that $\eta_1$ exists.. not that there are no $\eta_i$ with $i > 1$.
– Cat
Jul 24 '21 at 23:41
• It is my understanding that is what he meant. Also based on the overall context around the theorem, I am reasonably sure. I found the exposition in this text to be rather hard to follow, great as E Lehmann was. May he rest in peace. I wish I could have asked him directly. Jul 24 '21 at 23:54
• I found the original 1968 paper. I will give it a try. Jul 25 '21 at 0:00