Consider the family of normal distributions $$\{N( \mu,\sigma^2): \mu \in \mathbb{R}, \sigma >0\}$$ on $\mathbb{R}^n$. Its claimed in the book Statistical Inference by Silvey that if we restrict to the subfamily where $\mu = \sigma^2$, we have a nonzero function $\mathbb{R}^n\rightarrow \mathbb{R}$, namely $$f(x):= \overline{x}- \frac{1}{n-1}\sum_i(x_i-\overline{x})^2$$
with zero expectation w.r.t. every distribution in the subfamily. ($\overline{x}$ is supposed to be the average of the coordinates.) This is hard for me to see. Just in the two dimensional case I get a messy integral
$$\int_{\mathbb{R}^2} \bigg( \frac{x+y}{2} - \big[(x-\frac{x+y}{2})^2 - (y-\frac{x+y}{2})^2\big]\bigg) \cdot exp \big[\frac{-(x-\mu)^2-(y-\mu)^2}{2\mu}\big]dxdy$$
which even wolfram alpha doesn't want to do. Am I really supposed to hack through this integral? Or is this supposed to be more clear? I can see that in some sense $f$ is the difference of an average and a variance (although the $n-1$ rather than $n$ in the denominator puzzles me), but I don't know how to make sense of that.