# notation for point estimation in "All of Statistics" by Larry Wasserman

I'm currently reading All of Statistics by Larry Wasserman. The question is about the Delta Method, page 160 in the pdf (I have a hard copy, there it's page 133). The authors defines $\hat{\theta}$ or $\hat{\theta}_n$ as notation for the point estimator of $\theta$. In the Delta Method theorem he states (I quote):

If $\tau = g(\theta)$" where $g$ is differential and $g(\theta)\neq0 >$ then $$\frac{\hat{\tau}_n-\tau}{\hat{\operatorname{se}}(\hat{\tau})} \to > N(0,1)$$ in distribution, where $\hat{\tau}_n = g(\hat{\theta}_n)$ and $\hat{\operatorname{se}}(\hat{\tau}_n)=|g'(\hat{\theta})|\hat{\operatorname{se}}(\hat{\theta}_n)$.

Does the author really mean $|g'(\hat{\theta})| = |g'(\hat{\theta}_n)|$ etc? Why is he even mixing these up in the same expression? Is there any reason to do so?

• Did you have the curiosity to check what is the delta method entry on Wikipedia? We could have then maybe escaped this question. Oct 2, 2015 at 20:04
• @Xi'an I suspect the source of confusion is the use of the estimate instead of the population parameter which is the method of Wikipedia. Oct 2, 2015 at 20:06
• @JohnK yes its not about the theorem, just the authors notation. The book was suggested to me.
– math
Oct 2, 2015 at 20:09

I don't think it is a typo. It seems to me that the author uses the notation $\widehat{\theta}_n$ and $\widehat{\theta}$ interchangeably. The former just emphasizes the asymptotics. Recall that the mle is consistent and thus if $g(\theta)$ is continuous, by the CMT the standard error is estimated consistently.