Asymptotic equivalence of Likelihood Ratio Statistic and Wald Statistic? When we say the likelihood ratio statistic and the Wald statistic of a set of binomial distributions are asymptotically equivalent, do we mean that the sampling distributions of the two statistics will get close as we take larger samples of both statistics from a fixed set of binomial distribution?
Or does this mean that the two sampling distributions will converge as we increase the number of trials $n$ of the binomial distribution of interest?
I am slightly confused as to what $n$ (as in $n\rightarrow \infty$) stands for in this case.
 A: Either the number of binomial trials or the number of observations will do; usually we think of this result as applying more generally than binomial data and so think of the number of observations as the $n\to\infty$.
It's also important to note that the asymptotic equivalence is local.  Suppose 0 is the null value of $\theta$. If you set $\theta=\theta_A=\neq 0$ and take $n$ observations with that value of $\theta$, $n\to\infty$, there is no guarantee that the the test statistics will approach each other.   The standard result is that if you take have a sequence  values $\theta_n=h/\sqrt{n}$ and take $n$ observations with $\theta=\theta_n$, then as $n\to\infty$ the score, Wald, and likelihood ratio tests will converge in probability to the same random variable.
Here's the picture: on a graph with the score (derivative of loglikelihood) on the $y$-axis and $\theta$ on the $x$-axis, the Wald chi-squared statistic is twice the area of the blue triangle. The score chi-squared statistic is twice the area of the red triangle, and the likelihood ratio chi-squared statistic is twice the grey area under the curve.

With $n\to\infty$ and $\theta_n=h/\sqrt{n}$, we're zooming in to this picture. The curve locally asymptotically approaches a straight line and the three coloured areas become the same

But if you fix $\theta\neq 0$ and just increase $n$, the picture doesn't change. It still just looks like

and there's no asymptotic equivalence.
