Consider any arbitrary estimator called $\hat{M}$ (e.g., regression coefficient estimator or specific type of correlation estimator, etc) that satisfies the following asymptotic property:
$$\boxed{\sqrt{N}(\hat{M}-M) \overset{d}{\to}\mathcal{N}(0,\sigma^2)}\,\,\,\,\,\,\,\,\,\,\,\,(1)$$
which implies that our $\hat{M}$ is consistent. We also have a consistent estimator $\hat{\sigma}$, which gives rise to the asymptotic property:
$$\displaystyle \ \ \boxed{\frac{\sqrt{N}(\hat{M}-M)}{\hat{\sigma}} \overset{d}{\to}\mathcal{N}(0,1)}\,\,\,\,\,\,\,\,\,\,\,\,(2)$$
I'm wondering if I can use the $z$- or $t$-test just like normal for any such $\hat{M}$ that satisfies the above? Let $Q$ be defined as the test statistic:
$$\displaystyle \ \ \boxed{Q_\hat{M} = \frac{\hat{M}-M_{H_0}}{\sqrt{\frac{1}{N}\hat{\sigma}^2}}}\,\,\,\,\,\,\,\,\,\,\,\,(3)$$
My goal is to do the following hypothesis test:
$H_0: M = 0$
$H_a: M \not= 0$
yet the only information I have access to is $(1)$ and $(2)$, whence my question.
$$\underline{\text{Update}}$$
The current answers suggest that I can't always robustly $z$- or $t$-test for any such $\hat{M}$. I am reading the relevant sections of All of Statistics (Wasserman), as well as Statistical Inference (Casella & Berger). Both state that, if:
$$\displaystyle \ \ \frac{\sqrt{N}(\hat{M}-M)}{\hat{\sigma}} \overset{d}{\to} \mathcal{N}(0,1)$$
then "an approximate test can be based on the wald statistic $Q$ and would reject $H_0$ if.f. $Q < -z_{\alpha/2}$ or $Q > z_{\alpha/2}$" (in Casella & Berger, page 492, "10.3.2 Other Large-Sample Tests")
or, in (Wasserman, page 158, Theorem 10.13) "Let $Q = (\hat{M}-M_{H_0})/\hat{se}$ denote the observed value of the Wald statistic $Q$ $\big($where $\hat{se}$ is obviously equal to my $\sqrt{\frac{1}{N}\hat{\sigma}^2}$$\big)$. The p-value is given by:
$$p = 2\Phi(-|Q|)$$
This contradicts the existing advice since they do not state any other necessary assumptions to be able to do this legitimately (to the best of my ability to comprehend). Either;
- I have failed to understand existing answers.
- I have failed to express my original question clearly.
- I have failed to read these chapters properly.
- They are excluding thoroughness for pedagogical purposes.
I would appreciate some assistance on which option is correct. Thanks. $\big($Please go easy I am new to stats :)$\big)$.
Another dimension is that my intended application is $n = 3000$, so perhaps the finite sample problems are less relevant?