Background:
For the case $\eta_0$ known, we assume the existence of a function $S(\theta,\eta)$ such that
- $\tilde{\theta} = \theta_0 + Op(n^{-1/2})$
- $S(\theta,\eta)$ is differentiable in $\theta$ at $(\theta_0,\eta_0)$ with a derivative matrix $\Gamma$ of full rank
- $ S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0) = S_n(\tilde{\theta},\eta_0) - S_n(\theta_0,\eta_0) + op\left(n^{-1/2}\right)$
From 2), we get a Taylor expansion about $\theta_0$,
$$S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0) = \Gamma (\tilde{\theta} - \theta_0) + op(|\tilde{\theta} - \theta_0|)$$
Hence
$$ \tilde{\theta} - \theta_0 = \Gamma^{-1} \left( S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0) \right) + op(n^{-1/2})$$
From 3),
$$ \tilde{\theta} - \theta_0 = \Gamma^{-1} \left(S_n(\tilde{\theta},\eta_0) - S_n(\theta_0,\eta_0) \right) + op(n^{-1/2})$$
Note that assumption 3) is satisfied if assumption 4-6 and 7a found here are true.
To have an equivalent estimator when $\eta$ is unknown, we need to have an equivalent linearization.
Solution 1:
Assume that, in addition to 1-3,
A) $\hat{\theta} = \theta_0 + Op(n^{-1/2})$
B) $ S(\hat{\theta},\eta_0) = S(\tilde{\theta},\eta_0) + op\left(n^{-1/2}\right)$
Then we can write, from A),
$$ \hat{\theta} - \theta_0 = \Gamma^{-1} \left( S(\hat{\theta},\eta_0) - S(\theta_0,\eta_0) \right) + op(n^{-1/2})$$
From B),
$$ \hat{\theta} - \theta_0 = \Gamma^{-1} \left( S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0) \right) + op(n^{-1/2})$$
Solution 2:
If we assume 1-3, A) and
C) $\hat{\eta} = \eta_0 + Op(n^{-1/2})$
D) $S(\theta,\eta)$ is differentiable in $\eta$ at $(\theta_0,\eta_0)$ with a derivative matrix equals to zero
E) $S(\hat{\theta},\hat{\eta}) = S(\tilde{\theta},\eta_0) + op(n^{-1/2})$
Then we can perform the following Taylor expansion about $(\theta_0, \eta_0)$,
$$S(\hat{\theta},\hat{\eta}) - S(\theta_0,\eta_0) = \Gamma (\hat{\theta} - \theta_0) + op(|\hat{\theta} - \theta_0| + |\hat{\eta} - \eta_0|)$$
and thus
$$\begin{align} \hat{\theta} - \theta_0 &= \Gamma^{-1} \left( S(\hat{\theta},\hat{\eta}) - S(\theta_0,\eta_0)\right) + op(n^{-1/2}) \\ &= \Gamma^{-1} \left( S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0)\right) + op(n^{-1/2}) \end{align}$$
A sufficient condition for E) to hold is that 3) be true and
$\begin{align} S(\hat{\theta},\hat{\eta}) - S(\theta_0,\eta_0) &= S_n(\hat{\theta},\hat{\eta}) - S_n(\theta_0,\eta_0) + op(n^{-1/2}) \\ S_n(\hat{\theta},\hat{\eta}) - S_n(\tilde{\theta},\eta_0) &= op(n^{-1/2}) \end{align}$
Solution 3
If we assume 1-3, A) and
F) $\hat{\eta} = \eta_0 + op(1) $
G) $S(\theta,\eta)$ is uniformly differentiable in $\theta$ at $\theta_0$ on a neighborhood of $\eta_0$ with a derivative matrix $\Gamma(\eta)$
H) $\Gamma(\eta)$ is continuous and full rank at $\eta_0$, with $\Gamma = \Gamma(\eta_0)$
I) $S(\hat{\theta},\hat{\eta}) - S(\theta_0,\hat{\eta}) = S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0) + op(n^{-1/2})$
Then from G) we can perform the following Taylor expansion about $\theta_0$, which is valid with probability tending to one,
$$\begin{align}S(\hat{\theta},\hat{\eta}) - S(\theta_0,\hat{\eta}) &= \Gamma(\hat{\eta}) (\hat{\theta} - \theta_0) + op(|\hat{\theta} - \theta_0|) \\ &= \Gamma(\hat{\theta} - \theta_0) + op(|\hat{\theta} - \theta_0|) \end{align}\\$$
with the second line true because of F) and H).
Hence, with I)
$$\begin{align}\hat{\theta} - \theta_0 &= \Gamma^{-1}\left(S(\hat{\theta},\hat{\eta}) - S(\theta_0,\hat{\eta}) \right) + op(n^{-1/2}) \\ &= \Gamma^{-1}\left(S(\tilde{\theta},\eta_0) - S(\theta_0,\eta_0) \right) + op(n^{-1/2}) \end{align}\\$$
Note that a sufficient condition for $I$ to be true is that both E) be true and
I') $S(\theta_0,\hat{\eta}) = S(\theta_0,\eta_0) + op(n^{-1/2})$
Both conditions D) and I') are asymptotic orthogonality assumptions.