Parameter in expectation subscript

There are a few questions/answers out there about subscripts involving a random variable ($$E_X[...]$$) or a density ($$E_{f(X)}[...]$$) in expectations. I like this one.

But today I ran into a subscript involving a parameter of a density, and there is a point that confuses me.

The book of A. A. Tsiatis defines m-estimators $$\hat{\theta}_n$$ as the solution of:

$$\sum_{i=1}^nm(Z_i, \hat{\theta}_n)=0$$

where $$Z_1,...,Z_n$$ is an iid sample from $$p_Z(z,\theta)$$, $$\theta$$ is $$p$$-dimensional and, by definition, $$E_{\theta}[m(Z, \theta)]=0^{p \times 1}$$. When I first saw that zero-expectation condition I thought:

$$\int m(Z, \theta)p_\theta(\theta)d\theta$$

which is not weird in a Bayesian context, where parameters have distributions. But then I remembered that the likelihood in MLE is $$L(\theta;z)\equiv p_Z(z,\theta)$$, so I figured the $$E_{\theta}[\quad ]$$ was shorthand for:

$$\int m(Z, \theta)p_Z(z,\theta)d\theta$$

However, at the bottom of page 32 he says that this zero-expectation condition is equivalent to:

$$\int m(Z, \theta)p_Z(z,\theta)d\nu(z)=0 \quad \text{for all } \theta$$

where $$\nu(z)$$ is the dominating measure.

I would like to understand (1) what that subscript $$\theta$$ means in the expectation and (2) why the expectation is equivalent to this last integral + the "for all $$\theta$$" condition.

• The expectation is parameterized by $\theta$; it is taken with respect to the distribution given the parameter value $=\theta$. This is important because you want to make sure that the $\theta$ in $m(Z,\theta)$ has the same value as the parameter that's in $p_Z(z;\theta)$, and it's that latter $\theta$ that the subscript refers to. Dec 5 '19 at 15:29
• I think I understand what you're saying and it seems to also answer my second question. If you post it as an answer I will accept it. Dec 5 '19 at 15:40

In this case, the subscript means "the expectation is parameterized by $$\theta$$"; it is taken with respect to the distribution $$p$$ given $$p$$'s parameter value $$= \theta$$. This is important in this context because you want to make sure that the reader understands that $$\theta$$ in $$m(Z,\theta$$) has the same value as the parameter that's in $$p_Z(z;\theta)$$, and it's that latter $$\theta$$ that the subscript refers to.