2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$
    – jbowman
    Dec 5, 2019 at 15:29
  • $\begingroup$ 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. $\endgroup$
    – suckrates
    Dec 5, 2019 at 15:40

1 Answer 1

1
$\begingroup$

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.

It is a little confusing, as often the subscript on the expectation operator refers to what the expectation is taken with respect to, not what the parameters of the relevant probability distribution are.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.