Is there any significance for subscripts to E and Var?

For example, the risk function of an estimator $\delta(\mathbf x)$ of $\theta$ in my book is:

$$ R(\theta,\delta)=E_\theta[L(\theta,\delta(\mathbf X))]=Var_\theta [\delta(\mathbf X)]+\left(E_\theta[\delta(\mathbf X)]-\theta\right)^2=Var_\theta [\delta(\mathbf X)]+\left(\text{Bias}_\theta[\delta(\mathbf X)]\right)^2 $$

Does this signify anything else other than that the E.V., Vars, etc are functions of $\theta$?

  • 1
    $\begingroup$ Is $\theta$ a random variable in this context? $\endgroup$
    – mhdadk
    Nov 12, 2021 at 19:08
  • 1
    $\begingroup$ Although you don't describe your notation, the form of the left hand side suggests $\theta$ is a parameter, not a random variable, and that therefore the subscripts in the middle equation are partly superfluous and partly bad notation. They are needed in the right hand equation to show that both its terms (implicitly) depend on $\theta.$ Much better notation could have been devised... . It would be helpful for you to explain what kinds of mathematical objects $\theta$ (and perhaps $\mathbf X$) are intended to refer to. $\endgroup$
    – whuber
    Nov 12, 2021 at 19:24
  • 3
    $\begingroup$ I guess these indices indicate that the distribution of $\mathbf X$ depend on $\theta$, e.g., $$\mathbf X\sim f(x;\theta)$$ $\endgroup$
    – Xi'an
    Nov 12, 2021 at 20:07