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How does one test the efficiency and completeness of an estimator using monte-carlo simulation?

In particular, I want to use-montecarlo simuation to answer. Maybe the better question is how does one prove it mathematicallY?

1.Is the vcovcHAC estimator of standard errors from CRAN's "sandwich" heteroscedasticity and autocorrelation efficient and complete?

  1. Is the Huber-White estimator of standard errors heteroscedasticity efficient and complette?.

Efficiency is reaching rao-cramer bound.

I don't know how to find Information matrix of standard errors from huber-white of regression coefficients because I don't ever see parameters on real-data, so how do I do it empirically on a toy-example where I can simulate the random-variables?

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    $\begingroup$ Completeness is a theoretical property that I'd think cannot be tested empirically. Efficiency is empirical to the extent that if you can compute the Cramer-Rao bound theoretically for a given distribution, then you can simulate how close an estimator actually comes to it. But the first step is theory. You need to assume a model and be able to compute the CR-bound. $\endgroup$ Commented Sep 6, 2022 at 17:15
  • $\begingroup$ For establishing rao-cramer, How do you define the empirical variance of an estimator? @ChristianHennig $\endgroup$
    – user318514
    Commented Sep 6, 2022 at 17:21
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    $\begingroup$ There's nothing empirical in Cramer-Rao. You assume a model with certain parameters, and the Cramer-Rao bound is then theoretically computed from this. $\endgroup$ Commented Sep 6, 2022 at 17:29
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    $\begingroup$ If you know the theoretical Rao-Blackwell optimal variance $V_0$, you can check the empirical variance of the optimal estimator stays within two standard deviations of that value. $\endgroup$
    – Xi'an
    Commented Sep 6, 2022 at 17:30
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    $\begingroup$ @Germania Not sure whether this is what you're asking, but if you simulate several data sets from a distribution, the empirical variance of an estimator is the sample variance of all values the estimator takes over the different data sets. Obviously the estimator could be the optimal one, in which case the sample variance of its values estimates the true $V_0$ (the better the more data sets you simulate). $\endgroup$ Commented Sep 6, 2022 at 21:28

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Answering for completeness, you cannot.

From Wikipedia:

The statistic $T$ is said to be complete for the distribution of $X$ if, for every measurable function $g$ $$[\mathbb{E}_{\theta}[g(T)] = 0\ \forall \theta ] \implies [\mathbf{P}_{\theta}(g(T)=0) = 1\ \forall \theta]$$

There is an infinite number of measurable functions, and for many choices of distribution there will likewise is an infinite number of $\theta$'s. You're not going to check them all, and it is isn't at all clear what sampling strategy would provide meaningful results. You can't sample them with equal probability because a uniform distribution is not well-defined over an infinite set and any non-uniform sampling distribution would clearly just be favoring a conclusion weighted to those functions assigned a higher density/probability.

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