Suppose I have a continuous random variable whose distribution $f$ is some parametric form (normal, exponential, etc.) that is known to me. If I draw many independent samples $x_i$ from $f$, I can estimate the parameters of $f$ using these samples (using maximum likelihood estimation, for example), which gives me an "estimator distribution" $\hat{f}$ (which has the same parametric form as $f$). If I perturb my samples by small random amounts $\epsilon_i$, I would get another "estimator distribution" $\tilde f$. My question is: are there any sufficient conditions on $f$ and the estimator that I use, such that I can bound the difference between $\hat{f}$ and $\tilde{f}$ (measured in, say, the $L_1$ sense), as a function of the perturbations $\epsilon_i$? Are there any general theorems of this kind that talk about how sensitive an estimator distribution is to perturbations of the samples?
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$\begingroup$ are you familiar with robust statistics ? It uses the same setup, i.e., a contaminated sample, but it's focus is on stats that behave well even under contamination, not the performance of mle or other stats under contamination. $\endgroup$ – user83457 Oct 6 '16 at 6:50

$\begingroup$ You may have a better chance if you measure the difference between $\tilde{f}$ and $\hat{f}$ in the KomogorovSmyrnoff sense. $\endgroup$ – Mark Fischler Oct 6 '16 at 14:49