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This question asks about influence functions. Probabilityislogic's answer is a bit fuzzy to me, but I can make more sense of jayk's answer, as this was the way influence function was presented to me in class. Note that I have a bachelor's in math and doing my masters, though still haven't taken functional analysis or operator theory (which I think is part of why I can't make sense of influence functions).

In bold, jayk wrote:

An influence function tells you the effect that a particular observation has on the estimator.

I don't see how, unfortunately. To me the contaminated distribution $F_{\varepsilon}(x) = (1-\varepsilon)F(x) + \varepsilon \Delta(x)$ is just redistributing measures (creating a new distribution). Maybe there is something else to see in the contaminated distribution?

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    $\begingroup$ Would the examples, explanations, and code at stats.stackexchange.com/a/114363/919 suffice for an answer? The process of changing a data point is the empirical analog of the contaminated distribution. You don't need advanced mathematics to see this: draw graphs of some examples of $F_\varepsilon,$ especially where $\Delta$ is supported only on extremely large positive (or large negative) values. $\endgroup$
    – whuber ♦
    Commented Feb 15, 2023 at 19:02
  • $\begingroup$ @whuber I've been thinking about it for a bit. The best thing I came up with is just a shift in perspective of an outlier. I think I should see an outlier as a 'point' that is not 'supposed to be part of $F$' to begin with? So if $x$ is the outlier, $F(x+\delta) - F(x-\delta)$ should be very small, but in the contaminated distribution we just assign a positive measure to $x$ and I can kind of see how this translates to having an outlier present in the data. $\endgroup$
    – AyamGorengPedes
    Commented Feb 19, 2023 at 13:26
  • $\begingroup$ That looks like a good way to be thinking about it. $\endgroup$
    – whuber ♦
    Commented Feb 19, 2023 at 14:21
  • $\begingroup$ See an example at stats.stackexchange.com/questions/117950/… $\endgroup$
    – kjetil b halvorsen ♦
    Commented Feb 21, 2023 at 12:04

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