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This text is taken from Introduction to robust estimation and hypothesis testing. Wilcox R.

First I will write down the description that leads to definition of relative influence on $T(F)$ and then I will state my questions.

Consider a mixture of two distributions where an observation is randomly sampled from distribution $F$ with probability $1 − \epsilon$, otherwise sampling is from the distribution $\Delta_x$ . That is, with probability $\epsilon$, the observed value is $ x$. The resulting distribution is $F_{x,\epsilon} = (1 − \epsilon)F + \epsilon \Delta_x$. But $F$ and $\Delta_x$ are distributions, so |$F(y) − \Delta_x(y)| ≤ 1$. Consequently, the Kolmogorov distance between $F_{x,\epsilon}$, and $F$ is at most $\epsilon$. Moreover, $F(x)$, and $F$ can be made arbitrarily close by choosing sufficiently small. The relative influence on $T(F)$ of having the value $x$ occur with probability $\epsilon$ is

$$\frac{T(F_{x,\epsilon})-T(F)}{\epsilon}$$

and the influence function of $T$ at $F$ is $$IF(x) = \lim \frac{T(F_{x,\epsilon}) − T (F)}{\epsilon}$$

So, I can understand everything until the point where the relative influence on $T(F)$ of having the value $x$ occur with probability $\epsilon$ is defined. It is derivative at point $x$, but why the probability $\epsilon$ is at the same chosen to be the distance of change in $x$? How is it connected?

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