= Finite contamination bias of normal location
{title2=$b_T(\varepsilon,x)=T(F_{\varepsilon,x})-T(F)$}
Consider the population <statistical model> $F_{\varepsilon,x}=(1-\varepsilon)N(0,1)+\varepsilon\delta_x$. The following are deterministic <statistical functional> calculations, not observed contamination experiments. Compare the actual location bias with the <linear approximation> $\varepsilon\operatorname{IF}(x;T,F)$. The <mean> bias is exactly $\varepsilon x$. For positive $x$ exceeding the shifted <median> and $0\le\varepsilon<1/2$, the generalized-inverse <median> bias is exactly
$$
b_{\operatorname{Med}}(\varepsilon,x)=\Phi^{-1}\!\left(\frac{1}{2(1-\varepsilon)}\right).
$$
This formula applies to both $x=1$ and $x=8$ throughout the plotted interval $0\le\varepsilon\le0.1$. If the contaminating atom lies inside the <quantile> jump containing probability $1/2$, the generalized-inverse <median> instead equals that atom.
For the <Huber location estimator> with $s=1$ and $k=1.345$, solve its actual population equation, rather than substituting the <influence function>:
$$
(1-\varepsilon)g_k(t)+\varepsilon\psi_k(x-t)=0,\qquad g_k(t)=\mathbb E\psi_k(Z-t),\quad Z\sim N(0,1).
$$
Writing $a=t-k$ and $b=t+k$, integration of the <normal distribution> gives
$$
g_k(t)=\phi(a)-\phi(b)-t\bigl[\Phi(b)-\Phi(a)\bigr]+k\bigl[1-\Phi(b)-\Phi(a)\bigr],
$$
where $\phi$ is the <probability density function> of $N(0,1)$. Its <derivative> is $-[\Phi(t+k)-\Phi(t-k)]<0$, so the mixture score is strictly decreasing for $\varepsilon<1$. For $\varepsilon<1/2$, its limits at negative and positive infinity have opposite signs even with arbitrary distant point contamination. The displayed examples use the unique root, evaluated by the <bisection method> to numerical precision.
\Image[/normal-location-contamination-bias.png]
{title=Exact population contamination biases and first-order influence predictions for normal location at contaminating points 1 and 8}
At $x=8$ and $\varepsilon=0.01$, the exact <mean>, <median>, and <Huber location estimator> biases are $0.080000$, $0.012660$, and $0.016541$; the <influence function> predictions are $0.080000$, $0.012533$, and $0.016375$. At $\varepsilon=0.05$, they become $0.400000$, $0.066012$, and $0.086241$, versus predictions $0.400000$, $0.062666$, and $0.081875$. The bounded alternatives control the effect of the distant atom, but their actual response curves bend away from the tangent.
The nearby atom $x=1$ gives a useful comparison: at $\varepsilon=0.05$, the <Huber location estimator> bias is $0.060237$, slightly below its <linear approximation> $0.060874$, whereas the <median> bias is still $0.066012$. The direction and size of approximation error depend on the contaminating location. Even accurate local influence calculations must specify $F$, the scale, the cutoff, and the actual contamination fraction.
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