I have read this and I have stuck on page 4. It says that
By definition [a normal pseudo-residual] is precisely $N(0,1)$ distributed and its value is zero if $Y$ is equal to the median of its distribution. Thus these residuals measure the deviations from the median and not from the expectation.
I have managed to prove that it follows a $N(0,1)$ distribution, but I can't figure out the rest.
I know that this is probably a silly question, but I would appreciate any help you can provide.
In this paper, $Y$ is a random variable with continuous distribution function $F(y)=\Pr(Y \le y)$. One way to measure how extreme a small value of $Y$ may be is to report the "probability of observing an equal or more extremely (small) value under the model [$F$]": in other words, when $F(y)$ is close to $0$, $y$ is an extremely low value for $Y$.
Some people, in whom reasoning about Normal distributions (determined by the Standard Normal distribution function $\Phi$) is deeply ingrained, prefer to re-express $F(y)$ in terms of the number of standard deviations ("Z score") $z$ for which $\Phi(z) = F(y)$. If we assume that $F$ strictly increases, this can be solved to yield
$$Z(y) = \Phi^{-1}(F(y)),$$
producing a new random variable $Z(Y)$ with a standard Normal distribution.
$Z(y)=0$ if and only if $$1/2 = \Phi(0) = \Phi(Z(y)) = F(y).$$
That is the definition of the median of $F$: a value $y$ for which $F(y)$ is $50\%$.
If a distribution $F$ has a mean $\mu_F$, it is not necessarily equal to its median. When, for instance, the mean of $F$ exceeds its median, then $Z(\mu_F)$ must be greater than $0$. Consequently, $Z$ when thought of relative to $0$, which is the center of a Normal distribution according to any definition whatsoever, truly reflects deviations relative to the median of $F$, not its mean (and not any other particular central location of $F$).
In United States case law on discrimination, courts have been exposed to enough statistical experts to have heard about standard deviations and z-scores. Some case law has resulted in standards (to serve as evidence of discrimination) that are expressed in terms of "numbers of standard deviations;" that is, in terms of Z-scores. When the statistic of interest (such as a measure of discriminatory impact) does not have a normal distribution, some experts like to convert p-values into "numbers of standard deviations." (They hope the courts will thereby understand the p-values better.) These could be interpreted as the pseudo-residuals discussed in this paper.
