User:SamuelTheGhost/Reversion towards the Mean

Universal definition

Let X, Y be random variables with any joint distribution (discrete or continuous). Reversion towards the Mean is the property defined in the following theorem. Assume means exists and that X and Y have identical marginal distributions. Then for all c in the range of the distribution, so that

$P[X\geq c]\neq 0\,,$

we have that

$E[Y|X\geq c]\leq E[X|X\geq c]\,,$

with the reverse inequality holding for all $X\leq c\,\!.$

Proof

First we look at some probabilities. By elementary laws:

$P[X\geq c]=P[X\geq c\land Y\geq c]+P[X\geq c\land Y<c]$ and

$P[Y\geq c]=P[X\geq c\land Y\geq c]+P[X<c\land Y\geq c]$

But the marginal distributions are equal, which implies

$P[X\geq c]=P[Y\geq c]$

So taking these three equalities together we get

$P[X\geq c\land Y<c]=P[X<c\land Y\geq c]$

Going on the conditional probabilities we infer that

$P[Y<c|X\geq c]={\frac {P[X\geq c\land Y<c]}{P[X\geq c]}}={\frac {P[X<c\land Y\geq c]}{P[Y\geq c]}}=P[X<c|Y\geq c]$

Looking now at expected values we have

$E[Y|X\geq c]=$

$P[Y\geq c|X\geq c]\,\times \,E[Y|X\geq c\,\land \,Y\geq c]\;+\;P[Y<c|X\geq c]\,\times \,E[Y|X\geq c\,\land \,Y<c]$

But of course

$E[Y|X\geq c\,\land \,Y<c]<c$, so

$E[Y|X\geq c]\leq$

$P[Y\geq c|X\geq c]\,\times \,E[Y|X\geq c\,\land \,Y\geq c]\;+\;P[Y<c|X\geq c]\,\times \,c$

Similarly we have

$E[Y|Y\geq c]=$

$P[X\geq c|Y\geq c]\,\times \,E[Y|X\geq c\,\land \,Y\geq c]\;+\;P[X<c|Y\geq c]\,\times \,E[Y|X<c\,\land \,Y\geq c]$

and again of course

$E[Y|X<c\,\land \,Y\geq c]\geq c$, so

$E[Y|Y\geq c]\geq$

$P[X\geq c|Y\geq c]\,\times \,E[Y|X\geq c\,\land \,Y\geq c]\;+\;P[X<c|Y\geq c]\,\times \,c$

Putting these together we have

$E[Y|X\geq c]\leq E[Y|Y\geq c]$

and, since the marginal distributions are equal, we also have

$E[X|X\geq c]=E[Y|Y\geq c]$, so

$E[Y|X\geq c]\leq E[X|X\geq c]$

which concludes the proof.

Reference

• Myra L. Samuels (November 1991). "Statistical Reversion Toward the Mean: More Universal than Regression Toward the Mean". The American Statistician. 45 (4): 344–346. doi:10.2307/2684474. JSTOR 2684474.{{cite journal}}: CS1 maint: date and year (link)