Repeat sampling of extreme observations: regression to the mean revisited (Q626296)

Let there be two observations \(Y=T+\varepsilon\) and \(Y^*=T+\varepsilon^*\), where \(T\), \(\varepsilon\) and \(\varepsilon^*\) are independent r.v.s, and \(\varepsilon\) and \(\varepsilon^*\) are equally distributed. The author investigates the behaviour of the regression gradient \(\text{rg}(y)=E(Y^*| Y=y)=y\) as \(y\to\infty\) for different distributions of \(T\) and \(\varepsilon\). In the case of Gaussian \(T\) and \(\varepsilon\) this is the classical regression on the mean effects first observed by F. Galton. It is demonstrated that this behaviour depends on the tail behaviour of the \(T\) and \(\varepsilon\) distributions. Regularly-tailed continuous (RTC) distributions \(r\) are considered which satisfy \(\lim_{y\to\infty} r(y+h)/r(y)=\gamma^h\) for all \(h>0\) and some \(0\leq\gamma\leq 1\). Then, e.g., if \(E(| T| )<\infty\) and \(E(| \varepsilon| )<\infty\), \(\text{rg}(y)\to 0\) when \(\gamma=1\). Other values of \(\gamma\) and unbounded \(E(| \varepsilon| )\) are also considered.