Sorry if this is a stupid question, but is it true that p(X,Y)^2 has the degrees of freedom as you described? If X=Y is a uniform variable on [0,1] then p(X,Y)^2 = 1 but P(f(X),g(Y))^2 =/= 1 for (most) non-linear f and g.
In other words, I thought Pearson correlation is specifically for linear relationships so its variant under non-linear transformations.
Sorry if this is a stupid question, but is it true that p(X,Y)^2 has the degrees of freedom as you described? If X=Y is a uniform variable on [0,1] then p(X,Y)^2 = 1 but P(f(X),g(Y))^2 =/= 1 for (most) non-linear f and g.
In other words, I thought Pearson correlation is specifically for linear relationships so its variant under non-linear transformations.