I have a smooth continuous well-behaved function f(x), where f(x) is positive and mononically increasing with x, and x is positive real continuous variable.
Given the mean, variance, and correlation of two sequences of positive random numbers x and y, is there an estimate for the correlation between f(x)and f(y) in terms of function f and the mean, variance, and correlation of the two sequences x & y?
Yes, an estimate of the correlation of f(x) and f(y) is required.
The function f(x) is non-linear. The application is ultimately for estimating the expected value E of another function of two variables G(f(x),f(y)).
I have the estimate of E[G(a,b)] when a and b are random variables, expressed in terms of mean, variance, and correlation of a and b.
But now I need the estimate of E[G(f(x),f(y))]. I have the estimate for mean and variance of a=f(x) in terms of the mean and variance of x, and likewise the mean and variance of b=f(y) in terms of the mean and variance of y, but I still need an estimate of the correlation of a=f(x) and b=f(y) in terms of mean, variance, and correlation of x & y.
Thanks