# Least square estimation and nonlinear model

Suppoese X is the attribute and Y is the response as random variables. We observe (X,Y) jointly as a bi-variate normal variable, then least square estimation of Y as a regression function E(Y|X) is a linear function $$\omega_0 + \omega_1X$$. Now, when we take joint distribution of (X,Y) other than normal distribution (such as uniform), then corresponding least square estimate as a regression function $$E(Y|X)$$ may not be a linear function.

I have the folllowing questions:

1. How can perform linear approximation of the regression function $$E(Y|X)$$ when (X,Y) is having non-normal joint distribution?
2. What would be estimation error of the corresponding linear approximation?
• Sep 17 at 17:43
• I am looking for explicit examples such as if some how we came to know that the joint distribution of (X,Y) is unform, then the corresponding regression function is a piece-wise linear (i.e. a non-linear) function. Sep 18 at 13:30