What is sample regression function?

Answer 1

To take a simple case, let's suppose you have a set of pairs (x1, y1), (x2, y2), ... (xn, yn). You have obtained these by choosing the x values and then observing the corresponding y values experimentally. This set of pairs would be called a sample.

For whatever reason, you assume that the y's and related to the x's by some function f(.), whose parameters are, say, a1, a2, ... . In far the most frequent case, the y's will be assumed to be a simple linear function of the x's: y = f(x) = a + bx.

Since you have observed the y's experimentally they will almost always be subject to some error. Therefore, you apply some statistical method for obtaining an estimate of f(.) using the sample of pairs that you have.

This estimate can be called the sample regression function. (The theoretical or 'true' function f(.) would simply be called the regression function, because it does not depend on the sample.)