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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?
kjetil b halvorsen
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Lakshman
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  • Maybe have a look at https://stats.stackexchange.com/questions/381072/how-to-test-whether-the-association-between-two-continuous-variables-varies-by-a/381082#381082, https://stats.stackexchange.com/questions/123453/copulas-with-regression and https://stats.stackexchange.com/questions/308775/conditional-expectation-of-two-identical-marginal-normal-random-variables/308831#308831 – kjetil b halvorsen Sep 17 '21 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. – Lakshman Sep 18 '21 at 13:30

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