0
$\begingroup$

Consider a regression model $Y=E(Y|X)+Prediction \ Error$ i.e $Prediction \ error = Y-E(Y|X)$. Now, define an estimate of the regression function $E(Y|X)=\hat{Y}+ Fitted \ error$ i.e. Fitted error = $E(Y|X)-\hat{Y}$. Further, define residual = $Y-\hat{Y}$. Now, we have $Total \ error = Residual \ error + Fitted \ error$.

Therefore, the total error can be expressed as $Total \ error = Prediction \ error + 2 \times Fitted \ error$.

Is the above derivation regarding errors, is right or wrong?

Improve this question
$\endgroup$
3
  • $\begingroup$ Where did your $2$ come from? $\endgroup$
    – Henry
    Commented Sep 20, 2021 at 10:46
  • $\begingroup$ 1. Prediction error = Tru vale of response - predicted valeu of the response = True value of response - (estimated value of the prediction i.e. fitted value of y + fitted error) = True value - fitted value of y - fitted error => prediction error = residual - fitted error => residual = prediction error + fitted error. 2. Total error = residual error + fitted error => residual error = total error - fitted error. 3. From (1) and (2) total error = prediction error + 2 fitted error. $\endgroup$
    – Lakshman
    Commented Sep 20, 2021 at 14:40
  • $\begingroup$ I would have thought total error is in fact $Y - \hat Y$ (what you call residual error). This is $(Y- E[Y \mid X]) +(E[Y \mid X] - \hat Y)$ so the sum of what you call prediction error and fitted error. $\endgroup$
    – Henry
    Commented Sep 20, 2021 at 14:54

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.