1
$\begingroup$

Iterative reweighted least squares (IRLS) is used when errors are heteroscedastic. Let us assume that error comes from a distribution where its mean is zero and the variance is a function of the absolute value of the input. From what I have read, IRLS is applicable here and will give better results than OLS.

My question is can I solve this using MLE? Let us say that I define my output to be from a normal distribution, whose probability density is written as follows:

\begin{equation} \prod_{i=1}^n p(y_i | x_i, w_0, w_1, m, n) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp(-\frac{(y_i - (w_0 + w_1\cdot x_i))^2}{2\sigma^2} \end{equation}

where $\sigma = m|x| + n$, the absolute value of $x$ is denoted by $|x|$. For simplicity, $x, y \in \mathbb{R^1}$.

Now we can solve this by MLE to find values of $w_0, w_1, m, n$.

Is this better or worse than IRLS? I haven't seen much discussion about this method. So is there a problem with this method. The only disadvantage that comes to my mind is that we are assuming a functional form for the variance, which if turned out to be wrong, can affect the regression quality greatly. But then, even IRLS assumes a diagonal weight matrix where each entry is filled as follows (from here):

If a residual plot against a predictor exhibits a megaphone shape, then regress the absolute values of the residuals against that predictor. The resulting fitted values of this regression are estimates of σi. (And remember $w_i=\frac{1}{\sigma^2_i}$)

Another disadvantage of MLE is maybe it is more sensitive than IRLS.

This paper is the only thing that I found comparing MLE and IRLS, but was a little difficult to understand.

Any thoughts or is anyone aware of any studies. Also, I am still learning about this so please point out if there are any mistakes in my analysis.

$\endgroup$
  • $\begingroup$ Irls is a way of computing the mle $\endgroup$ – kjetil b halvorsen Aug 18 '17 at 20:28
  • $\begingroup$ @kjetilbhalvorsen Okay. Thanks. Can you please elaborate on how it estimates MLE. I would have followed your explanation if the noise had come from a normal distribution. How does that extend here? Second question is, is it even correct to model the variance as a linear function of absolute value and then estimate those parameters by MLE? $\endgroup$ – Stats_student Aug 18 '17 at 23:11
  • $\begingroup$ I already wrote an answer here (might be a duplicate): stats.stackexchange.com/questions/236676/… $\endgroup$ – kjetil b halvorsen Aug 21 '17 at 7:22

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.