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I am trying to determine if two regression estimates are different. The first is obtained by ordinary least squares (OLS) and the second is obtained by M-estimation.

As a minimum example, fits for the "Stack Loss Data" (Montgomery et al. 2001 - Introduction to Linear Regression Analysis, pg. 397) are:

OLS: $\hat{y} = -39.9 +0.72x_1 + 1.30x_2 - 0.15x_3$ and M-estimation (Andrews Sine, a = 1.5): $\hat{y} = -37.2 + 0.82x_1 + 0.52x_2 - 0.07x_3$.

Is there a hypothesis test that I could use? From what I've read so far, an ordinary t-test isn't applicable to robust estimates. Please bear with me, I'm a geophysicist who has learned a lot of statistics on the fly.

To clarify, to show that they are different means that they come from different populations, right? From what I understand, M-estimates are asymptotically normal.

Jordan Bishop
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  • What is it you're trying to compare exactly? estimates of which quantity? – Glen_b Jun 22 '18 at 03:54
  • I'm trying to determine if I can use an approach like this: https://stats.stackexchange.com/questions/151916/are-two-linear-regression-models-significantly-different if one of my regression lines is not the result of OLS. – Jordan Bishop Jun 22 '18 at 19:09
  • In very large samples you should still be able to make use of asymptotic normality to derive a chi-squared test from the coefficients and their estimated variance-covariance matrices (assuming you have such estimates) in similar fashion to the answer at the question you point to. – Glen_b Jun 23 '18 at 10:05

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