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I don't know if the question is worded weirdly, but I'm having difficulties understanding its logic. I have the solution, but if possible, can someone explain the reason behind it?

We have two models (assume random sampling): $E[y]=\alpha$ and $E[y|x]=\alpha$

How can we estimate for both models, $\alpha$ consistently and as efficiently as possible;

And from using these estimates, how can we test the hypothesis $H_0: \alpha= 0$

The solution given is: For model 1, which is a regression of y on a constant, the only consistent estimator is the analog estimator $\hat{\alpha}=\bar{y}$. Model 2 is a regression on x and assuming that x is not constant with a very tight regression function. The regression parameters (here only one) can be efficiently (as the true conditional density is unknown) estimated through GLS: enter image description here

Maybeline Lee
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  • Interesting question. You might find it helpful to contemplate the simplest situation where the two models differ: namely, $x$ can take on only two distinct values (which might as well be $0$ and $1$). In this case you have a bunch of $y$ observations associated with $x=0$ and another bunch associated with $x=1.$ *What do you do when the former set is much more scattered than the latter?* Answer: [construct a weighted mean with weights in inverse proportion to the group variances.](https://stats.stackexchange.com/questions/12251) – whuber Mar 17 '21 at 18:19
  • I am sorry, I am still not following. Is $x=1$ in the second model for the GLS to look like that? – Maybeline Lee Mar 17 '21 at 19:05
  • @MaybelineLee you have to think beyond regression and look into likelihood. You would need the EM algorithm to estimate the MLE in whuber's example. – AdamO Mar 17 '21 at 19:41
  • @Adam The quotation does not seem to be pushing for MLE, as evidenced by its reference to "nonparametric regression." – whuber Mar 17 '21 at 19:52
  • @whuber I would think GLS is a rather a *parametric* regression routine, and a convenient off-the-shelf way of estimating the covariance structure (via EM). But is the question how do we go to the next step and construct the hypothesis test? – AdamO Mar 17 '21 at 21:05
  • @Adam What makes this procedure nonparametric is the need to estimate the error variances from the data. If this is not done in a robust manner, the entire program fails (as I have learned firsthand). The use of a parametric algorithm in *part* of a procedure does not necessarily make the procedure parametric! – whuber Mar 19 '21 at 14:17

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