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Consider a classic linear regression model.

I want to show that the Wald, Likelihood Ratio (LR) the Lagrange Multiplier (LM) test statistics for the null hypothesis $H_0: R\beta =r$ for a constant vector $r$ are all monotonic functions of the standard F statistic.

Please suggest a book or document which show this theoretical proof.

Update: I found the article which exactly proves the statement in this question.

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    $\begingroup$ Does this answer your question? Likelihood ratio, Wald, and Score are equivalent? $\endgroup$
    – Dave
    Commented Sep 19, 2022 at 22:03
  • $\begingroup$ Exactly no. I want to theoretically prove that these tests follows F distribution (with mathematical proving ways) @Dave $\endgroup$
    – 1190
    Commented Sep 19, 2022 at 22:09
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    $\begingroup$ I thought that answer addressed the inequality. It’s in a follow-up. $\endgroup$
    – Dave
    Commented Sep 19, 2022 at 23:00
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    $\begingroup$ @Dave I found its proof. sciencedirect.com/science/article/abs/pii/… Thank you. $\endgroup$
    – 1190
    Commented Sep 20, 2022 at 1:46
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    $\begingroup$ You might consider typing up an answer to close out this question. Self-answers are not particularly common, but they are not frowned-upon. (I have done it a couple of times.) That also gives you a chance to evaluate your knowledge: if the post gets downvotes or other negative feedback, it could signal a gap in need of filling. $\endgroup$
    – Dave
    Commented Sep 20, 2022 at 1:53

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