# the Wald, Likelihood Ratio (LR) the Lagrange Multiplier (LM) test statistics Are monotonic function of F statistics

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.

• Does this answer your question? Likelihood ratio, Wald, and Score are equivalent?
– Dave
Commented Sep 19, 2022 at 22:03
• Exactly no. I want to theoretically prove that these tests follows F distribution (with mathematical proving ways) @Dave
– 1190
Commented Sep 19, 2022 at 22:09
• I thought that answer addressed the inequality. It’s in a follow-up.
– Dave
Commented Sep 19, 2022 at 23:00
• @Dave I found its proof. sciencedirect.com/science/article/abs/pii/… Thank you.
– 1190
Commented Sep 20, 2022 at 1:46
• 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.
– Dave
Commented Sep 20, 2022 at 1:53