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Partial answer for linear models:

Partial answer for linear models (I do not argue things are different for nonlinear models, just do not present results):

Remark 1: if we knew the error variance, the result would be true, as Wald and LR then are identical, cf. Exact equivalence of LR and Wald in linear regression under known error variance)

Remark 5: Since $\mathcal{W}_{ij}=O_p(1)$ also under local alternatives, as the statistics are noncentral chi-square distributed, Remark 3 should also go through under such local alternatives.

Remark 1: If we knew the error variance, the result would be true, as Wald and LR then are identical - no differences can arise from estimating the variance in different ways. See also Exact equivalence of LR and Wald in linear regression under known error variance)

Remark 5: $\mathcal{W}_{ij}=O_p(1)$ also under local alternatives, as the statistics are noncentral chi-square distributed (see e.g. Sampling distribution of Coefficient of determination in general for a related discussion for the F statistic). Therefore, Remark 3 should also go through under such local alternatives.

Remark 5: Since $\mathcal{W}_{ij}=O_p(1)$ also under local alternatives, Remark 3 should also go through under such local alternatives.

Remark 5: Since $\mathcal{W}_{ij}=O_p(1)$ also under local alternatives, as the statistics are noncentral chi-square distributed, Remark 3 should also go through under such local alternatives.