Timeline for Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$
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| Nov 13, 2018 at 20:44 | comment | added | dohmatob | BTW, using your bounds I get, $$R_n(\theta) \rightsquigarrow \begin{cases}\frac{1}{n}\chi^2_{(r)},&\mbox{ if }\mathbb E_{P}[g(Z;\theta)] = 0,\\\mathbb E_{P}[\|g(Z;\theta)\|^2] + \mathcal N(0,\frac{C}{n}),&\mbox{ else,}\end{cases}$$ where $r=r(\theta)$ is the rank of the covariance matrix of $g(Z;\theta)$ and $C \ge 0$ is an absolute constant. | |
| Nov 13, 2018 at 20:09 | comment | added | dohmatob | @IosifPinelis Indeed. I thought of it but it was too good to be true. Thanks again. | |
| Nov 13, 2018 at 19:00 | comment | added | Iosif Pinelis | If $\theta$ is fixed, then this question was answered, at the end of the answer linked in your post -- just replace $z_i$ there by $g(z_i;\theta)$. | |
| Nov 5, 2018 at 12:16 | comment | added | dohmatob | Turns out the $\alpha_n(\theta)$ matches the so-called empirical likelihood functional (Owen 1990, Qin & Lawless 1994). If $\hat{\theta}_n$ minimizes $\alpha_n(\theta)$, then $\alpha_n(\hat{\theta}_n)-\alpha_n(\theta^*) \rightsquigarrow \chi^2_{(p)}$. This is theorem 2 of Qin & Lawless 1994. It's not much of an insight, but I don't know if I should add this as an "observation" under the question or a separate answer. | |
| Nov 5, 2018 at 12:06 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Nov 4, 2018 at 19:01 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Nov 2, 2018 at 12:54 | history | asked | dohmatob | CC BY-SA 4.0 |