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Consider the parameter space $\boldsymbol{\Theta}=\boldsymbol{\Theta_{1}}\times \boldsymbol{\Theta_{2}}$, which is a compact subset of $R^{n}$, and contains the true parameter vector $\boldsymbol{\theta^{*}}=(\boldsymbol{\theta_{1}'^{*}},\boldsymbol{\theta_{2}'^{*}})'$.

Consider also the parameter $\boldsymbol{\theta}=(\boldsymbol{\theta_{1}}',\boldsymbol{\theta_{2}')'}$, where $\boldsymbol{\theta_{1}}\in\boldsymbol{\Theta_{1}}\subset R^{n_{1}}$, $\boldsymbol{\theta_{2}}\in\boldsymbol{\Theta_{2}}\subset R^{n_{2}}$.

Define $\boldsymbol{\Theta_{r}}$ to be the restriction of the parameter space $\boldsymbol{\Theta}$ to $\boldsymbol{\Theta_{r}}=\boldsymbol{\Theta_{1}}\times N_{r}(\boldsymbol\theta_{2}^{*})$, where $N_{r}(\boldsymbol\theta_{2}^{*})=\{\boldsymbol\theta_{2}\in \boldsymbol\Theta_{2}:\lVert\boldsymbol\theta_{2}-\boldsymbol\theta_{2}^{*}\rVert\leq r\}$ for some small $r>0$.

Is it "topologically" correct to say that $\boldsymbol{\Theta_{r}}$ defines a local (nonzero measure) neighbourhood of the true parameter space?

I then want to consider a neighbourhood $B_{\epsilon}(\boldsymbol\theta^{*})\subset \boldsymbol{\Theta_{r}}$ with compact compliment $B_{\epsilon}^{c}(\boldsymbol\theta^{*})$ in $\boldsymbol{\Theta_{r}}$, and $\epsilon>0$, in order to prove $\limsup_{N\to\infty}max_{\boldsymbol{\theta}\in B_{\epsilon}^{c}(\boldsymbol\theta^{*}) }E[L_{N}(\boldsymbol\theta)-L_{N}(\boldsymbol\theta^{*})]<0$, where $L$ is my function.

For this to make sense as far as I can see I can't have $N_{r}(\boldsymbol\theta_{2}^{*})\subset B_{\epsilon}(\boldsymbol\theta^{*})$ because then $N_{r}(\boldsymbol\theta_{2}^{*})\cap B_{\epsilon}^{c}(\boldsymbol\theta^{*})=\varnothing$. So I guess what I really need is part of $N_{r}(\boldsymbol\theta_{2}^{*})$ to lie in $B_{\epsilon}(\boldsymbol\theta^{*})$ and part of it in $B_{\epsilon}^{c}(\boldsymbol\theta^{*})$. Is this "topologically" feasible?

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  • $\begingroup$ I presume this problem comes from statistics? $\endgroup$ – David Roberts 2 days ago
  • $\begingroup$ Yes that is right. $\endgroup$ – user0735 2 days ago
  • $\begingroup$ What is the "true parameter space"? You haven't defined it, only "some parameter space". $\endgroup$ – David Roberts 2 days ago
  • $\begingroup$ The true parameter space is the space where the true parameter vector lives hence Theta. $\endgroup$ – user0735 2 days ago
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    $\begingroup$ Well, clearly $\Theta_r$ is the closure of the relatively open subset $\Theta^0_r := \Theta_1 \times N^0_r(\theta^*_2)$ of $\Theta$, where $N^0_r(\theta^*_2)$ is the open ball of radius $r$ around $\theta^*_2$. So $\Theta^0_r$ is definitely an "open neighborhood'' of $\theta^*$ in $\Theta$. And $\Theta_r$ can also be said to be a "neighborhood'' of $\theta^*$ in $\Theta$ according to the convention of at least one topology text I can think of. Is it of nonzero measure? That depends on the measure of course. $\endgroup$ – auniket 2 days ago

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