# Local neighbourhood

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?

• I presume this problem comes from statistics? – David Roberts 2 days ago
• Yes that is right. – user0735 2 days ago
• What is the "true parameter space"? You haven't defined it, only "some parameter space". – David Roberts 2 days ago
• The true parameter space is the space where the true parameter vector lives hence Theta. – user0735 2 days ago
• 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. – auniket 2 days ago