# Homework Problem Consistancy of Estimator

We have to show:

Let $$\theta_0$$ be a k-dim vector. Show, that the following statements are equivalent:

(1) $$\hat{\theta}_n$$ is consistent for $$\theta_0$$.

(2) For each component $$i= 1,...,k$$: $$\hat{\theta}_{n,i}$$, is $$\hat{\theta}_{n,i}$$ consistent for $$\theta_{0,i}$$, where $$\hat{\theta}_{n,i}$$ and $$\theta_{0,i}$$ are the i'th component of the respective vector.

Hint: $$||x||_\infty \leq ||x||_2 \leq \sqrt{k}||x||_\infty$$ as well as: $$P(A+B \geq \varepsilon) \leq P(A\geq \frac{\varepsilon}{2}) + P(B\geq \frac{\varepsilon}{2})$$

Any suggestions would be greatly appreciated.

So an estimator is consistent if:

$$\lim_{n \rightarrow \infty} P(|\hat{\theta}_n - \theta| > \varepsilon) = 0$$, however using the inequality: $$P(A+B \geq \varepsilon) \leq P(A\geq \frac{\varepsilon}{2}) + P(B\geq \frac{\varepsilon}{2})$$ doesn't make much sense here (from my understanding).

• Could you give some context about what you have tried so far? Apr 13 '21 at 18:53
• Just edited my question. Apr 13 '21 at 18:55
• I think there is a small typo - it should be $i = 1,\ldots,k$. Based on the hint, I am going to assume that $\hat \theta_n$ is consistent for $\theta_0$ means specifically that for fixed $\varepsilon, \delta$, there eixsts $N$ such that $\mathbb P(||\hat\theta_n - \theta_0||_2 > \varepsilon) < \delta$ for $n \geq N$. Can you show that (2) is the same as the above statement holding, but with the $L-\infty$ norm in place of the $L2$ norm? Apr 13 '21 at 18:59
• you mean like: $P(||\hat{\theta}_n - \theta||_\infty \geq \frac{\varepsilon}{\sqrt{k}})$? Apr 13 '21 at 19:07
• Ok, so the argument is, that if $max(|\hat{\theta}_{n,i} -\theta_{n,i}| > \frac{\varepsilon}{\sqrt{k}})$ the so are all the others, and hence the statement is proved, correct? Apr 13 '21 at 19:15