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Iosif Pinelis
Iosif Pinelis
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I have already asked that Question on Cross Validated: Link

Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do statistical inference. Assume that there is a asymptotical (weak) consistency result, i.e. there is an estimator $\theta_{n}(X_{1},\ldots,X_{2})$, s.t. $\theta_{n}(X_{1},\ldots,X_{2})\overset{\mathbb{P}}{\longrightarrow}\theta$.

Now, I want to go one step further, calculating a weak limit theorem allowing for testing and confidence on $\theta$. This means, I am looking for a deterministic sequence $\ell_{n}$, s.t. $$\ell_{n}\cdot\left(\theta_{n}(X_{1},...,X_{n}\right)-\theta)\longrightarrow V$$ in distribution, with some r.v. $V$, s.t. $\mathbb{P}_{V}$ is a well known distribution.

Question 1: Is there a way to find the optimal convergence rate $\ell_{n}$?

Question 2: Is it possible to change the limit distribution from $V$ to $V'$ to get a different sequence $\ell_{n}'$? Of course, both distributions, $\mathbb{P}_{V}, \mathbb{P}_{V'}$ are assumed to be non-degenerateddegenerate.

I have already asked that Question on Cross Validated: Link

Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do statistical inference. Assume that there is a asymptotical (weak) consistency result, i.e. there is an estimator $\theta_{n}(X_{1},\ldots,X_{2})$, s.t. $\theta_{n}(X_{1},\ldots,X_{2})\overset{\mathbb{P}}{\longrightarrow}\theta$.

Now, I want to go one step further, calculating a weak limit theorem allowing for testing and confidence on $\theta$. This means, I am looking for a deterministic sequence $\ell_{n}$, s.t. $$\ell_{n}\cdot\left(\theta_{n}(X_{1},...,X_{n}\right)-\theta)\longrightarrow V$$ in distribution, with some r.v. $V$, s.t. $\mathbb{P}_{V}$ is a well known distribution.

Question 1: Is there a way to find the optimal convergence rate $\ell_{n}$?

Question 2: Is it possible to change the limit distribution from $V$ to $V'$ to get a different sequence $\ell_{n}'$? Of course, both distributions, $\mathbb{P}_{V}, \mathbb{P}_{V'}$ are assumed to be non-degenerated.

I have already asked that Question on Cross Validated: Link

Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do statistical inference. Assume that there is a asymptotical (weak) consistency result, i.e. there is an estimator $\theta_{n}(X_{1},\ldots,X_{2})$, s.t. $\theta_{n}(X_{1},\ldots,X_{2})\overset{\mathbb{P}}{\longrightarrow}\theta$.

Now, I want to go one step further, calculating a weak limit theorem allowing for testing and confidence on $\theta$. This means, I am looking for a deterministic sequence $\ell_{n}$, s.t. $$\ell_{n}\cdot\left(\theta_{n}(X_{1},...,X_{n}\right)-\theta)\longrightarrow V$$ in distribution, with some r.v. $V$, s.t. $\mathbb{P}_{V}$ is a well known distribution.

Question 1: Is there a way to find the optimal convergence rate $\ell_{n}$?

Question 2: Is it possible to change the limit distribution from $V$ to $V'$ to get a different sequence $\ell_{n}'$? Of course, both distributions, $\mathbb{P}_{V}, \mathbb{P}_{V'}$ are assumed to be non-degenerate.

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stochic
stochic
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How to find the optimal convergence rate?

I have already asked that Question on Cross Validated: Link

Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do statistical inference. Assume that there is a asymptotical (weak) consistency result, i.e. there is an estimator $\theta_{n}(X_{1},\ldots,X_{2})$, s.t. $\theta_{n}(X_{1},\ldots,X_{2})\overset{\mathbb{P}}{\longrightarrow}\theta$.

Now, I want to go one step further, calculating a weak limit theorem allowing for testing and confidence on $\theta$. This means, I am looking for a deterministic sequence $\ell_{n}$, s.t. $$\ell_{n}\cdot\left(\theta_{n}(X_{1},...,X_{n}\right)-\theta)\longrightarrow V$$ in distribution, with some r.v. $V$, s.t. $\mathbb{P}_{V}$ is a well known distribution.

Question 1: Is there a way to find the optimal convergence rate $\ell_{n}$?

Question 2: Is it possible to change the limit distribution from $V$ to $V'$ to get a different sequence $\ell_{n}'$? Of course, both distributions, $\mathbb{P}_{V}, \mathbb{P}_{V'}$ are assumed to be non-degenerated.