# Tagged Questions

Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.

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### Asymptotic Distribution of the Wald Test Statistic

I am trying to understand the asymptotic distribution of the Wald test statistic, specifically under the alternative hypothesis which I've found little reference to. For clarity, the binary ...
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### How to symmetrize a given function (U-statistics)?

Van der Vaart's "Asymptotic Statistics" (Ch 12) contains the quote: Given a known function $h$, consider the estimation of the "parameter" $$\theta = Eh(X_{1},\cdots,X_{r})$$ In order to simplify ...
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### What's the convergence rate in the context of convergence in probability?

A sequence $z_n$ with $\lim_n z_n = z$ is said to have $Q$-linear convergence if a constant $r\in (0,1)$ exists such that $\displaystyle |z_{n+1} - z| \leq r \, |z_n - z|$, where $r$ is called the ...
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### Are Maximum Likelihood Estimators asymptotically unbiased?

I can follow the proofs in which the asymptotic normal-distribution of a maximum likelihood estimator $\tilde{\theta}_n$ is derived. however, does this already imply that the maximum likelihood ...
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### CLT and Wald-Wolfowitz runs test asymptotic distribution

I need help finding a theorem which could be used to prove that the Wald-Wolfowitz runs test is asymptotically normal. Let me formalize my question. We have a random sample $\{X_0,X_1,...,X_n\}$ (if ...
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### Problem involving Scheffe's theorem and asymptotic distribution

If $\{ X_n \}$ are independently and identically distributed $U(0,1)$ random variables and $V_n = n(1 - X_{(n)})$ (where $X_{(n)}$ denotes the $n$th or largest order statistic), then how do I derive ...
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### a question on Edgeworth Expansion

I'm working Edgeworth Expansion. I couldn't understand one thing . Can you help me about that please. $$Z= \frac{\sqrt {n} (\bar {x} -\mu)}{\sigma}$$ converges in distribution to N(0,1) I have ...
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### Cramer-Rao Lower Bound for the estimation of Pearson correlation

Given a bivariate Gaussian distribution $\mathcal{N}\left(0,\begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}\right)$, I am looking for information on the distribution of $\hat{\rho}$ ...
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### $p$-value for non-standard asymptotics

Suppose I have an asymptotic result like $$\sqrt{n}(T_n - \theta) \overset{D}{\to} \sum_{i=1}^k \lambda_i X_i$$ where $X_i$ are independent $\chi^2_1$. i.e. some test statistics $T_n$ is ...
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### MLE asymptotic properties in non-regular families

I am working with asymptotic results about the MLEs and I know that if the family of distributions to whom the pdf of my sample belongs is exponential the regularity conditions for the asymptotic ...
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### What is the asymptotic distribution of the variance of the error term (in MLE linear regression)

In most treatments of the MLE linear regression, the author focuses on the asymptotic normality of $\hat \beta_{MLE}$. To estimate $Var(\hat \beta_{MLE})$, which relies on $\sigma^2$, they also show ...
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### What can go wrong using lagged terms as instrumental variables?

Can anybody give one example of when the set of all lagged $X$ can (or can't) be a good choice of IV's for $X_{t}$?
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### Observed Fisher Info as an estimator of Expected Info

When I construct an asymptotic confidence interval for a parameter $\theta$, taken from a sample iid distributed with a generic pdf/pmf, I usually implement the mle $\hat\theta$ instead of $\theta$, ...
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### Using asymptotic distribution under the null when the null hypothesis is false

This is a general question about hypothesis testing in statistics. A standard hypothesis test as I see it is based on 3 things: Stating a null hypothesis Computing a test statistic for the ...
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### How do weak instruments violate full rank condition?

So I know that if the instruments $(Z_i)$ in 2SLS regression are weak i.e. $X_i$ and $Z_i$ are uncorrelated, then full rank condition for matrix $\Sigma_{ZX}=E[Z_iX_i]$ will fail to hold, but how can ...
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### Is it better to use a MLE or a MME to build an asymptotic confidence interval for a real parameter $\theta$?

I thought that the answer was pretty straightforward given that the MLEs possess some strong asymptotic properties, i.e. normality, efficiency and consistency. But then, I have found that also MME (...
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### Asymptotic normality for nonsmooth objective functions

Assume that $f ({\bf x}; \theta): \mathbb{R}^p \times \Theta \to \mathbb{R}$, where ${\bf x}$ is the vector of inputs (with some distribution) and $\theta$ is the vector of parameters. Also, assume ...
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### Asymptotically exact confidence interval

I am trying to answer the following question: I have modeled it using a Poisson distribution, and from this calculated the Fisher information as $I(\lambda)= \frac{n}{\lambda}$. Then, when I ...
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### Likelihood Ratio Test for the variance of a normal distribution

I've found that the asymptotic LR test is used in simple vs bilateral hypothesis test in which it is impossible to actually compute the rejection region, or better, in which we would need to find a ...
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### To determine asymptotic distribution, when to ignore lower-order $O_p$ terms?

Let $A_{m,n} + B_{m,n}$ be such that: $A_{m,n} \overset{d}{\rightarrow} A$ as $m,n \rightarrow \infty$ $A_{m,n} = O_p(\sqrt{nm(n+m)})$ $B_{m,n} = O_p(\sqrt{mn})$. Under what conditions would the ...
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### Negative population variable importance

The question is: Is it possible for the population variable importance to be negative ? The variable importance is defined below in the context it comes from: random forests. But no knowledge is ...