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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35 views

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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24 views

Consistency and asymptotic normality of two-dimensional parameter

In a textbook exercise, for a sequence of iid variables, I have calculated the score function to be $$\begin{bmatrix} - \frac{n}{2\lambda} + \sum_{i=1}^n \frac{( x_i - \mu)^2}{2\mu^2 x_i}& \sum_{i=...
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Why doesn't the CLT work for $x \sim poisson(\lambda = 1) $?

So we know that a sum of $n$ poissons with parameter $\lambda$ is itself a poisson with $n\lambda$. So hypothetically, one could take $x \sim poisson(\lambda = 1) $ and say it is actually $\sum_1^n ...
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14 views

Discrete asymptotically decreasing function with mass 1?

I'm looking for asymptotically decreasing functions are there where the sum (probability mass) of the value corresponding to positive integers (x=1, 2, 3, ...) is 1 in the limit. Two extras would be ...
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24 views

Non-linear least squares and the distribution of an estimator

I have been trying to find the asymptotic normality of the non-linear least squares estimator. If I start with $0=X_t(\beta)'(y_t-x_t(\beta))$. I know that I have to perform Taylor expansion around ...
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35 views

Reference Books on Asymptotic theory of Statistics and Probability

Can anyone suggest me some good reference books on Asymptotic Theory of Statistics and Probability for students pursuing a post-graduate degree in Statistics ? It would be very much helpful if the ...
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Why does the continuity correction (say, the normal approximation to the binomial distribution) work?

I wish to better understand how the continuity correction to the binomial distribution for the normal approximation was derived. What method was used to decide we should add 1/2 (why not another ...
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26 views

Simplification in proof of OLS inconsistency

I'm a little confused right now regarding the LLN "jump" from probability limits to expectations and variances/covariances: Say we have a linear regression model of the form with $S$ observations: $$...
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24 views

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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Tail Bound from Asymptotics of an Estimators

Consider an non-parametric estimator for a random variable $X$ is $\hat{X}$. We know an asymptotic convergence for this estimator, in which $N^{1/4}\left(\hat{X} - X \right)\rightarrow\mathcal{N}(0,\...
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Presenting finite sample examination of asymptotics

From statistical theory, we often obtain results such as $\sqrt n (\theta - \hat \theta) \rightarrow_d N(0, \sigma)$ ie we have a normal limiting distribution. Because this formula says nothing ...
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Asymptotic conditional expectation

Problem Setup Let $\{X^d_1, X^d_2, \cdots, X^d_n\}$ be a $d-$dimensional zero-mean, i.i.d. random variables. Let $S_n^d$ be $$ S^d_n = \frac{\sum_{i=1}^n X_i^d}{\sqrt{n}} $$ Let $Y^d$ be a zero-...
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What happens to integration over a term that converges to zero in probability?

I have to do integration like this $\int h(x) [\hat{g}_n(x) - g(x)] dx$ ,where $\hat{g}_n(x)$ is a non-parametric estimator of $g(x)$ and $\hat{g}_n(x) - g(x) = o_p(1)$; $h(x)$ is an arbitrary ...
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show asymtotic normality

Let $x=(x_1,x_2,...,x_n)$ be a sample from a multivariate normal distribution, with mean vector $\mathbf{\mu}$ (n by 1 column vector, all elements equal to $\mu$) and covariance matrix $\Sigma$ (...
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Central Limit Theorem when the dimension size increases with the sample size

Let $X_1, X_2,\ldots, X_n \in \mathcal{R}^d$ and be zero-mean, unit variance random variables. Here the dimension ($d$) is a function of the sample size($n$) i.e, $d=f(n)$. For example $d = \sqrt{n}$. ...
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Definition of asymptotic variance

Upon studying the ML estimator this concept still confuses me. First define an asymptotic covariance matrix for the MLE estimator (just as an example, we have two parameters $\beta$ and $\sigma^2$, $...
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26 views

Asymptotic normality and rate of convergence of the mean

Is there any results on the rate of the convergence of the mean of a random variable being asymptotically normally distributed. For example, let $X_n$ be such that $$\sqrt{n} (X_n - \mu) \to \mathcal{...
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35 views

Rate of expected value of $\mathcal{O}_p$

This is certainly very basic but what is the rate of the expected value of a random variable that is bounded in probability. For example, let $X_n = \mathcal{O}_p (a_n)$ is it true that $\mathbb{E} [...
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Asymptotics of the estimator y'y/y'x in a linear model

I am trying to learn to understand how to derive asymptotic distributions. In an exercise, I am trying to analyze the asymptotic behaviour of the estimator $\hat{\beta} = \frac{y'y}{y'x}$, where $y = ...
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Asymptotic distribution of MLE of iid exponentials?

If $Y_1, ..., Y_n$ are iid exponentially distributed with mean $\zeta$ then the Fisher information is $n/\zeta^2$, the MLE estimator is $\sum Y_i /n$, and the variance of $Y_i$ is $\zeta^2$. By a ...
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48 views

Speed of convergence of probability

I posted this on mathstackexchange several days ago, but it may be more appropriate to ask this here. Let $r_n$ be some real sequence converging to zero. Let $X_n$ be a sequence of random variables. ...
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39 views

How to find the asymptotic variance of a UMVUE?

Suppose that $X_1, \ldots , X_n$ are iid observations from an exponential distribution with a pdf: $p(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. Suppose that my interest is in estimating the ...
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30 views

Another representation of the law of large numbers

Recently I saw the new representation of LLN like below $n^{-1}\sum X_{i}=E(X) + O_{p}(n^{-1/2}\text{Var}(X)^{-1/2})$ Does anyone verify this? In the paper, $K()$ is a kernel function
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65 views

Convergence in probability in high-dimensional settings

I am trying to prove the following. $$ (\hat{\theta} - \theta)^T c \rightarrow 0 \;\;\; \text{ (in probability)}$$ where $\theta \in R^p$, $c$ is a vector of constants such that $\sup_i (|c_i|) &...
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Asymptotic distribution of a weighted sum of chi squared variables beyond CLT?

I have a sum $$ S = \sum_{i=1}^{n} d_i X_i^2, $$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum ...
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162 views

Does the standard error of the mean approach 0 as the number of samples increases?

The standard error of the mean (SEM) is often given as $s / \sqrt n$, where $s$ is the standard deviation and $n$ the number of samples. Does this mean that if we were to calculate the SEM with more ...
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1answer
37 views

Expectation of squared sample t statistics

Given i.i.d data points $X_{1},...,X_{n}$ from unknown smooth distribution $f(x)$ with $EX=\theta$ with parameter dimension 1. Let $\bar{X}=\sum_{i=1}^{n}X_{i}/n$, and $\sigma_{n}^2=\sum_{i=1}^{n}(X_{...
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Expectation of log likelihood ratio

Given that $X_{1},...,X_{n}$ are i.i.d random variables with joint distribution $f(x\mid \theta) $ with 1 dimensional parameter $\theta$, let $\hat\theta$ be the maximum likelihood estimator of $\...
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15 views

Instantaneous drift of a stochastic process

(Posted also on math.stackexchange) Let $\mu_t$ and $\sigma_t$ be strictly positive bounded predictable processes and $W_t$ a Wiener process. Consider for $\Delta>0$ $$ X_{\Delta} = \int_0^{\...
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32 views

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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37 views

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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40 views

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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39 views

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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1answer
32 views

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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87 views

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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29 views

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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37 views

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 ...
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57 views

Is finite and strictly positive variances all that is needed for the CLT Lindeberg condition?

I cannot understand what I am doing wrong here, so please somebody, point it out from me. The issue? I keep finding that the Lindeberg sufficient condition for the Central Limit Theorem for ...
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38 views

simplification of variance formula about linear combination of order statistics

In the book APP. Math. Stat (https://books.google.com/books?id=enUouJ4EHzQC&pg=PA266&lpg=PA266&dq=serfling+variance+of+L+estimate&source=bl&ots=ehRxuMmiQ5&sig=...