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

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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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30 views

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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43 views

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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37 views

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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21 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 ...
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1answer
28 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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11 views

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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2answers
77 views

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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1answer
42 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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1answer
29 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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23 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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31 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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30 views

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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1answer
151 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
34 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 ...
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53 views

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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14 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} = ...
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1answer
27 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 $\hat{\rho}$ ...
4
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1answer
26 views

$p$-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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1answer
29 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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1answer
36 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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1answer
86 views

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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26 views

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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1answer
35 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 ...
4
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82 views

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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24 views

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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19 views

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 ...
2
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1answer
68 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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1answer
27 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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34 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 ...
4
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1answer
53 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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118 views

Chebychev’s Weak Law of Large Numbers

This theorem is on Econometric Analysis (7th edition) by Greene (2012), Page 1071. It states that "If $x_i$, $i=1,2,...,n$ is a sample of observations such that $E(x_i)=\mu_i<\infty$ and ...
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38 views

Is Chi-Square test when divided by variance valid?

The Pearson chi-square test is $$ \sum_{i=1}^k \dfrac{(X_i - \text{E}[X_i])^2}{\text{E}[X_i]}$$ A good link as to why it is divided by the expected value (rather than variance) is ...
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1answer
107 views

Law of large numbers and convergence

Suppose $X{\sim}N(\mu_1,\sigma_1^2)$ and $Y{\sim}N(\mu_2,\sigma_2^2)$, $x_1,...x_n$ and $y_1,...,y_n$ are i.i.d samples from X and Y, respectively. Consider the estimator ...
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110 views

How to formally prove two formulas are asymptotically equivalent?

I have two formulas about variance estimation of linear combination of order statistics given order statistics $X_{1}\leqslant X_{2} \leqslant ...\leqslant X_{n} $ from a random sample of arbitrary ...
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Mathematical definition of Infill Asymptotics

I am writing a paper that uses infill asymptotics and one of my reviewers has asked me to please provide a rigorous mathematical definition of what infill asymptotics is (i.e., with math symbols and ...
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how to find asymptotic joint distribution of two linear combination of order statistics?

Suppose I have n order statistics from some unknown continuous distribution funciton F(x), $X_{1}\leqslant X_{2}\leqslant...\leqslant X_{n}$. And I have two linear combination of these order ...
5
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73 views

Why do statisticians prove asymptotic normality?

In many statistics papers, authors suggest a new data analysis methodology and prove its properties such as consistency or asymptotic normality. I think it's a kind of tradition or custom. I ...
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128 views

Asymptotic normality of order statistic of heavy tailed distributions

Background: I have a sample which I want to model with a heavy tailed distribution. I have some extreme values, such that the spread of the observations are relatively large. My idea was to model this ...
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0answers
28 views

Does the asymptotic distribution of sample median follow from statistical functionals?

I know that if $F_n$ is the empirical distribution function and $F$ is the true distribution function, then, with $T$ being any statistical functional (satisfying the von-Mises derivative conditions), ...
3
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2answers
59 views

Gaussian Process Infill Asymptotics

I had asked a question recently about what happens to the predictive variance of a Gaussian process as you let $n\rightarrow\infty$ and have realized what that the name of these type of asymptotic ...
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1answer
25 views

Distribution of weighted Cox regression coefficients

I know that under typical conditions, the coefficients of a Cox model, the coefficients are asymptotically normally distributed. I plan to weight my Cox model by inverse probability of treatment ...
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149 views

Fisher information for $\rho$ in a bivariate normal distribution

I have seen many times people using the Delta method in order to find the asymptotic distribution of $r$, the sample correlation coefficient, for bivariate normal data. This distribution is given by ...
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1answer
34 views

Asymptotics of a quadratic form with growing vector / matrix dimensions

Let $\ {\bf x}_n=\big\{x_1,x_2,...,x_n\big\}$ be a vector of random variables with ${\bf m}_n=\big\{\mu_1,\mu_2,...,\mu_n\big\}$ and $\ n^{1/2}({\bf x}_n-{\bf m}_n)\rightarrow N(0,\Omega_n)$ Assuming ...
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2k views

Latin Hypercube Sampling Asymptotics

I am trying to construct a proof for a problem I am working on and one of the assumptions that I am making is that the set of points I am sampling from is dense over the entire space. Practically, I ...
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Understanding Borel-Cantelli in an “asymptotic” setting

Let $X_1(p),...,X_j(p),\dots$ be a sequence of random variables that depend on a parameter $1\le p \le \infty$. Suppose that the $n$-th variable satisfies $Pr [X_n>t] < c_n/p^n$ for some ...
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37 views

Asymptotic Distribution of Sample Quantile

My question is related to the proof below. Why can we say that $F^{-1}(Y_n(x))=X_{[np]}$? I would like to understand better the reason for that. Any help would be appreciated. I've also asked ...
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Proof of the asymptotic normality of the sample estimator $\rho_k$ and of the Bartlett formula

I was reading the two above proofs from Brockwell and Davis (1991), Time series theory and methods Theorems 7.2.1 and 7.2.2, but there's a lot of formalism that I don't get very well... Can somebody ...