Questions tagged [asymptotics]

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

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3
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1answer
32 views

Central limit theorem for the function of an iid random variable

Given an iid random variable $X$, instead of the distribution $\sqrt{n}(n^{-1}\sum{X_{i}}-E[X])$ which is the result that the central limit theorem provides , I am interested in the distribution of $\...
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0answers
18 views

large-n ols simple linear regression assumption “no perfect linear collinearity”

Suppose I have a model: $$ y = \beta_{0} + \beta_{1}X + \epsilon $$ where X is a binary dummy, either 0 or 1. Suppose all other conditions are satisfied (linearity, $y_{i}$ $x_{i}$ iid, ...
2
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1answer
27 views

Mean and variance of the Gaussian resulting from Central Limit Theorem

Let $\{x_i\}$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $\frac{1}{n}\sum_{i=1}^n x_i$ is asymptotically normal. What do we know about the mean and ...
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12 views

Asymptotics of the maximum of k subsample means

Suppose we have $n$ i.i.d samples $X_1, X_2, \cdots, X_n$ from some real-valued distribution $P$. Let $\alpha \in [0,1]$ be a fixed constant. Select an uniformly random subset $S_1 \subset \{1,2,\...
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1answer
32 views

Does $X_n = O_P(a_n)$ and $a_n \to 0$ imply $X_n \stackrel{a.s.}{\to} 0$?

My attempt at this is: $$X_n = O_P(a_n) \implies P(|X_n| > C a_n) < \epsilon$$ for some $0 < C < \infty$ Then taking the limit inside the probability, we get $$P(\lim_{n \to \infty} |X_n| &...
6
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2answers
144 views

If $X_n - \mu = O_p(a_n)$ does that imply that $X_n^{-1} - \mu^{-1} = O_p(a_n)$?

If a random variable $X_n$ converges in probability to a constant $\mu$, we know by the rules for probability limits that its inverse converges to the inverse of the constant, i.e. $X_n^{-1} \stackrel{...
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1answer
16 views

Gauss-Markov and Asymptotic Properties

Is it true that Gauss-Markov assumptions (i.e. linearity, full rank, strict exogeneity, and $\sigma^2 I$) can imply "consistency" and "asymptotic normality" of the OLS estimator? ...
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7 views

$\limsup$ in proof that $X_n = o_p(Y_n)$ and $Y_n = O_p(1)$ then $X_n = o_p(1)$

In the proof where we have $$ P(|X_n| \geq \varepsilon) \leq P\left(|\frac{X_n}{Y_n}| \geq \frac{\varepsilon}{B}\right) + P(|Y_n| > B) $$ why do we need to take the $\limsup$ to show $X_n = o_p(1)$ ...
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1answer
37 views

Asymptotic distribution of $\sum X_{i}^2$

We have $X_{1},X_{2},...,X_{n}$ as the independent standard normal random variables. Let us define: $T_{n} = \sum X_{i}^2$ then what will be the asymptotic distribution of $\sqrt{n}(\frac{T_{n}}{n} - ...
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20 views

Convergence of sum of (1) random variable that converges in distribution to Normal and (2) degenerate random variable that diverges to infinity?

Say that we have $\sqrt{n}(\hat{\mu} - \mu_0)$, which we can equivalently write as $\sqrt{n}(\hat{\mu} - \mu) + \sqrt{n}(\mu - \mu_0)$, where $\mu$ is the population mean, $\hat{\mu}$ is the sample ...
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1answer
38 views

Is a p-value of < 2.2e-16 in r the same as a p-value that is asymptotically 0? [duplicate]

I am getting a p-value of < 2.2e-16 for my coefficient in R, but I was wondering if I can say in my write up that it is asymptotically zero? Do these have the same meaning? What would I need to ...
1
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1answer
25 views

$O_p$ and orders in probability

How can we relate the notions of the order of some term with his expectation and variance? I was reading a paper which aims to find the order of some random sequence $X_n$ and it says (it doesn't ...
1
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1answer
38 views

Failing to obtain $\chi^2(1)$ asymptotic distribution under $H_0$ in a likelihood ratio test: example 2

I have a large sample (a vector) $\mathbf{x}$ from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null ...
2
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1answer
35 views

Failing to obtain $\chi^2(1)$ asymptotic distribution under $H_0$ in a likelihood ratio test: example 1

I have a large sample (a vector) $\mathbf{x}$ from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null ...
1
vote
1answer
14 views

Asymptotic null distribution of the LR statistic with point null and point alternative

I have a large sample (a vector) $\mathbf{x}$ from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null ...
2
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1answer
19 views

Wilks' theorem when dimension of submodel is not well defined

Suppose $\{f(\cdot,\theta) : \theta \in \mathbb{R}^p\}$ is a statistical model satisfying the conditions for Wilks' theorem, and that we have a hypothesis test of the form: $$H_0: \theta_p >0$$ $$...
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0answers
27 views

Limiting distribution of iterative applications of Bayes' rule

The question Suppose we iteratively use the posterior as the prior on the same data.* What is the limiting distribution of the posterior? Let's make that precise. The data $X$ and the likelihood ...
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0answers
25 views

Slutsky's theorem applied to a sample mean conditional on a Bernoulli variable?

Let $(Z_{i},Y_{2i})$, $i=1,2,\ldots,N$ be iid random vectors, where $Y_{2i}$ is the outcome vector and $Z_{i}\sim\operatorname{Bernoulli}(\delta)$. Assume that $E(Y_{2i}|Z_{i})=\mu_{2}+\beta Z_{i}$ ...
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0answers
31 views

Are stationary markov chains iid random variables?

Let $\{X_t\}_{t=1}^{\infty}$ be a Markov Chain. An initial marginal distribution $\pi^T$ for a markov chain is a stationary distribution if $\pi^TP = \pi^T$. My understanding of this is that if the ...
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0answers
14 views

Why does consistency of cluster robust standard error depend on the number of clusters?

I've seen many hand wavy explanations about it, but when I read White's book for the original reference, the math is too dense. Could someone help me derive this result in terms of the asymptotic ...
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1answer
20 views

Using asymptotic confidence intervals in practice

I would like to know the correct way of using the following result in practice: Let $G_n$ be some function of $n$ i.i.d. samples $X_1,\dots,X_n$, and say we have that $$ \sqrt{n} (G_n - \theta) \...
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1answer
41 views

Can a Markov Chain have a limiting distribution and more than 1 stationary distribution?

Can a Discrete-Time Markov Chain have a limiting distribution and more than 1 stationary distribution?
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1answer
21 views

Is Bias Affected By Dataset Size?

I am trying to understand the concept of asymptotic unbiasedness. I understand that an estimator is said to be asymptotically unbiased if, when the size of our data increases to infinity, the bias of ...
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0answers
9 views

Question about consistent estimators and asymptotic distributions

Lets say you have an estimator that is consistent, and you do not have any information on the asymptotic distribution, what can you do with such an estimator? Also when using aysmptotic distribtuions, ...
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1answer
64 views

Confused about conditions of the weak and strong laws of large numbers

I am a little confused by what conditions need to hold for the weak law of large numbers (WLLN) and the strong law of large numbers (SLLN) to be true. It seems different sources give me different ...
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0answers
69 views

What's the asymptotic distribution of $\exp(X_n)$, if $X_n$ is a sequence of asymptotically normally distributed random variables?

Let $(X_n)_{n\in\mathbb{N}} $ be a sequence of asymptotically normally distributed random variables, such that $\lim\limits_{n\to\infty}\sqrt{n}X_n\sim N(0,1)$. What's the asymptotic distribution of $...
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0answers
11 views

consistency of weight matrix

Take the model $Y = X'\beta + e$ with $\mathbb{E} [Ze] = 0$. Let $\tilde{e}_i = Y_i - X'_i \tilde{\beta}$ where $\tilde{\beta}$ is consistent for $\beta$ (e.g. a GMM estimator with some weight matrix)....
3
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0answers
36 views

What conditions are needed for $a_n = O_p(n^d) \implies E[a_n] = O(n^d)$?

Let $X_n$ be a uniformly integrable sequence of random variables. In a recent question I asked about the possibility of converting Big $O_p$ convergence in probability of the sequence $X_n$ to Big $O$ ...
6
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1answer
119 views

Can we go from $X_n = \mu + O_p(n^{-1})$ to $E[X_n] = \mu + O(n^{-1})$?

Let $X_n$ be a uniformly integrable (UI) sequence of random variables. If we have $$ X_n = \mu + O_p(n^{-1}), $$ then for $0 \le \delta < 1$ this implies $$ X_n = \mu + o_p(n^{-\delta}) \quad \quad ...
2
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0answers
31 views

Sampling/Asymptotic Distribution of Estimated Coefficients of Logistic Regression

If I understand correctly, in a logistic regression, we have that $Y_i \mid X \sim Bern(S(X\beta))$ where $S(x)$ is the sigmoid function. Suppose we estimate $\beta$ using MLE and get $\hat \beta$. ...
1
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1answer
50 views

Is $\frac{\mu_X + O_p(n^{-1})}{\mu_Y + O_p(n^{-1})} = \frac{\mu_X}{\mu_Y} + O_p(n^{-1})$?

Let $X_n = \mu_X + O_p(n^{-1})$ and let $Y_n = \mu_Y + O_p(n^{-1})$ where $\mu_X$ and $\mu_Y$ are constants. Then is it possible to write $Z_n = X_n/Y_n$ in a simplified form? E.g. I have $$ \begin{...
2
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1answer
80 views

Rate of convergence of $\hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1}$ to the probability limit?

Consider the simple linear regression model. $$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n. $$ Let $\mu_x$ and $\sigma_x^2$ represent the mean and variance of ...
4
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1answer
64 views

Is $\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ uniformly integrable (UI)? What assumptions make it UI?

$\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ Let $\mathbf{X}_n$ be the usual data matrix in standard multiple regression where I have used the subscript $n$ to indicate the number ...
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0answers
61 views

OLS estimator is consistent if the smallest eigenvalue of $X^TX$ goes to infinity as $n\to\infty$

I want to show that if $\lambda_{min}(X^T X)$ (i.e., the smallest eigenvalue of $X^TX$) goes to infinity as $n\to\infty$, then $\hat{\beta}$ is a consistent estimator of $\beta$. My approach is the ...
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2answers
51 views

Does a vast amount of probability and statistical literature make a mistake when they make use of CLT/asymptotic normality?

Suppose we toss a fair coin $N$ times and we are interested in the probability that we get at least $cN$ heads for $c\in [0,1]$. We can model this situation by letting $S_N = \sum_{i=1}^N X_i$ where $...
1
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1answer
117 views

Validity of approximating a covariance matrix by making use of a probability limit?

I want to know can we approximate the covariance matrix of a random vector by making use of a probability limit. Define the linear regression model in matrix form as $$ \mathbf{Y} = \mathbf{X} \beta + ...
3
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2answers
45 views

What is difference between $\hat{X}_n \overset{p}{\to} \bar{x}$ and $(\hat{X}_n - \bar{x}) = o_p(1)$?

Let $\{\hat{X}_n\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, which I take to mean that, for any $\epsilon > 0$, $\lim \limits_{n \to \infty} \Pr(|\hat{X}...
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0answers
39 views

asymptotic normality of z-estimator

I'm working on Problem 5.4.1 in Bickel and Docksum's Mathematical Statistics Let $X_1, \dots, X_n$ be i.i.d. random variables distributed according to $P\in\mathcal{P}$. Suppose $\psi:\mathbb{R}\to\...
2
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2answers
48 views

Concentration inequalities for estimated least squares regression coefficients?

I would like to know what is the best concentration inequality we can use for the estimated least squares regression coefficients. Let $\hat \beta_0, \hat \beta_1$ be the estimated regression ...
1
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0answers
16 views

verifying Asymptotic Distributions using simulation methods [closed]

I've created 10 thousand simulated time series with sample size $T = 200$, simulated with a given autoregressive parameter ($\theta_0$ = 0.3) and for each I've estimated the autoregressive parameter ...
0
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0answers
23 views

Finding the limiting distribution of $\sqrt{n}(\hat{\tau}-\tau)$ where $\hat{\tau}$ is the difference in means?

In a similar manner to this problem: Asymptotic distribution of $\sqrt{n}\left(\hat{\sigma_{1}^{2}}-\sigma^2\right)$ I'm a little confused as to how the solution to this problem would change if: $\tau ...
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0answers
25 views

Consistency and asymptotic unbiasedness?

I understand the differences between the two concepts, but they look similar so I was searching for some theorems which tie them. I found that a sufficient condition for an estimator $T_n$ to be ...
1
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0answers
11 views

The asymptotic properties of $V$-statistic for mixing multivariate process

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel ...
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0answers
41 views

If a random variable $Y$ converges in distribution, can we use the parameters of the asymptotic distribution as if they are the parameters of $Y$?

Let $Y_n$ be a sequence of random variable such that $$ \sqrt{n}(Y_n-\mu) \stackrel{d}{\to} \mathcal{N}(0, \sigma^2), $$ and thus we can say $Y_n$ is asymptotically normally distributed as $$ Y_n \...
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0answers
26 views

If $X_n-X=o_p(N^{-\alpha})$, $f(\cdot)$ is smooth, do we have $f(X_n)-f(X)=o_p(N^{-\alpha})$?

If $X_n-X=o_p(N^{-\alpha})$ with $\alpha>0$, $f(\cdot)$ is smooth, do we have $f(X_n)-f(X)=o_p(N^{-\alpha})$? I guess this is true as $f(X_n)-f(X)=f'(X)(X_n-X)+o_p((X_n-X))$, which has the same ...
0
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1answer
25 views

The nonparametric estimation in generalized regression model

Let $Y_t \in \mathbb{R}$ be a response variable and $X_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X_1, Y_1), \cdots, (X_n, Y_n)$. \begin{equation} Y_{t} = \mu(...
1
vote
1answer
36 views

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$?

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$? According to Slutsky theorem, I can directly get $X_nY_n\overset{d}{\...
4
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0answers
62 views

Why is the observed Fisher information defined as the Hessian of the log-likelihood?

In an MLE setting with probability density function $f(X, \theta)$, the (expected) Fisher information is usually defined as the covariance matrix of the fisher score, i.e. $$ I(\theta) = E_\theta \...
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0answers
15 views

In kernel regression, what are the common theoretical motivations for using a kernel that is Lipschitz continuous?

I read a few papers on Nadaraya-Watson kernel regression in which I saw assumptions that require the kernel function being Lipschitz continuous without explanation ( and without citation of such ...
1
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1answer
47 views

What will U converge to?

Hi all, I have several queries below: Would the X-bar and Y-bar be fixed in value? Would the denominators become exceedingly large as n increases? Would the numerator become exceedingly large as ...

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