Questions tagged [asymptotics]
Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.
634
questions
3
votes
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 $\...
0
votes
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
votes
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 ...
0
votes
0answers
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,\...
1
vote
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
$$P(\lim_{n \to \infty} |X_n| &...
6
votes
2answers
144 views
If $X_n - \mu = O_p(a_n)$ does that imply that $X_n^{-1} - \mu^{-1} = O_p(a_n)$?
If a random variable $X_n$ converges in probability to a constant $\mu$, we know by the rules for probability limits that its inverse converges to the inverse of the constant, i.e. $X_n^{-1} \stackrel{...
1
vote
1answer
16 views
Gauss-Markov and Asymptotic Properties
Is it true that Gauss-Markov assumptions (i.e. linearity, full rank, strict exogeneity, and $\sigma^2 I$) can imply "consistency" and "asymptotic normality" of the OLS estimator?
...
0
votes
0answers
7 views
$\limsup$ in proof that $X_n = o_p(Y_n)$ and $Y_n = O_p(1)$ then $X_n = o_p(1)$
In the proof where we have
$$
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)$ ...
2
votes
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} - ...
0
votes
0answers
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 ...
-1
votes
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
vote
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
vote
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
votes
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
votes
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$$
$$...
0
votes
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 ...
2
votes
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
$E(Y_{2i}|Z_{i})=\mu_{2}+\beta Z_{i}$ ...
0
votes
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 ...
0
votes
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 ...
1
vote
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) \...
1
vote
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?
0
votes
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 ...
0
votes
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, ...
1
vote
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 ...
1
vote
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 $...
0
votes
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
votes
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
votes
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
votes
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
vote
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
$$
\begin{...
2
votes
1answer
80 views
Rate of convergence of $\hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1}$ to the probability limit?
Consider the simple linear regression model.
$$
y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n.
$$
Let $\mu_x$ and $\sigma_x^2$ represent the mean and variance of ...
4
votes
1answer
64 views
Is $\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ uniformly integrable (UI)? What assumptions make it UI?
$\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$
Let $\mathbf{X}_n$ be the usual data matrix in standard multiple regression where I have used the subscript $n$ to indicate the number ...
1
vote
0answers
61 views
OLS estimator is consistent if the smallest eigenvalue of $X^TX$ goes to infinity as $n\to\infty$
I want to show that if $\lambda_{min}(X^T X)$ (i.e., the smallest eigenvalue of $X^TX$) goes to infinity as $n\to\infty$, then $\hat{\beta}$ is a consistent estimator of $\beta$.
My approach is the ...
0
votes
2answers
51 views
Does a vast amount of probability and statistical literature make a mistake when they make use of CLT/asymptotic normality?
Suppose we toss a fair coin $N$ times and we are interested in the probability that we get at least $cN$ heads for $c\in [0,1]$. We can model this situation by letting $S_N = \sum_{i=1}^N X_i$ where $...
1
vote
1answer
117 views
Validity of approximating a covariance matrix by making use of a probability limit?
I want to know can we approximate the covariance matrix of a random vector by making use of a probability limit.
Define the linear regression model in matrix form as
$$
\mathbf{Y} = \mathbf{X} \beta + ...
3
votes
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}...
0
votes
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
votes
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
vote
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
votes
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 ...
0
votes
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
vote
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 ...
0
votes
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
$$
Y_n \...
0
votes
0answers
26 views
If $X_n-X=o_p(N^{-\alpha})$, $f(\cdot)$ is smooth, do we have $f(X_n)-f(X)=o_p(N^{-\alpha})$?
If $X_n-X=o_p(N^{-\alpha})$ with $\alpha>0$, $f(\cdot)$ is smooth, do we have $f(X_n)-f(X)=o_p(N^{-\alpha})$?
I guess this is true as $f(X_n)-f(X)=f'(X)(X_n-X)+o_p((X_n-X))$, which has the same ...
0
votes
1answer
25 views
The nonparametric estimation in generalized regression model
Let $Y_t \in \mathbb{R}$ be a response variable and $X_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X_1, Y_1), \cdots, (X_n, Y_n)$.
\begin{equation}
Y_{t} = \mu(...
1
vote
1answer
36 views
If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$?
If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$? According to Slutsky theorem, I can directly get $X_nY_n\overset{d}{\...
4
votes
0answers
62 views
Why is the observed Fisher information defined as the Hessian of the log-likelihood?
In an MLE setting with probability density function $f(X, \theta)$, the (expected) Fisher information is usually defined as the covariance matrix of the fisher score, i.e.
$$
I(\theta) = E_\theta \...
1
vote
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
vote
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 ...