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

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4
votes
1answer
105 views

Likelihood function as number of observations increases

If we have $n$ iid observations from some $X \sim p(\cdot|\theta)$, what happens to the likelihood function $p(x_1,\dots,x_n|\theta)$ as $n\rightarrow \infty$? I plotted the product of several $\...
2
votes
1answer
36 views

What, if any, asymptotic arguments are used in moving between various statements of the central limit theorem?

What, if any, asymptotic arguments are used in moving between the various statements of the central limit theorem (e.g. in terms of sample means compared with standardised sample means)? Context. My ...
0
votes
1answer
39 views

Reference for regularity conditions for asymptotic of MLE

I wonder if there is a complete list of regularity conditions for MLE asymptotic normality. I read this post and found a list of 6 conditions but the answer does not include any reference. I read the ...
2
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2answers
26 views

Clarification on $m$ and $n$ I'n the $m$-out-of-$n$ bootstrap

I've asked questions on the $m$-out-of-$n$ bootstrap here before. Responses have been quite valuable, but one key aspect still sparks some confusion from non-statisticians. Chernick (2007, 2011) ...
3
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2answers
116 views

Asymptotic distribution of $ \sqrt{n} (θ_n - θ)$ then CI of $θ$

Suppose we have $X_1,...,X_n$ iid with distribution: $f(x)=xe^{−(\frac{x^2 − θ^2}{2})},x≥θ, θ > 0$ By calculated the median of $f(x)$, $X_{\frac{1}{2}}$ equals $ \sqrt{θ^2 + \log4}$ we obtain an ...
3
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1answer
35 views

Delta Method around zero is a N(0, 0)

I have this problem: $\sqrt N \hat{\theta} \sim N(0, V)$ where $E(\hat{\theta}) = \theta_{0} = 0$. I must find the asymthotic distribution of $\frac{N}{V}\hat{\theta}^{2}$ but if I use the Delta ...
4
votes
1answer
87 views

Asymptotic distribution of OLS standard errors

Consider the linear regression model: $y_i = \beta + u_i$. We might write this as: $$Y = X\beta + U\text{ with }Y = \begin{pmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{pmatrix}, X = \begin{pmatrix}1 \\ 1 \\...
0
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0answers
12 views

Standard Error of ERGM Coefficients

I am trying to calculate the standard error of ERGM coefficients, which is estimated by MCMC sample. For an ERGM $P(y;\eta) = \exp[\eta^\top g(y) - \psi(\eta)]$, denote $\eta$ as the true parameter, $\...
3
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1answer
123 views

Deriving the limiting distribution of the Hodges-Le Cam estimator in Bickel and Doksum (2015)

I am trying to better understand the Hodges-Le Cam estimator, and am having difficulty rendering explicit some of the asymptotic arguments in the derivation of the estimator's limiting distribution. I ...
3
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1answer
100 views

Asymptotic normality of MLE

We know under regularity conditions the MLE is asymptotically normal. Usually, it is said that in practice it's hard to check these assumptions. However, I wondered whether we can claim that these ...
2
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0answers
20 views

Asymptotic Variances of MLE with Orthogonality Constraints

I have a model that is parameterized by orthogonal matrix $\boldsymbol \Gamma$ and other unconstrained parameters $\theta$ that are closely related to $\boldsymbol \Gamma$ within the model. ...
2
votes
1answer
63 views

Asymptotic chi-squared distribution of likelihood ratio statistic in regression problem

There is a famous result, going back to Wilks (1938) "The large-sample distribution of the likelihood ratio for testing composite hypotheses" (Ann. Math. Stat., 9, 60-62) that states that ...
1
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0answers
25 views

Asymptotic efficiency of estimators of autoregressive models

Are OLS or MLE estimators of autoregressive model asymptotically efficient if errors are i.i.d? Consider the case of an AR(1) model $$x_t=\alpha x_{t-1} + \epsilon_t$$ with $\epsilon_t$ ~ $i.i.d. N(0,\...
1
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1answer
25 views

Normal approximation and Hoeffding bound

Hoeffding bound for any $\epsilon>0$ is: $$P_F(|\bar{X}_n-\mu(F)|\geq \epsilon)\leq 2 \exp\{-\frac{n\epsilon^2}{2}\}=h(\sqrt{n}\epsilon)$$ wherever $|X|<1$. Now I want to have a comparison ...
5
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1answer
64 views

Convergence of uniformely distributed random variables on a sphere

I am reading "Asymptotic Statistics" by A.W van der Vaart and I am stuck with an exercise of chapter 2. Here is the question : for each $n \in \mathbb{N}$, let $U_n$ be uniformly distributed ...
1
vote
1answer
35 views

Why approximate delta-method Variance isn't multiplied by $\frac{1}{n}$?

I'm reading Casella-Berger chapter 10, where they introduce asymptotic evaluations. I don't seem quite to understand how the factor $\sqrt{n}$ works when we are using asymptotic evaluations in order ...
0
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0answers
12 views

Local Data Generating Process in Semiparametric Statistics

I am a bit confused about the LDGP assumption that is mentioned in books on semiparametric statistics. For example, in Semiparametric Theory and Missing Data by Tsiatis, the LDGP is defined as follows:...
0
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1answer
17 views

OLS Consistency for Count Data?

Why would we choose Poisson / NB regression (GLM) over OLS for fitting count data? Is there a way to show that OLS estimator would lost it consistency and asymptotic normality for count data? I'm ...
0
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0answers
36 views

relation between wald statistic and f statistic

$Y=X_i\beta_i+X_2\beta_2+\epsilon$ Y and $\epsilon$ are n*1 matrix $X_1$ is $n*k_1$ matrix $X_2$ is $n*k_2$ matrix $\beta_1$ is $k_1*1$ matrix $\beta_2$ is $k_2*1$ matrix F is f statistic for testing $...
1
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0answers
24 views

Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...
1
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0answers
34 views

If $E[|X_n|] = O(n)$ is $E[|X_n|^2] = O(n^2)$?

Let $X_n$ be a random variable that depends on $n$ and suppose $E[|X_n|] = O(n)$. Then can we say $E[|X_n|^2] = O(n^2)$? If it doesn't hold in general, are there particular interesting cases where it ...
0
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0answers
11 views

Asymptotic distribution of extremum of two objective functions with the same limit

Suppose $Q^1_T(\theta) \rightarrow Q(\theta)$ and $Q^2_T(\theta) \rightarrow Q(\theta)$ in probability uniformly and $\theta_0$ is the unique maximiser of continuous function $Q(\theta)$ over a ...
0
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0answers
40 views

How can I find $ARE_{\hat{\theta}_{n}^{(1)},\hat{\theta}_{n}^{(2)}}$ (in terms of $\tau_1$, $\tau_2$ and $\alpha$)?

Given that $n^{\alpha}[\hat{\theta}_{n}^{(1)}−\theta_0] \xrightarrow{L} \tau_1 H$ and $n^{\alpha}[\hat{\theta}_{n}^{(2)}−\theta_0] \xrightarrow{L} \tau_2 H$, find $ARE_{\hat{\theta}_{n}^{(1)},\hat{\...
0
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0answers
22 views

Theoretical properties of joint maximum likelihood estimator on returns and options when fitting an option pricing model

Suppose we have a simple GARCH option pricing model $$ R_t = \sqrt{h(t)} z(t)$$ $$ h(t) = \omega + \alpha z(t-1) + \beta h(t-1)$$ where $R_t$ is the daily log return, $h(t)$ is the conditional ...
3
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0answers
105 views

As $n \to \infty$, can we 'ignore' a matrix in an expectation that does not depend on $n$?

Let $A_n = \sum_{i=1}^n X_iX_i^T$ and $B = X_1X_1^T$ be random matrices of dimension $m\times m$. Note that the elements of $B$ are dependent on elements of $A$. Note that the vectors $X_i$ are iid ...
4
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0answers
46 views

Asymptotic recovery of sparse coefficient vector with lasso

Given $\beta^* \in \mathbb{R}^d$, suppose $X_n \sim N(\mathbf{0}, I_d)$ are iid and $Y_n = X_n\beta^* + \epsilon_n$, where $\epsilon_n \sim N(0, 1)$ and are iid. Let $\hat{\beta}_n$ be the solution to ...
1
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2answers
36 views

Approximating $E[g(\overline X_n)]$ and want to bound the remainder using some form of CLT or Berry-Essen Theorem

If we have a set $X_1,\dots,X_n$ of iid random variables with finite mean $\mu$ and variance $\sigma$, the CLT says that $\sqrt{n}(\overline X_n - \mu) \stackrel{d}{\to} \mathcal{N}(0,\sigma^2)$. If ...
2
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0answers
34 views

Let $h(t)$ be a continuous function. Show that if $X_n \xrightarrow{D} X$, then $h(X_n) = O_p(1)$

My initial thought on proving this was to use the continuous mapping theorem to conclude $h(X_n) \xrightarrow{D} h(X)$ and then use the fact that convergence in distribution implies tightness in ...
0
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0answers
10 views

Asymptotic properties of between-group estimates?

Suppose my data consists of the mean of several separate groups (sizes may vary) from an iid sample of an outcome variable Y and an independent variable X. The data generating process is $Y = \beta X +...
0
votes
1answer
33 views

If $X_n \xrightarrow{D} X$, then $X_n = O_P(1)$

I've seen this result in several places, however, I've yet to find a proof for it and I'm struggling to come up with one on my own. So far I know that I want to show that for all $\epsilon > 0$ ...
1
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0answers
25 views

Convergence of credible regions on simple Bayesian model

Consider a basic Bayesian model : $$ \begin{array}{rcl} \theta &\sim &\pi(\theta)\\ X_1, \cdots, X_n&\overset{IID}{\sim} &\mathcal{N}(\theta, I_d)\\ \end{array} $$ where $d$ is the (...
0
votes
0answers
75 views

For Y∼ Uniform (−1,1) Prove (a) $Y_{n}\xrightarrow{L}Y$

For $Y \sim$ Uniform$(-1,1)$ $ Y_{n}= \begin{cases} \text{Y if } > \vert Y \vert \leq 1-\frac{1}{n}\\ \text{n if } \vert Y \vert > >1-\frac{1}{n}\\ \end{cases} $ Prove (a) $ Y_{n}\...
1
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0answers
41 views

Asymptotic distribution of $S_n$

I am reviewing some exams from Hansen's webpage and came across this question that unfortunately doesn't have a suggested solution. In the regression/projection model $$y_i=x_i'\beta+e_i$$ $$E(x_ie_i)=...
0
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0answers
27 views

Is K-W test statistics a U statistics?

The K-W H test statistic is given by: $$ H=(N-1) \frac{\sum_{i=1}^{g} n_{i}\left(\bar{r}_{i \cdot}-\bar{r}\right)^{2}}{\sum_{i=1}^{g} \sum_{j=1}^{n_{i}}\left(r_{i j}-\bar{r}\right)^{2}}, \text { where:...
1
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0answers
25 views

Does $E[\hat \Sigma^{-1}] \to \Sigma^{-1}$ still hold for samples drawn from a non-normal population?

For a sample of observations $\{x_i\}_{i=1}^n$ where $x_i=(x_{i1},\dots,x_{ik})^T$ of a population random vector $X=(X_1,\dots,X_k)^T$, the population covariance is $$ \Sigma = E[(X-E[X])(X-E[X])^T], $...
1
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0answers
28 views

Unconditional and conditional models

I don't know if the question is worded weirdly, but I'm having difficulties understanding its logic. I have the solution, but if possible, can someone explain the reason behind it? We have two models (...
4
votes
1answer
36 views

Law of Large Numbers for geometrically decaying sequences

Let $(X_n)_n$ be a sequence of i.i.d. random variables, and let $\rho \in (0,1)$. Is there any asymptotic Theorem for the following random variable: $$ Y = \lim_{n\rightarrow \infty}\sum_{i=0}^n \rho^...
0
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0answers
14 views

Using functional delta method to obtain the asymptotic distribution of survival function

I am trying to obtain the asymptotic distribution of survival function $S(t|Z)$. Suppose $\sqrt{n}(\widehat{\Lambda}_0(t)-\Lambda_0(t))$ converges to a mean zero Gaussian process with asymptotic ...
3
votes
0answers
47 views

Joint asymptotic distribution and measures of fit

I have been looking for something similar but couldn't find any, unfortunately. Some help would be much appreciated. Consider a simple bivariate linear mean regression $$y=\beta x + e$$ where $E[e|x]=...
0
votes
0answers
35 views

Consistency and asymptotic distribution

This problem is from Hansen's econometrics textbook, chapter 7. The model is $Y = X'\beta+e$ with $E[e|X] = 0.$ An econometrician is worried about the impact of some unusually large values of the ...
1
vote
1answer
43 views

Why does the order of expectation plus the square root of order of variance equal the big $O_p$ rate of convergence in probability?

Midway through p. 154 of this paper, the authors compute the expectation and variance of a variable $\sum_{i,j=1}^nG_{i,j}^l$ and get \begin{align} E\bigg[\sum_{i,j=1}^nG_{i,j}^l\bigg] & = O(n^2h^{...
3
votes
1answer
82 views

Asymptotic order of the $L_\infty$ norm of asymptotically normally distributed random variables [closed]

Let $\mathbf{X}_n \in R^p$ be a random variable and $\mathbf{s} \in \mathcal{S} = \{x \in R^p \ s.t. \ ||x||_2 = 1\}$. Then, suppose that $p$ and $n$ are allowed to diverge and that we have $$\sqrt{n} ...
1
vote
2answers
40 views

does asymptotic normality imply convergence of expectations?

Suppose $\sqrt{n}(X_n-\mu)\to N(0,\sigma^2)$. I know this implies $X_n\to \mu$ in probability. Does it imply $E(X_n)\to \mu$? I heard about a converse to the lindebergh CLT (but couldn't find it on ...
3
votes
1answer
44 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
20 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
40 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
13 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
38 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
153 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{...
2
votes
0answers
29 views

How to use Consistency and bounded variance to imply asymptotically unbiasedness?

I saw this question here https://math.stackexchange.com/questions/239146/consistency-and-asymptotically-unbiasedness Let $Y_1,Y_2,\dots$ be a consistent sequence of estimators for a random variable $X$...

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