Questions tagged [asymptotics]

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

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Asymptotic standard errors vs exact standard errors

I am getting confused about the derivation of standard errors for the OLS estimator $\widehat{\beta}$. I have seen two different ways to derive standard errors: (i) from the exact covariance matrix of ...
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Asymptotic normality implies consistency

I'm trying without success to solve the following exercise in my econometric textbook: Show that $\sqrt{N}\left(\widehat{\beta_1} - \beta_1 \right) \xrightarrow{d} \mathcal{N}(0,a^2)$, where $a^2$ is ...
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Local Linearity vs Regularity Conditions for the asymptotic distribution of the Likelihood Ratio

In his book 'Asymptotic Statistics,' Aad van der Vaart when discussing the asymptotic distribution of the log-likelihood-ratio says: "The most important conclusion of this chapter is that, under ...
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OLS vs WLS in a Heteroskedastic Trending Regression

Suppose we have the following heteroskedastic trending regression model: $$y_i=bi+a_i u_i$$ for some sequence of non-zero constants $a_i$ and $u_i$ an i.i.d. sequence of mean 0 and variance 1 random ...
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Asymptotic Normality for GEE Parameters

In the famous Liang and Zeger 1986 paper on GEEs https://www.jstor.org/stable/2336267?seq=9, they sketch a proof using the standard m-estimator arguments: (unstated) regularity conditions + first-...
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Is this a typo on P.75, Theorem 5.52 of the book "Asymptotic Statistics" by Van der Vaart?

Let $\Theta$ be a compact metric space, $\theta \in \Theta.$ Let $m_{\theta}:\mathbb{R}^d\to \mathbb{R}: x\mapsto m_{\theta}(x)$ be a family of measurable function indexed by $\theta \in \Theta.$ Let $...
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Asymptotics of $\mathbb E[-\log(p)]$ in a one-sample t-test as $n\to\infty.$

Consider a one-sample two-sided t-test, i.e. $X_1, \ldots, X_n$ are iid. $N(\mu, \sigma)$ random variables and we want to test $H_0\colon \mu=0$ versus $H_A\colon \mu\neq0$. The $t$-statistic is ...
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References: convergence rates of kernel regression, exchangeable data

I have been studying Kernel estimation; in particular, the Nadaraya-Watson estimator. I am interested in studying the rate of convergence in L^p of the NW (or similar) estimators for subgaussian ...
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Prove that the Deviance and the Generalised Pearson Statistic are asymptotically equivalent

I am reading the paper Exponential Dispersion Models from Jørgesen and at page $137$ I have encountered a claim that I don't know how to prove. The author claims that the Generalised Pearson Statistic,...
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Finding the limiting distribution of $\sqrt{n} (\hat{\tau} - \tau)$ as $n \rightarrow \infty$ for $N(\mu, \mu^2 \tau)$

Let $X_i$ for $i = 1, ..., n$ be a random sample from the distribution $N(\mu, \mu^2 \tau)$ with unknown parameters $\mu \in (\infty, 0) \cup (0 ,\infty), \tau > 0$. Find and justify the mle $\hat{\...
Stats_Rock's user avatar
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Why is the asymptotic bias of the maximum likelihood estimate $b(\theta) = \frac{b_1(\theta)}{n}+\frac{b_2(\theta)}{n^2}+...$?

Firth (1993) states in his introduction that for a $p$-dimensional parameter $\theta$ the asymptotic bias of the maximum likelihood estimate $\hat{\theta}$ may be written as: $b(\theta) = \frac{b_1(\...
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Conditions for existence of KL divergence and unique minimum

Consider two probability density function g(y) and f(y: $\theta$), $\theta \in \Theta$. The KL divergence of f and g is defined by $$ D_{KL}(g|f) := \int \log \frac{g(y)}{f(y: \theta)} \, dy = \...
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Asymptotic distribution of $n^{\frac{1}{2}}(\hat{\gamma},\gamma_0)$

I am struggling to understand a proof from Browne M. (1984) Asymptotically distribution-free methods for the analysis of covariance structures. Given $\boldsymbol{\delta_s}=n^{\frac{1}{2}}(\boldsymbol{...
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Standard Error of Truncated Fisher's z Transform

I have a question regarding data following the truncated Fisher's z-transform. I currently understand that the correlation coefficient data, after the unbounded Fisher's z-transform, has a standard ...
Thea Ng's user avatar
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When stochastic boundedness $O_p(1)$ does not hold

The formal definition of stochastic boundedness $O_p(1)$ of a sequence of random variables $\{X_n\}$ goes $$\{X_n\} = O_p(1) \implies \forall \varepsilon >0, \quad \exists\, N_{\varepsilon}, \...
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Statistical significance tests for neural networks

Horel & Giesecke 2020 developed a statistical signficance test for feature variables in a single-layer feedforward neural network. Namely, fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})...
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Estimators that are superefficient on a dense set

In Chapter 8 of van der Vaart's Asymptotic Statistics, it is shown that (under weak regularity conditions) an estimator can be "superefficient" on at most a set of Lebesgue measure zero (...
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Is convergence in probability implied by consistency of an estimator?

Every definition of consistency I see mentions something convergence in probability-like in its explanation. From Wikipedia's definition of consistent estimators: having the property that as the ...
Estimate the estimators's user avatar
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How can we maintain asymptotic normality with slight change?

If $(X_n-\mu_n)/\sigma_n\rightarrow_{d} N(0,1)$ (i.e., $X_n$ is $AN(\mu_n,\sigma_n^2)$), I want to show the following two statements: (1) $X_n$ is $AN(\bar{\mu}_n, \bar{\sigma}_n^2)$ if and only if $\...
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How to know when to use non-parametric coefficient confidence interval estimates for regression?

Say I have either logistic regression or simple linear regression and I am not sure if I have a moderate number of observations, $n = 40$. How do I know when to switch to using a non-parametric ...
Estimate the estimators's user avatar
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How do we know the distribution of regression coefficients

I'm reading up on asymptotics and hypothesis testing and was thinking about how they link together with regression coefficients. I have read that the CLT shows that the standardised sample mean ...
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Intuition of Influence Function and Score function: $E[IF(X)S_{\beta}(X; \theta_0)]$

Question I find a theorem regarding influence function and score function \begin{align*} E\left\{IF(Z) S_\beta\left(Z, \theta_0\right)\right\}&=1\\ E\left\{IF(Z) S_\eta^T\left(Z, \theta_0\right)\...
mayu's user avatar
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What is the asymptotic bound for the ratio of sample mean and expectation?

For an i.i.d. observations $X_1,\cdots,X_n$ (bounded), we have the Hoeffding's inequality that establishes the upper bound for the tail probability of $|\bar{X_n}-\mathbb{E}[X_1]|$. I would like to ...
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Consistency of the pooled standard deviation estimate

Suppose that $X_{ik}\sim\mathcal N(0,\sigma^2)$ for $k = 1,2,\dots, n_i$ are independent and identically distributed for each $i \in\{ 1,2\}$. Note that I assume equal means ($0$) and variances ($\...
Syd Amerikaner's user avatar
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Exact Likelihood ratio statistic for discrete distribution

Suppose that the random variables in a sample $Y_1, Y_2, \ldots, Y_n$ are iid with values in $[0,1]$, and that an investigator knows that the underlying probability density $f_Y(y)$ has the form $f_Y(...
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How to prove an asymptotical converge estimations to the MLE estimations?

To estimate the parameters of copulas, both the classical maximum likelihood method (MLE) and alternative methods can be used, for example, the natural sampler and the efficient importance sampler. ...
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Compare beta coefficients from two different Poisson models

I would like to compare two beta coefficients from two different Poisson models which have the same variables and applied to different samples. I would like to test their difference to check if the ...
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Could the numerical value of mean square error (or root mean square error) tell us something about the rate of convergence?

Suppose I have an estimator $\widehat{\theta}$ for $\theta_0$ that is root-n consistent and asymptotically normal. In the monte carlo simulations of many papers that I've read, consistency is usually ...
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Asymptotic distribution of $n^r \frac{U_{(1)}}{U_{(n)}}$: figuring possible $r>0$

Consider the i.i.d. sample $U_1, U_2 \cdots, U_n$ from the uniform distribution $U(0, 1)$. I should find a possible values of $r>0$ to have an asymptotic distribution of $$ n^r \frac{U_{(1)}}{U_{(n)...
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How can $V(k)-V(r)=O_p(C_{NT}^{-2})$ imply $V(k)/V(r)=1 + O_p(C_{NT}^{-2})$, where $C_{NT}=\min \{ \sqrt{N} , \sqrt{T} \}$?

I am reading Bai and Ng (2002). In their proof of Corollary 1, they said “Next, consider k > r. Lemma 4 implies that $V(k)/V(r)=1 + O_p(C_{NT}^{-2})$”. However, Lemma 4 just states that $V(k)-V(r)=...
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What does the likelihood function converge to when sample size is infinite?

Let $\mathcal{L}(\theta\mid x_1,\ldots,x_n)$ be the likelihood function of parameters $\theta$ given i.i.d. samples $x_i$ with $i=1,\ldots,n$. I know that under some regularity conditions the $\theta$ ...
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proof of asymptotic distribution of autocorrelation [closed]

1.How to prove the asymptotic distribution formula? 2.How to derive Ljung Box statistic test?
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Inference on asymptotically biased estimators

For simple estimators (e.g. sample mean, OLS), we typically establish asymptotic normality by appealing to some type of Central Limit Theorem $$ \sqrt{n}(\hat{\beta} - \mathbb{E}[\hat{\beta}]) \to^d \...
Adam's user avatar
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Were missing values in data augmentation procedure parameters?

Suppose I have a data set with $(Y,X)$ where $Y$ has some missing entries(say 20% missing). From $p(Y,X|\lambda)$ and prior $p(\lambda)$, I can conduct MCMC by data augmentation to impute missing $Y$'...
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Confusion about asymptotic distribution of the MLE and of the MAP

It's well known that the MLE $\hat{\theta}$ maximizes $f(y\mid\theta)$ and under regularity conditions has asymptotic distribution $$N\left(\theta, \frac{I(\theta)}{J^2(\theta)} \right)$$ where $I(\...
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Compare an $O_p(1)$ with some growing numbers

There is a known positive sequence $M_n>0$ (e.g., $M_n = \log(n)$), where $M_n\to\infty$ as $n \to \infty$. If a sequence $X_n$ is $O_p(1)$, then can I claim $\Pr(|X_n| > M_n ) \to 0$ as $n \to \...
Jim's user avatar
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Asymptotic normality in central limit theorem

I am a bit confused by Classical CLT section of the central limit theorem on Wikipedia. It basically says at the sample size gets larger, the difference between the sample mean and true mean ...
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Results on the variance of the MAP estimate of the random effects in a GLMM?

As part of a student project, I'm working on explaining this paper (link https://www.jstor.org/stable/43305585 ) "A simple test for random effects in regression models". For context, it ...
ThighCrush's user avatar
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Does asymptotic normality of the likelihood function follow from the central limit theorem?

In the book Mathematical Methods for Physics and Engineering it is said that the likelihood function tends to a Gaussian (centred on the maximum-likelihood estimate) in the large sample limit. The way ...
Ghorbalchov's user avatar
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Is there an optimality result for the two-sample Wilcoxon-Mann-Whitney test?

Is there any mathematical result that states that the Wilcoxon-Mann-Whitney (WMW) test is optimal in some sense, for a specific testing problem that is a subproblem of the general problem the WMW test ...
Christian Hennig's user avatar
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1 answer
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Proving uniform convergence of moment restriction score function in GMM asymptotic normality proof

I am asked in a homework question to prove asymptotic normality for the generalized method of moments estimator. The assumptions (which i think are necessary to solve this particular subproblem) given ...
Jeppe Pilgaard's user avatar
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If $n=2$ and $N$ gets large, show that the MLE $\hat{\sigma}^2$ converges in probability but is not consistent

Suppose that for $i=1,2, \ldots, N$ and $j=1, \ldots, n$, the r.v.'s $Y_{i j} \sim \mathcal{N}\left(\vartheta_i, \sigma^2\right)$ are mutually independent, where the parameters $\{\vartheta_i\}_{i=1}^...
Stats_Rock's user avatar
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What is the non-asymptotic counterpart of the mutual information?

The mutual information of a joint probability distribution $p(x,y)$ tells us, if we send $n$-letter messages with each letter drawn from the marginal distribution $p_X(x)$, that we can use roughly $2^{...
glS's user avatar
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Asymptotic behavior of $\prod_{i=1}^s (1-\alpha x^2_i)^2$ for Gaussian $x_i$

Suppose $x_i$ comes from standard Normal. For a given $\alpha$, I'm interested the following random variable: $$f(s)=\log \prod_{i=1}^s (1-\alpha x^2_i)^2$$ For $\alpha=2.422$, empirical distribution ...
Yaroslav Bulatov's user avatar
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If sample average converges a.s. in an iid sample, must it converge to the mean?

SLLN tells us that if $X_1,...,X_n$ are iid, with $X_1$ having finite mean $\mu$, then their sample average converges almost surely to $\mu$. Suppose instead we know that $X_1,...,X_n$ are iid and ...
Golden_Ratio's user avatar
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1 answer
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If sample average converges in an iid sample, must it converge to the mean?

WLLN tells us that if $X_1,...,X_n$ are iid, with $X_1$ having finite mean $\mu$, then their sample average converges in probability to $\mu$. Suppose instead we know that $X_1,...,X_n$ are iid and ...
Golden_Ratio's user avatar
1 vote
1 answer
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How to find asymptotically normal estimator if I know probability density function [closed]

I have $X_1, X_2,\ldots,X_n$ be a random sample of size n from a distribution with probability density function: $$p(x) = \theta^2xe^{-\theta x}I (x > 0).$$ How can I find an asymptotically normal ...
Karlos Margaritos's user avatar
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Asymptotic normality of penalized MLE?

Let $L(\theta;X)$ denote the log-likelihood of a model and I maximize the following to estimate $\theta$, $$ \arg\max_{\theta}L(\theta;X)+\lambda\theta^2 $$ If $\lambda$ is 0, then the asymptotic ...
cookiemonster's user avatar
5 votes
1 answer
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Why does the "infeasible" Waugh-Frisch-Lovell estimator agree with the usual one?

In Lemma 1 of these lecture notes, Chernozhukov and Fernández-Val write that partialing out with the Frisch-Waugh-Lovell theorem has an "adaptivity" property. Namely, suppose we regress $Y$ ...
Colin Aitken's user avatar
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Asymptotic distribution of $\bar{X}_n^k$

Suppose $(X_n)$ are iid with mean $\mu$ and variance $\sigma^2$. Then by CLT $\sqrt{n}(\bar{X}_n-\mu) \overset{D}{\rightarrow} N(0,\sigma^2)$ and use delta method we can get the asymptotic ...
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