Questions tagged [asymptotics]

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

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Show the ergodicity of a random sum of ergodic processes

We say that a mean stationary stochastic process $(X_t)_{t \in \mathbb N}$ - i.e. $E[X_t]= \mu_X$ for all $t$ - is ergodic mean if \begin{equation}\tag{I} \frac 1 T \sum_{t=1}^T X_t \overset {pr} \...
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But what if the 2-th absolute moments converge in probability?

I'm trying to understand a kind of convergence. I had posted another question, but I think it got too polluted and I decided to delete it and simplify it a bit. We know that $X_n \to X$ in mean square ...
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What actually is the definition of asymptotic normality for an estimator? Some inconsistencies

In most books, the result that the MLE is asymptotically normal is given, and that is used as the definition of asymptotically normal, with no mention of what the actual definition is for a general ...
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Interpretation of likelihood ratio test for MANOVA model using an asymptotic distribution

I'm studying multivariate linear models and I wanna test a hypothesis on the form \begin{equation} \begin{gathered} H_0: \textbf{CB = 0} \\ \text{vs.} \\ H_A: \textbf{CB $\neq$ 0}, \end{gathered} ...
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Independence/ Asymptotic independence of asymptotic normal random variables

Let $\{X_{n}\}_{n\in I}$ be a sequence of random variables, where $X_{n}$ takes value $\{-c_n,c_n\}$, each with probability $1/2$, $|c_{n}|\leq \alpha \in \mathbb{R}$ and $I$ denotes the index set ...
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When do we find convergence in distribution to independent variables?

Let $\{X_{i}\}_{i=1}^{n}$ and $\{Y_{i}\}_{i=1}^{n}$ are two sequences of random variables such that $\bar{X} = \sum_{i=1}^{n} X_{i}$ and $\bar{Y} = \sum_{i=1}^{n} Y_{i}$ asymptotically converges in ...
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Is asymptotic unbiasedness different from unbiasedness in practice?

Given some estimator T for a parameter θ, by definition T is unbiased if its bias B(T) is 0. It is asymptotically unbiased if B(T) is not 0, but some value that tends to 0 as n goes to infinity. My ...
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Asymptotic behaviour of product of normal r.v.s

Let $X \sim N(\mu ,1)$ and $Y \sim N(\mu, 1)$ where we have $\mu >0.$ I'm trying to evaluate asymptotically the tail distribution function of product of these two random variables. Let $x>0$, ...
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7 votes
2 answers
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How to prove unbiasedness/consistency/normality of an estimator that doesn't have a closed form?

My estimator looks like this: $$ \hat\theta(X) = \arg\max_{\theta} \frac1N \sum_{n=1}^N f(x_n|\theta) $$ Here, $f(x_n|\theta)$ is some arbitrary function: it's not a logarithm, and the sum is not a ...
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Asymptotic variance of identically distributed but non-independent random variables

I have a question in computing the asymptotic variance of a sequence of random variables that is identically distributed but are not independent. Suppose we have a sequence of i.i.d. random variables $...
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From univariate to joint convergence in distribution

Let $X_n \rightarrow_d X$ and $Y_n \rightarrow_d Y$ where $X$ and $Y$ are i.i.d standard exponential random variables. However, I do not have that for any $n$, $X_n$ and $Y_n$ are independent. Can I ...
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1 vote
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Central limit theorem for asymptotically i.i.d. random variables

I observe a sequence of r.v. $X_1, X_2, \dots$ where each $X_i$ is a function of the sample size $n$. When $n \rightarrow \infty$ I have the following result: $X_1 \rightarrow^d E_1, X_2 \rightarrow^d ...
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2 votes
1 answer
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How to show linear combination of independent, but non-identically distributed Bernoulli's is asymptotically normal?

Summary I am curious about whether there exists theoretical justification to say a linear combination of a sufficiently large number of independent (but not identically distributed) Bernoulli random ...
2 votes
1 answer
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Do residuals in linear regression converge to $\mathcal{N}(0, \sigma^2 I)$?

As argued in this answer, we can compute the residuals in the standard linear regression: \begin{align} y &= X\hat{\beta} + e,\\ e &= y - X(X^\top X)^{-1}X^\top y = (I-H)y, \end{align} where $...
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1 answer
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Does there always exist for n small, a non-chi-squared test-statistic for the likelihood-ratio (neyman-pearson, karlin-rubin), score, and wald-tests?

An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT).[6] LRTs have several desirable ...
18 votes
3 answers
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What's the point of asymptotics?

I presume this is a rather stupid question, but I hope some of you can find a bit of time to entertain it. Looking at asymptotic behavior of estimators/test statistics etc means looking at their ...
1 vote
1 answer
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A question about the delta method in asymptotic distributions

I am reading up on the delta method from its Wikipedia page. Under the heading Univariate delta method the statement of the method is as follows: If $$\sqrt{n}[X_n - \theta]\xrightarrow{\text{D}} \...
2 votes
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An asymptotic statistics calculation

I have a question about an apparently routine asymptotic calculation. Suppose that we have variables $X \in \mathbb{R}^{p}, Z \in \mathbb{R}^{q}, Y \in \mathbb{R}$. Write $E[Y \mid X] = f(X; \beta_0)$ ...
2 votes
1 answer
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Comparing efficiency between estimators

Suppose that $\hat \theta_1, \hat \theta_2$ are two estimators of $\theta$. Furthermore, assume that \begin{align} \sqrt{n}(\hat \theta_1-\theta)\overset{d}{\to}N(0,V_1)\\ \sqrt{n}(\hat \theta_2-\...
2 votes
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Asymptotic distribution of minimizer of sum of squared pairwise differences

Suppose I have iid random variables $X_1,\dots,X_n$ and some parametric function of $\theta,$ $f_{\theta},$ which is linear or nonlinear in $\theta$ (I am more interested in the nonlinear case, but if ...
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9 votes
1 answer
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What are "poor finite sample properties"?

In MacKinlay’s (1997) well-known article about event study methodologies, he states that there are two ways event clustering can be accommodated in event studies. Event clustering means that event ...
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2 votes
1 answer
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Asymptotic normality of OLS estimators in practice

Apologies in advance because I notice quite a few questions that have similar titles but I didn't see one that answered my specific curiosity: Supposedly OLS coefficient estimators are asymptotically ...
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1 vote
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asymptotic distribution of correlation coefficient in bivariate normal distribution [duplicate]

Suppose \begin{align} \begin{pmatrix} X_i\\ Y_i \end{pmatrix} \sim_{iid} N \begin{pmatrix} \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \begin{pmatrix} \sigma^2_1 & \rho \sigma_1 \sigma_2 \\ \...
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2 votes
2 answers
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What does asymptotic efficiency mean in statistic?

I reads some comparison articles, and always find " asymptotic efficiency", "asympototically less efficient", and "asympotoically normal". I really confused about the ...
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Design points of local polynomial regression

Let the random set $\{(Y_{t},X_t)\}_{t=1}^n$ follow the model: $$Y_t=m(X_t)+\epsilon_t,\quad t\in\{1,\cdots,n\}\quad (1)$$ where $\epsilon_t$ is a random error term and $m(\cdot)$ is an unknown smooth ...
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Schemes for increasing the joint coverage probability of confidence intervals with sample size

I'm estimating a fixed number of parameters and I want to have joint confidence intervals for them. However, for reasons that would take most of a paper to describe, I'm looking for a scheme that ...
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Asymptotic variance of relative risk

I'm interested in the asymptotic variance of relative risk (without sample size). I know that in an application, the variance is $$ \frac{1-p_1}{n_1p_1} + \frac{1-p_0}{n_0p_0} $$ When removing $n$ ...
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2 votes
0 answers
37 views

Asymptotic distribution of t-test with two coefficients

I have the following CEF: $$ y_{i}=\beta_{1} d_{i}+\beta_{2}\left(1-d_{i}\right)+e_{i} $$ And the following assumptions: $$ d_{i} \perp\left(y_{i}(0), y_{i}(1)\right) $$ $$ \mathrm{E}\left[e_{i} \mid ...
3 votes
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Do asymptotic statistics "solve" the Behrens-Fisher problem?

The Behrens-Fisher problem concerns comparing two means from independent (maybe multivariate) samples in a way robust to heteroskedasticity in the populations being compared. It seems that if one ...
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Asymptotic Distribution of Likelihood Ratio under Nonlinear Hypothesis

Suppose we are testing $\mathbf h(\boldsymbol\theta) = \mathbf 0$ versus $\mathbf h(\boldsymbol\theta) \neq \mathbf 0$ for a vector of parameters $\boldsymbol\theta \in \boldsymbol\Theta\subset \...
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1 answer
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Is there a statistic such that for large sample sizes $a_n (\hat{\theta} - \theta) \sim N(0, \Sigma)$ approximately but $a_n \neq n^{1/2}$?

Various central limit theorems are of the form $a_n(\hat{\theta}-\theta)\sim N(0, \Sigma)$ approximately as $n \to \infty$ and usually $a_n = n^{1/2}$. Are there central limit theorems for statistics ...
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2 votes
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Issue with bounded in probability

I have tried to prove the following problem that I read in the lecture and it seems not transparent to me. Suppose that $Y_{i}$ be independent random variables (with $i=1,2,3, \dotsc$). Each has the ...
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Find fisher information matrix for optimization estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$ and we have ...
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2 votes
2 answers
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What does $\hat{\theta}_n = \theta + O_p(n^{-1/2})$ mean in terms of rates?

For $\hat{\theta}_n = \theta + O_p(n^{-1/2})$ we have $$\hat{\theta}_n - \theta = O_p(n^{-1/2})$$ Therefore, we have for any $\epsilon > 0$, there exists a finite $M > 0$ and finite $N > 0$ ...
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Correcting for systematic bias in test size

Let $X_1,\dots,X_n$ be iid random variables with mean $0$ and variance $\sigma^2$, and let $$\xi:=\frac{1}{\sqrt{n}}\sum_{i=1}^n (X^2_i-\sigma^2).$$ Then, by the standard CLT, we have $\xi\Rightarrow ...
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exponential limit of experiments

I'm trying to understand the theory of "limits of statistical experiments" as explicated in Chapter 9 of Van Der Vaart's text, "Asymptotic Statistics". For some models, the ...
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5 votes
2 answers
642 views

Expectation of the ratio of sum (XY) and sum(X)

I want to know (mathematically) how the following expression changes as $M$ increases but still have no clue after thinking about it for a while. Any suggestions or comments will be much appreciated. $...
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The exact distribution of the conditional distribution of the OLS estimator

This is the problem that I have tried figuring it out for a while, and I still need some advice because there is no explicit derivation in the textbook that I have seen so far. The problem looks easy ...
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0 answers
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Use MLE to construct a 90% asymptotic confidence interval for $\alpha\beta$

Given $X_1,...,X_n$ are i.i.d random vectors from $p_{\alpha,\beta}$, $\alpha,\beta\in(0,\infty)$ the Fisher information matrix is $I(\alpha,\beta)=\begin{pmatrix} \beta/\alpha&0\\0&\alpha^2\\\...
5 votes
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Nuisance parameters and $o_p(n^{-1/4})$ convergence: citation

I'm looking for an original reference to a proof idea. Suppose we have $n$ iid observations $(X_i,Y_i)$ and an estimating function $$\bar U(\beta;\alpha)=\frac{1}{n}U(\beta;\alpha; X_i,Y_i)$$ where we ...
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1 answer
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Asymptotic normality with weighted sum of objective function $\min_{x} \; f_n(x) + g_n(x)$

Suppose $f_n(x)$, $g_n(x)$ are convex functions w.r.t. $x$ the optimal point of the two problems $\min_x f_n(x)$ and $\min_x g_n(x)$ have asymptotic normality as $n \rightarrow \infty$ they converge ...
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a good estimator in 2-stages least-squares

I am now studying the 2-stages least-squares method and have been curious about the following circumstances. Suppose that I have $Y_i = X^{T}_{i}β +e_{i}$ with $\mathbb{E}(e_{i}X_{i}) ̸\ne 0$, that ...
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Using CLT to get Confidence Interval for Mean in MA(1)

Self-study: Let $$X_t = \mu + a_t + \theta a_{t-1}$$ $a_t$ is white noise with mean 0 and variance $\sigma^2$. Given $\bar x = c$ for a sample size of 100. Find confidence interval for $\mu$. My ...
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8 votes
4 answers
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How would a bayesian estimate a mean from a large sample?

What would a bayesian do if she wanted to do inference for the mean with a large sample but has no idea of the underlying distributions? A frequentist statitician would use the sample mean as a point ...
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Determine the asymptotic distribution $\sqrt{n}(\hat{\theta}_n-\theta) $ for trinomial distribution over the group sizes $(x,y,z)$

A random sample of $n$ individuals are classified into three groups, with probabilities $\theta^2$, $2\theta(1-\theta)$, and $(1-\theta)^2$ respectively, yielding the trinomial distribution over the ...
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1 vote
1 answer
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Asymptotic distribution of a linear combination

If $X_1$ and $X_2$ are independent and follow asymptotic standard normal distribution as $\min(n_1,n_2)\to\infty$, how do I show that $\frac{\sqrt {n_1}X_1+\sqrt {n_2}X_2}{\sqrt{n_1+n_2}}$ also has an ...
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1 vote
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What's the asymptotic variance for OLS estimates of intercept and slope of homoskedastic simple linear regression?

Suppose data is generated by $Y_i=\beta_0+\beta_1X_i+U_i$ satisfying $E(U_i|X_i)=0$ and $E(U_i^2|X_i)=\sigma^2$. Suppose I have a random sample $\{Y_i,X_i\}_{i=1}^{n}$, and obtained OLS estimates $\...
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1 answer
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What is the big $O_p$ of the product between a $O_p(a_n)$ term and a uniformly bounded function?

Suppose $\frac{1}{n}\sum_{i=1}^n \hat{\theta}_i^2 = O_p(a_n)$ and $||f(X)||_{\infty}$ is bounded. What is the big $O_p$ of $\frac{1}{n}\sum_{i=1}^n (\hat{\theta}_i f(X_i))^2$? The way I understand ...
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1 vote
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Multivariable asymptotic regression model

Is there a way to extend following univariable asymptotic regression model to include additional variables? $$ Y = Asym + (R0 - Asym)* e^{(-lrc * T)} $$ $Asym$ = maximum attainable value of $Y$ $R0$ =...
0 votes
1 answer
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Suppose $\hat{\theta}_1 = O_p(n^{-1/2})$ and $\hat{\theta}_2 = O_p(n^{-1/2})$, what is $\sqrt{\hat{\theta}_1\hat{\theta}_2}$?

Suppose $\hat{\theta}_1 = O_p(n^{-1/2})$ and $\hat{\theta}_2 = O_p(n^{-1/2})$, what is the big $O_p$ for $\sqrt{\hat{\theta}_1\hat{\theta}_2}$? I think $\hat{\theta}_1\hat{\theta}_2 = O_p(n^{-1/2})O_p(...
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