Questions tagged [asymptotics]

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

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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 ...
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Negative multinomial distribution

The pmf of negative multinomial distribution is \begin{align*} P(\boldsymbol{\rm{X}}=\boldsymbol{\rm{x}})=\frac{\Gamma\left(x_0+\sum_{i=1}^{m}x_{i}\right)}{\Gamma\left(x_0\right)}p_0^{x_0} \prod_{i=1}...
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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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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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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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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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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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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\\\...
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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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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
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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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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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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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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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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 $Y$ $R0$ =...
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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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Suppose $E[( \theta-\hat{\theta}_n)^2] = O(n^{-1/2})$. Show that $\frac{1}{n}\sum_{i=1}^n (\theta-\hat{\theta}_n)^2 = O_p(n^{-1/2}).$

Assume that $E[( \theta-\hat{\theta}_n)^2] = O(n^{-1/2})$. How can I show that $\frac{1}{n}\sum_{i=1}^n (\theta-\hat{\theta}_n)^2 = O_p(n^{-1/2})?$ What I'm trying to ask is: if the expected value of ...
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1 vote
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Assume that $X_n\sim Beta(n,n)$, how to drive the limiting distribution of $ 2\sqrt{2n}(X_n-\frac{1}{2})\to^d N(0,1) $

Assume that $X_n\sim Beta(n,n)$. Use the Delta method to show that $$ 2\sqrt{2n}(X_n-\frac{1}{2})\to^d N(0,1) $$ Since $X_n=\frac{Y_n}{Y_n+Z_n}$ where $Y_n$ and $Z_n$ are indepedndent Gamma$(n,1)$ ...
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Are there any theoretical guarantees about the log-likelihood's inverse Hessian when the observations are not i.i.d.?

Let $X=[X_1...X_n]$ be some random variables in which $X_i$ are not independent. For example, you may envision the observations came from some stochastic process. Let $\ell(\theta; X)$ be the log-...
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A question about the test statistic for testing the difference in two population proportions

For independent samples $X_1,\cdots,X_n $from $\textit{Bernoulli }(p_1)$ and $Y_1,\cdots,Y_m$ from $\textit{Bernoulli }(p_2)$,where $n$ and $m$ are large. Then,the Central Limit Theorem tell us $\frac{...
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1 answer
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Derive the variance of the standardized normal of sample mean

I have solved the following problem and felt a little bit uncertain about the my answer. Here is the problem. Let $Y_i \in L_{2}$ for $i=1,2,...,N$ be a scalar random variable with iid with $\mu_Y=E(...
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0 answers
51 views

Why should we care about DAGs for causal inference? [duplicate]

I am not acquainted with Pearl's approach for causal inference. From what I have seen so far, the causality is inferred from directed acyclic graphs(DAGs). Rubin's Causal Inference Sec 7.5 proved a ...
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How do we select model for causal inference?

I am reading Rubin's Causal Inference Sec 7.5 in context of completely randomized experiment. It says performing linear regression will produce asymptotically unbiased estimate of causal effect, ...
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How to interpret the asymptotic normality of Gaussian Kernel estimator

I am trying to estimate the return of AAPL with Gaussian kernel estimator code for reproducing: ...
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1 answer
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For $r>s\geq1$, convergence in $s^{\text{th}}$ mean does not imply convergence in $r^{\text{th}}$ mean

I need a counterexample for the problem: if $r>s\geq1$, convergence in $s^{\text{th}}$ mean does not imply convergence in $r^{\text{th}}$ mean. The definition for convergence in mean is as follows: ...
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Consistency for the estimator in a mixture of objective function

Current we have two discrepancy functions $f_1(x_1,x_2,y_1,y_2)$ and $f_2(x_1,y_1)$. $f_1$ reaches minimum when $x_1=y_1$, $x_2=y_2$; $f_2$ reaches minimum when $x_1=y_1$. We consider an objective ...
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2 votes
2 answers
85 views

Asymptotic MLE Distribution With Two Random Samples

I'm studiyng for an exam, and I found this problem which I can not managed to solve... I will be really grateful if someone can help me, thanks you. Let $\left\{X_{1}, \ldots, X_{n}\right\} \sim^{...
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3 votes
1 answer
69 views

Asymptotic value of an integral related to distances in a unit n-ball

In trying to find out the pdf of the range $T$ of euclidean distances of $m$ randomly and uniformly chosen points from the origin in an $n$-dimensional unit ball, I have obtained the following : $$...
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3 votes
2 answers
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Does the Bernstein von Mises theorem require that the true posterior is actually Gaussian?

Reading up on the Bernstein von Mises Theorem, it says that in the infinite data limit, the posterior converges to a Multivariate Gaussian. Just a sanity check which I cannot find anywhere...This is ...
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Is an asymptotic approximation to the standard error asymptotically equal to the standard error?

Suppose that $(\hat{\theta}_n -\theta)/b_n \stackrel{d}{\to} N(0,1)$. Does this imply $b_n/se(\hat{\theta}_n){\to}1 $? $b_n$ here is any sequence, but the example I have in mind is where $b_n$ is $1/\...
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Observing tails in an i.I.d. data sample

Is there a result that says that in an I.I.d. data sample one shouldn’t observe tails “too soon”? I am trying to prove consistency of a certain MLE and want to exclude scenarios of the form: In $\...
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Probability of correctly identifying the Bayes class

Consider $X$ being a random variable taking value $\{1, \ldots, K\}$, with probability $p_1 = \frac{1}{K} + \varepsilon$ and $p_k = \frac{1}{K} - \frac{\varepsilon}{K-1}$ for all $k \neq1$. Taking i.i....
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2 votes
0 answers
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Large sample properties of classical estimator for single scale parameter

This question was first posted on Math Stackexchange and I was told in the comment it would be a good question on Stats Stackexchange, since it comes from the well-established theory of point ...
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Delta method for $\bar X$^2

I have a question about the delta method. The question is: $X_1,...,X_n \sim N(\mu,\sigma^2)$, where $\sigma^2=V(x)$ and $\mu=E(X)$ let T=$\bar X^2$ be an estimate for $\mu^2$. Find the asymptotic ...
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2 votes
0 answers
34 views

Does $o_p(f_n)O_p(f_n) = o_p(f_n)$

I know $o_p(1)O_p(1) = o_p(1)$ because assuming $X_n = o_p(1)$ and $Y_n = O_p(1)$ $$ \begin{align}P(|X_nY_N|< \epsilon) &= P(|X_nY_n|<\epsilon\cap|Y_n| < M_\epsilon) + P(|X_nY_n|<\...
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1 vote
0 answers
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Method of Sieves with Data Driven Basis Functions

Consider a nonparametric regression problem with i.i.d. sampled data $(y_1,x_1), (y_2,x_2),\ldots, (y_n,x_n)$ and regression function $$y_i = g_0(x_i) + \varepsilon_i,\quad \mathbb E[\varepsilon_i | ...
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  • 1,461
0 votes
0 answers
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Positive definiteness of integral of matrix

I was reading a paper, and did not understand a statement that the author made without further explanation. The author derives the limiting distribution of a non-linear least-squares estimator and ...
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0 votes
1 answer
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Question regarding asymptotic assumptions and hypotesis testing paradox for large samples

Suppose we would like to verify if a r.v $X$ follows a distribution with cumulative density as $F$, if $n$ goes to $\infty$ I'm able to use komogorov test which states reject $H_0$ (stating that $X$ ...
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99 views

MLE of log-normal distribution with one parameter

Suppose we have a sample of random variables $X_1, \cdots, X_n$ that are from a log-normal distribution, where $\log X_i \sim N(\mu,\mu^2)$ (same variance and mean). Find the MLE $\tilde{\mu_n}$ of $\...
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1 vote
0 answers
90 views

is inverse of 1 + small op(1) equal to 1 + small op(1)?

One rule for small op and big Op is $$ (1+o_p(1))^{-1} = O_p(1) $$ (on page 13 of Vaart, A. W. van der. (1998). Asymptotic statistics. Cambridge University Press.) I am curious whether it is true to ...
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1 vote
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Bias of MLEs increases rather than decreases in n

In the context of conducting simulations to assess the performance of MLE point estimates for truncated data, I am encountering surprising settings in which the bias of MLEs is clearly non-monotonic ...
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  • 3,491
8 votes
2 answers
380 views

MLE and non-normality

What is a non-trivial example of an identifiable model whose MLE is consistent, but the MLE's asymptotic distribution is not normal? Parametric setting and IID sample would be desirable.
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