# 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 ...
45 views

### 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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### 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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### 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 ...
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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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### 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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### 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 $Y$ $R0$ =...
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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(...