# 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 sample variance of non-normal sample

This is a more general treatment of the issue posed by this question. After deriving the asymptotic distribution of the sample variance, we can apply the Delta method to arrive at the corresponding ...
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### Why does the continuity correction (say, the normal approximation to the binomial distribution) work?

I wish to better understand how the continuity correction to the binomial distribution for the normal approximation was derived. What method was used to decide we should add 1/2 (why not another ...
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### Is there a statistical application that requires strong consistency?

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is ...
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### Approximation error of confidence interval for the mean when $n \geq 30$

Let $\{X_i\}_{i=1}^n$ be a family of i.i.d. random variables taking values in $[0,1]$, having a mean $\mu$ and variance $\sigma^2$. A simple confidence interval for the mean, using $\sigma$ whenever ...
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### Large sample asymptotic/theory - Why to care about?

I hope that this question does not get marked "as too general" and hope a discussion gets started that benefits all. In statistics, we spend a lot of time learning large sample theories. We are ...
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### Does a Bayesian interpretation exist for REML?

Is a Bayesian interpretation of REML available? To my intuition, REML bears a strong likeness to so-called empirical Bayes estimation procedures, and I wonder if some kind of asymptotic equivalence (...
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### Derivation of normalizing transform for GLMs

$\newcommand{\E}{\mathbb{E}}$How is the $A(\cdot) = \displaystyle\int\frac{du}{V^{1/3}(\mu)}$ normalizing transform for the exponential family derived? More specifically: I tried to follow the ...
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### Why is the asymptotic relative efficiency of the Wilcoxon test $3/\pi$ compared to Student's t-test for normally distributed data?

It is well-known that the asymptotic relative efficiency (ARE) of the Wilcoxon signed rank test is $\frac{3}{\pi} \approx 0.955$ compared to Student's t-test, if the data are drawn from a normally ...
780 views

### Do third order asymptotics exist?

Most asymptotic results in statistics prove that as $n \rightarrow \infty$ an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood ...
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### Product and sum of big $O_p$ random variables

I seems that people often use the following properties $O_p(a_n)O_p(b_n) = O_p(a_nb_n)$ and $O_p(a_n)+O_p(b_n) = O_p(a_n+b_n)$. I'm wondering, if these are true for any sequences $a_n,b_n$. The reason ...
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### What is the asymptotic covariance matrix?

Is it true that the asymptotic covariance matrix is equal to the covariance matrix of parameter estimates? If not, what is it? And what is the difference between the covariance matrix and the ...
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### Is MLE estimation asymptotically normal & efficient even if the model is not true?

Premise: this may be a stupid question. I only know the statements about MLE asymptotic properties, but I never studied the proofs. If I did, maybe I woulnd't be asking these questions, or I maybe I ...
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### Hessian of profile likelihood used for standard error estimation

This question is motivated by this one. I looked up two sources and this is what I found. A. van der Vaart, Assymptotic Statistics: It is rarely possible to compute a profile likelihood ...
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### How many of the biggest terms in $\sum_{i=1}^N |X_i|$ add up to half the total?

Consider $\sum_{i=1}^N |X_i|$ where $X_1, \ldots, X_N$ are i.i.d. and the CLT holds. How many of the biggest terms add up to half the total sum ? For example, 10 + 9 + 8 $\approx$ (10 + 9 + 8 $\dots$ +...
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### Can MCMC iterations after burn in be used for density estimation?

After burn-in, can we directly use the MCMC iterations for density estimation, such as by plotting a histogram, or kernel density estimation? My concern is that the MCMC iterations are not ...
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### Density of robots doing random walk in an infinite random geometric graph

Consider an infinite random geometric graph in which the node locations follow a Poisson point process with density $\rho$ and edges are placed between the nodes that are closer than $d$. Therefore, ...
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### A continuous function of a sequence of random vectors converges in probability to the function of the limit

Proposition: If $\{ X_n \}$ is a sequence of k-dimensional random vectors s.t. $X_n \overset{p}{\to} X$ and if $g: R^k \rightarrow R^m$ is a continuous mapping, then $g(X_n) \overset{p}{\to} g(X)$. ...
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### Conjecture related to Kolmogorov 0-1 Law (for events)

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. There ...
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### Does the definition of regular estimator depend on the rate of convergence? If not, should it?

The definition of regular estimator in my lecture notes is: Let $X_1^{(n)}, \dots, X_n^{(n)} \overset{iid}{\sim} P_n \sim \mathcal{P}(\Theta)$ where $\mathcal{P}(\Theta)$ is a regular parametric ...
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### Practical usefulness of pointwise convergence without uniform convergence

Motivation In the context of post-model-selection inference, Leeb & Pötscher (2005) write: Although it has long been known that uniformity (at least locally) w.r.t. the parameters is an important ...
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### Example of CLT when moments do not exist

Consider $X_n = \begin{cases} 1 & \text{w.p. } (1 - 2^{-n})/2\\ -1 & \text{w.p. } (1 - 2^{-n})/2\\ 2^k & \text{w.p. } 2^{-k} \text{ for } k > n\\ \end{cases}$ I need to show that ...
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### Asymptotic normality of order statistic of heavy tailed distributions

Background: I have a sample which I want to model with a heavy tailed distribution. I have some extreme values, such that the spread of the observations are relatively large. My idea was to model this ...
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### Asymptotic distribution of censored samples from $\exp(\lambda)$

Let $X_{(1)}, \ldots, X_{(n)}$ be the order statistic of an i.i.d. sample of size $n$ from $\exp(\lambda)$. Suppose the data is censored so we see only the top $(1-p) \times 100%$ percent of the data, ...