Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [asymptotics]

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

Asymptotic theory is concerned with the properties of estimators and test statistics in large samples which are assumed to tend towards infinity in size. This allows to obtain complicated estimators and tests which would not be available in small samples. Note that asymptotic theory is only an approximation with small samples, and it is not always a good approximation. Examples of asymptotic properties of estimators are consistency, regularity or their asymptotic distribution. Frequently used concepts in asymptotic theory include the weak and strong law of large numbers, the central limit theorem, certain classes of expansions (e.g. Taylor, Edgeworth, von Mises) and the Delta method.

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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 distribution for the standard deviation. Let a…
Alecos Papadopoulos
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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 essential for the application and the application…
chris
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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 number?). Any explanation (or a link to suggested…
Tal Galili
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Is there a result that provides the bootstrap is valid if and only if the statistic is smooth?

Throughout we assume our statistic $\theta(\cdot)$ is a function of some data $X_1, \ldots X_n$ which is drawn from the distribution function $F$; the empirical distribution function of our sample is $\hat{F}$. So $\theta(F)$ is the statistic viewed…
orizon
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Why doesn't Wilks' 1938 proof work for misspecified models?

In the famous 1938 paper ("The large-sample distribution of the likelihood ratio for testing composite hypotheses", Annals of Mathematical Statistics, 9:60-62), Samuel Wilks derived the asymptotic distribution of $2 \times LLR$ (log likelihood…
ratsalad
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Intuitive understanding of the difference between consistent and asymptotically unbiased

I am trying to to get an intuitive understanding and feel for the difference and practical difference between the term consistent and asymptotically unbiased. I know their mathematical/statistical definitions, but I'm looking for something…
StatsStudent
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Cauchy Distribution and Central Limit Theorem

In order for the CLT to hold we need the distribution we wish to approximate to have mean $\mu$ and finite variance $\sigma^2$. Would it be true to say that for the case of the Cauchy distribution, the mean and the variance of which, are undefined,…
JohnK
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Why $\sqrt{n}$ in the definition of asymptotic normality?

A sequence of estimators $U_n$ for a parameter $\theta$ is asymptotically normal if $\sqrt{n}(U_n - \theta) \to N(0,v)$. (source) We then call $v$ the asymptotic variance of $U_n$. If this variance is equal to the Cramer-Rao bound, we say the…
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When the Central Limit Theorem and the Law of Large Numbers disagree

This is essentially a replication of a question I found over at math.se, which didn't get the answers I hoped for. Let $\{ X_i \}_{i \in \mathbb{N}}$ be a sequence of independent, identically distributed random variables, with $\mathbb{E}[X_i] = 1$…
Alecos Papadopoulos
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Asymptotic consistency with non-zero asymptotic variance - what does it represent?

The issue has come up before, but I want to ask a specific question that will attempt to elicit an answer that will clarify (and classify) it: In "Poor Man's Asymptotics", one keeps a clear distinction between (a) a sequence of random variables…
Alecos Papadopoulos
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What are the regularity conditions for Likelihood Ratio test

Could anyone please tell me what the regularity conditions are for the asymptotic distribution of Likelihood Ratio test? Everywhere I look, it is written 'Under the regularity conditions' or 'under the probabilistic regularities'. What are the…
Kingstat
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Can the empirical Hessian of an M-estimator be indefinite?

Jeffrey Wooldridge in his Econometric Analysis of Cross Section and Panel Data (page 357) says that the empirical Hessian "is not guaranteed to be positive definite, or even positive semidefinite, for the particular sample we are working…
Jyotirmoy Bhattacharya
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Is bootstrap problematic in small samples?

In "3 Things That Bother Me" (1988), Ed Leamer writes: Bootstrap estimates of standard errors are based on the assumption that the observed sample is the same as the true distribution, which is OK asymptotically. But a sample of size $n$ implies a…
Richard Hardy
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Observed information matrix is a consistent estimator of the expected information matrix?

I am trying to prove that the observed information matrix evaluated at the weakly consistent maximum likelihood estimator (MLE), is a weakly consistent estimator of the expected information matrix. This is a widely quoted result but nobody gives a…
dandar
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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 it is known, is given by $$ P( | \bar X - \mu| >…
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