Questions tagged [asymptotics]

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

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What are the degress of freedom in the summary output for GLMs in R?

I am currently self-studying GLMs with the book "Generalized Additive Models An Introduction with R" and I am a bit confused regarding the degrees of freedom in the summary output for GLMs ...
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Small sample MLE vs OLS efficiency

MLE estimates are asymptotically efficient. Both MLE and OLS estimates are asymptotically normal and for many distributions their limiting variances coincide (information for one observation being the ...
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What can we say about the variance of the posterior mean?

In Bayesian inference, there's one famous theorem, Bernstein–von Mises theorem (see the Wikipedia or this lecture notes, page 35), states that in front of sufficiently large samples, that is ...
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Construct transformations of random variables that are "more normal"

I am reading this page in the Encyclopedia of Mathematics about transformations of random variables. I am puzzled about the Example 2: Let $X_1,...,X_n,...$ be independent random variables, each ...
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Is OLS asymptotically the best estimator even without gaussian error?

It is known that MLE is consistent and asymptotically efficient. OLS under certain assumptions is asymptotically normal. If the errors are gaussian, then OLS is equivalent to MLE. If the errors are ...
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What is the meaning of $\asymp$ and $\lesssim$ in Martin wainwright's high dim textbook? [closed]

Unfortunately, this text book did not provide a table of notations he used. Can anyone provide me with a definition of $\asymp$ and $\lesssim$ and few examples? For an example in the book, in display (...
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Asymptotic unbiasedness + asymptotic zero variance = consistency?

Here, Ben shows that an unbiased estimator $\hat\theta$ of a parameter $\theta$ that has an asymptotic variance of zero converges in probability to $\theta$. That is, $\hat\theta$ is a consistent ...
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Computing the limiting distribution of the Bayes estimator for exponential data with a Gamma prior (by using consistency?)

Let data be $X_i \sim \text{Exp}(\theta)$ iid, $i=1,...,n$. Let the prior be $\text{Gamma}(\alpha, \beta)$. The posterior is then of course $\text{Gamma}(\alpha + n, \beta + \sum X_i)$. The Bayes ...
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Asymptotic Distribution and Describe Sources of Increasing Power in an hypothesis testing problem

I am currently dealing with the following problem in a past exam (with no solution): Suppose $S$ follows the Poisson distribution with mean $2\lambda>0$, here $\lambda$ is a parameter. Another two ...
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Asymptotics of $\mathbb E[-\log(p)]$ in a one-sample t-test as $n\to\infty.$

Consider a one-sample two-sided t-test, i.e. $X_1, \ldots, X_n$ are iid. $N(\mu, \sigma)$ random variables and we want to test $H_0\colon \mu=0$ versus $H_A\colon \mu\neq0$. The $t$-statistic is ...
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References: convergence rates of kernel regression, exchangeable data

I have been studying Kernel estimation; in particular, the Nadaraya-Watson estimator. I am interested in studying the rate of convergence in L^p of the NW (or similar) estimators for subgaussian ...
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Prove that the Deviance and the Generalised Pearson Statistic are asymptotically equivalent

I am reading the paper Exponential Dispersion Models from Jørgesen and at page $137$ I have encountered a claim that I don't know how to prove. The author claims that the Generalised Pearson Statistic,...
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How to know when to use non-parametric coefficient confidence interval estimates for regression?

Say I have either logistic regression or simple linear regression and I am not sure if I have a moderate number of observations, $n = 40$. How do I know when to switch to using a non-parametric ...
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How do we know the distribution of regression coefficients

I'm reading up on asymptotics and hypothesis testing and was thinking about how they link together with regression coefficients. I have read that the CLT shows that the standardised sample mean ...
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Intuition of Influence Function and Score function: $E[IF(X)S_{\beta}(X; \theta_0)]$

Question I find a theorem regarding influence function and score function \begin{align*} E\left\{IF(Z) S_\beta\left(Z, \theta_0\right)\right\}&=1\\ E\left\{IF(Z) S_\eta^T\left(Z, \theta_0\right)\...
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For an i.i.d. observations $X_1,\cdots,X_n$ (bounded), we have the Hoeffding's inequality that establishes the upper bound for the tail probability of $|\bar{X_n}-\mathbb{E}[X_1]|$. I would like to ...
Suppose that $X_{ik}\sim\mathcal N(0,\sigma^2)$ for $k = 1,2,\dots, n_i$ are independent and identically distributed for each $i \in\{ 1,2\}$. Note that I assume equal means ($0$) and variances (\$\...