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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Unbiased and consistent estimator with positive sampling variance as n approaches infinity? (Aronow & Miller) [duplicate]
In Aronow & Miller, "Foundations of Agnostic Statistics", the authors write on p105:
[A]lthough unbiased estimators are not necessarily consistent, any unbiased estimator $\widehat{\...
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Normal approximation for posterior distribution
I am reading the example 4.3.3 of "The Bayesian Choice" by Christian P. Robert and I was wondering if it is possible to obtain a normal approximation in this case to estimate the posterior. ...
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Unit-Root Asymptotics
I am using the book "Time-series-based econometrics" by Hatanaka to learn about asymptotic theory of unit roots. However, it is quite technical, so I am also using Hamilton's "Time ...
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Sum of asymptotically independent random variables - Convergence
Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$.
If $X_i\sim Ber(\pi_i)$, I ...
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Almost sure convergence using exponential tail bound
I have a question about a theorem in the following set of lecture notes 'A Gentle Introduction to Empirical Process theory' (http://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf). In ...
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In linear regression, what's the asymptotic distribution of the error variance estimator?
Suppose $$Y_i=X_i'\beta+\epsilon_i$$ with $E(\epsilon_i|X_i)=0$ and $E\epsilon^2_i=\sigma^2$ and I estimate $\sigma^2$ using $s^2=\frac{1}{n}\sum_{i=1}^n (Y_i-X_i'\widehat{\beta})^2$, where $\widehat{...
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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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Hypothesis testing by asymptotic distribution
Consider the following hypothesis testing problem:
under $H_0$: $(X_1,\cdots,X_n) \sim P_n,$
under $H_1$: $(X_1,\cdots,X_n) \sim Q_n.$
We want to show that the minimum testing error goes to zero when $...
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