Questions tagged [asymptotics]

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

458 questions
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Hausman test statistic - to multiply or not to multiply by n

I am having some serious doubts regarding the formula of the Hausman statistic for the case in which I compare OLS and IV estimates. I am getting confused with what my references are giving me. What ...
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Limiting Distribution of $n\left[Y_n\right]$ where $Y_n$ is the minimum of a sample of size n from Uniform$\left(0,\theta\right)$ distribution

Suppose $X_1,X_2,\dots,X_n$ is a random sample from Uniform$(0,\theta)$ for some unknown $\theta > 0$. Let $Y_n$ be the minimum of $X_1,X_2,\dots,X_n$. (a) Suppose $F_n$ is the CDF of $nY_n$. Show ...
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Why are oracle inequalities called that way?

The oracle property is an asymptotic property of an estimator, and is about variable selection: An estimator $\hat \beta_n$ satisfies the oracle property if in the limit of $n\to \infty$, the ...
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Let $X$ be a $p$-dimensional vector that is asymptotically normal such that $$\sqrt{n}(X - \mu_X) \stackrel{d}\longrightarrow N(0, \Sigma)$$, and let $H$ be a random $p\times p$ symmetric matrix, ...
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Distribution of variance of $X(X^TX)^{-1}X^TZ$?

I had a question that I couldn't quite wrap my mind around. Essentially, it's this. Suppose $x_i$ is a random $k$-vector, and $X$ is an $n\times k$ vector that is $n$ i.i.d. copies of this $x_i$ "...
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Counting the number of tuples $(i,i',k,k')$ satisfying some conditions

I'm struggling to find the cardinality of a set that will be presented below, as well as an upper bound for it. It is an engaging problem and perhaps not trivial. I will give now the baseline of the ...
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Generalizing Bayesian methods by assuming a “distribution of distributions” instead of a prior

Bayesian methods assume a prior distribution with several hyperparameters. Unfortunately, this is asymptotically incorrect, because distributions in the real world are never exact. For example, the ...
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Finding a test using asymptotic theory. for Poisson $(\lambda)$

If we have a sample of Poisson $(\lambda)$ (a) Find a test for $H_0: \lambda =2$ vs $H_a: \lambda =\lambda_1> 2$ (b) Find a test using asymptotic theory. (c) Compare the results in en (a) y (...
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Asymptotic relationship between tests of compact composite nulls and one sided tests

A motivating example Let $x_t\sim N(\mu,1)$ for $t=1,\dots,T$. Consider testing the null $H_0: \mu\in[-1,1]$ against the alternative $H_1:\mu\in\mathbb{R}\setminus [-1,1]$. The natural (UMPU?) test ...
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Understanding simple LRT test asymptotic using Taylor expansion?

I am trying to understand the proof that the LRT test for $$H_0: \theta = \theta_0 \quad vs \quad H_1: \theta \neq \theta_0$$ is asymptotically chisquared distributed with one degree of freedom. I ...
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confidence interval for the variance with unknown distribution

Let $X_i \sim \text{iid}(\mu, \sigma^2)$ (the question does not specify whether or not $\mu$ and $\sigma^2$ are known). I have to show the confidence interval for the variance. Since $X_i$ is not ...
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When is the bias of a statistic of the form a/n + b/n^2 + c/n^3 +

In many books the bias-correction of the Jackknife resampling method is being prooved under the assumption, that the bias has a special form, namely a/n + b/n^2 + c/n^3 * ... Sometimes it's written "...
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What's an example of an estimator that is $o_p(n^{-1/2})$?

For $X_i$ i.i.d. normal with mean $\mu < \infty$ and variance 1, by the law of large numbers, we have $\bar{X} \overset{\mathcal{P}}{\rightarrow}\mu$, i.e. $\bar{X} - \mu = o_p(1)$. Similarly, by ...
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Approximating the Kullback-Leibler Divergence with a Laplace approximation

Suppose I wish to compute the (asymptotic) Kullback-Leibler Divergence (KLD) between the exact Bayesian posterior $$q_{n}(\theta|x_{1:n}) \propto \pi(\theta)\prod_{i=1}^n p(x_i|\theta)$$ and the ...
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Bound of the bias of a Gaussian Process by its standard deviation in Gaussian Process Regression

In Gaussian Process Regression (GPR), intuitively, the bias of the conditioned Gaussian Process (posterior) at a location $x^*$ gets smaller if the variance at $x^*$ is getting smaller, for example in ...
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Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

Suppose $(X,Y)$ has the pdf $$f_{\theta}(x,y)=e^{-(x/\theta+\theta y)}\mathbf1_{x>0,y>0}\quad,\,\theta>0$$ Density of the sample $(\mathbf X,\mathbf Y)=(X_i,Y_i)_{1\le i\le n}$ drawn from ...
Suppose that $X_1, ..., X_n$ are independent and identically distributed Poisson($\lambda$) random variables. What is a good approximating distribution for \$\sum_{i = 1}^{200} \frac{(X_i - \lambda)^2}...