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

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Convergence of Bernoulli sampling procedure

Let $X$ be a variable which has density $f_{X}$ and mean $E[X]=\mu$ Define $B$, a Bernoulli variable, which has a probability $P(b_{i}=1)=p_{i}=\frac{1}{N^\gamma}$ Consider the following ...
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What is the intuition behind asymptotic normality of a parameter vector

I'm studying statistical inference and am trying to understand what it means for the estimate of a parameter vector to be asymptotically normal. I can understand the univariate version of this but ...
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13 views

Conjecturing asymptotic normality for a sum of dependent random variables

I am hunting for the asymptotic distribution of a scaled partial sum of pair-wise equi-correlated, identically distributed continuous random variables $$W =n^{-\delta}\sum_{i=1}^nY_i(n),\;\; ...
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19 views

Theoretical properties of Gaussian Process Emulator

I am studying Guassian Prcess Emulator (GPE) to approximate computationally expensive computer models. Basically, we suppose the computer model, or simulator, is denoted by $f(x)$, where $x$ is the ...
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72 views

Intuition for difference in asymptotic distribution

I am looking for some intuition on the following: Assume a sequence of random variables $\{X_n\}$ for which we know that the following hold: $$X_n \xrightarrow{p} c \neq 0$$ $$Z_n \equiv \sqrt{n}(X_n ...
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34 views

Why are the Wald standard errors often very poor estimates of the uncertainty of variances?

As from this post as @Ben Bolker pointed out that : .. note that these (as often pointed out by Doug Bates) the Wald standard errors are often very poor estimates of the uncertainty of variances, ...
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Testing the variance component in a mixed effects model

Say $y=X\beta+ Zu +\epsilon$ is our mixed effects model where $u=(u_1,..,u_r)$ and $u_{j} \stackrel{i.i.d.}{\sim} N(0, \sigma^2_{a})$ for $j=1,...,r$ and $\epsilon=(\epsilon_1,...,\epsilon_n)$ are ...
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24 views

How to compare orders of magnitude?

In Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties", they provided a proof to Theorem 1. The very last part of the proof is as follows. Some ...
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25 views

Cross-validating the tbats/bats function in forecast

Is there a way to cross validate the tbats/bats function in the forecast package in R? I have been trying to get CV weighted parameters which then I can pass to a function for revised estimates. ...
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Does a “pruned” i.i.d. multivariate sample behave as the i.i.d. sample?

Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
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43 views

How to prove test statistic has $\chi^2$ distribution using minimum chi-square estimator?

This is for iid case where the test statistic is sum of $(observed - expected)^2/expected$. It is supposedly shown on page 5 of Fisher (1924): ...
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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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12 views

In what class of estimators is CRLB a lower bound of asymptotic variance?

In what class of estimators is CRLB a lower bound of asymptotic variance? Asymptotically efficient is defined as asymptotically normal and asymptotic variance achieving CRLB. And asymptotically normal ...
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88 views

Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used

I am trying to understand a proof of quite a long theorem that I report completely for the sake of completeness. This is From Jensen and Rahbek Asymptotic Inference for Nonstationary GARCH (2004). My ...
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54 views

Identification of asymptotic distribution of a function of the MLE

Let $X_1, X_2, \dots, X_n$ be Bernoulli$(\theta)$ and let $\hat{\theta}$ be the MLE of $\theta$. I am attempting to identify the asymptotic distribution of the odds ratio. I believe that I ...
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32 views

Show Pearson Chi-Square Statistic is Score Statistic for Multinomial Data

I've read in many textbooks that the Pearson Chi-Square statistic is a score statistic in the multinomial setting (as well as others). I thought it would be a good exercise to derive this, but I am ...
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Regarding convergence in probability

Let $\{X_n\}_{n\geq 1}$ be a sequence of random variables s.t $X_n \to a$ in probability, where $a>0$ is a fixed constant. I'm trying to show the following: $$\sqrt{X_n} \to \sqrt{a}$$ and ...
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98 views

Central limit theorem question

I am thinking about the rate of convergence in central limit theorem (CLT) for different distributions. Let's assume we have a set of i.i.d random variables, $X_1,X_2,\ldots$ which follow an unknown ...
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61 views

Approximate distribution of product of N normal i.i.d.? Special case μ>10σ, σ>0

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and $|\mu_X|\geq10\sigma_X$, $\sigma > 0$, looking for: accurate closed form distribution approximation of ...
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66 views

Approximate distribution of product of N normal i.i.d.? General case

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and NO assumptions about $\mu_X$ and $\sigma_X$, looking for: accurate closed form distribution approximation of ...
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208 views

Approximate distribution of product of N normal i.i.d.? Special case μ≈0

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and $\mu_X \approx 0$, looking for: accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic ...
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58 views

Limiting variance of normal mean

In Casella's Statistical Inference,in Example 10.1.8 on page 470, it says that the limiting variance of normal mean $\bar X_n$, is $\lim_{n\to\infty}\sqrt n\text{Var}\bar X_n=\sigma^2$. However, since ...
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99 views

asymptotic distribution of a statistic

Say we have iid sample of size $n$ with $X_i \sim Exp(\lambda)$ and the task is to find asymptotic distribution of the statistic $$T_n := \frac{\bar{X}}{s}$$, where $s^2$ is the unbiased sample ...
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Class of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
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63 views

An example of delta method performing poorly

Q: Is there an example of the delta method being used in applied statistics where the sample size needed for a good approximation is practically unfeasible? I am looking for an example from applied ...
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130 views

Explaining Consistency of estimators to a non-statistical audience

How would you demonstrate to a non-statistical audience (pictorially) that the consistency of an estimator matters? The idea is the following. I have proved that a multivariate estimator people are ...
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90 views

Do I get the nice asymptotic properties of MLE when I restrict the parameter space?

I would like to know if the MLE is still consistent, asymptotically normal, and efficient when I put restrictions on the parameter space. I think my confusion stems from the definition of the ...
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117 views

A question on the paper by Littell and Folks (1971)

I have been reading this very interesting paper but have come across something that I do not quite understand, namely how the asymptotic limit of the quantity $-\frac{1}{n} \log \left(1-F_n^{F} ...
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23 views

What is the condition for $Var(\sqrt{n}(\hat\theta-\theta))\to V$ as $n\to \infty$ where $V$ is the asymptotic variance [duplicate]

What is the condition for $Var(\sqrt{n}(\hat\theta-\theta))\to V$ as $n\to \infty$, where $V$ is the asymptotic variance of $\sqrt{n}(\hat\theta-\theta)$. That is $\sqrt{n}(\hat\theta-\theta))\to^D ...
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Convergence rate: $E\|\hat f - f\|^2 = O(\psi_n)$ vs $\|\hat f - f\| = O_p(\psi_n^{1/2})$

I have seen two types of results on convergence rates for some estimator $\hat f$: $E\|\hat f - f\|^2 = O(\psi_n)$ and $\|\hat f - f\| = O_p(\sqrt{\psi_n})$. The first result seems to be stronger, ...
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150 views

Asymptotic distribution of $\chi^2_{(1)}$

Let $z_1,\dots,z_n$ be i.i.d. $\chi^2$ with $k$ degrees of freedom (Neither $n$ or $k$ are fixed). Is anything known about the distribution of $\min_i(z_i)$ asymptotically, when say $n/k\to ...
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How to show consistency and asymptotic normality of estimators from solving an estimating equation?

Estimating equation is another way to obtain estimators when likelihood can not be written out explicitly. Generally we will solve a set of estimating equation. And the estimators are consistent and ...
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41 views

Showing that moment estimates are asymptotically bi-variate normal

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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39 views

What is first order asymptotic distribution?

I am stuck with this term in a research paper(Hansen, 1999). Its written (p 351) "Hansen (1996) shows that a bootstrap procedure attains the first-order asymptotic distribution, so p-values ...
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50 views

Asymptotic chi square tables?

I just need to calculate the value of the right hand side of the test statistic which is an asymptotic chi square distribution. How can I find those values? All I have is the normal chi square ...
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49 views

Finding a test statistic when you don't know the distribution?

I am working on this problem from my class and it has stumped me for a while now. I will show a picture of the problem and then my work/thoughts: Now we are not given the individual distribution of ...
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221 views

Asymptotic distribution of the max (min) of IID binomial variables

I would like to know the limiting distribution when $k \uparrow \infty$ and $k/n \rightarrow \lambda$ of $$ \max(X_1, \ldots, X_k), \text{ where $X_i$ are IID $B(n,p)$}.$$ This is most likely a ...
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43 views

What is the optimal bandwidth in this problem

Suppose that I have some estimator $\hat f$ for the parameter $f$. I obtain the following convergence rate $$E\|\hat f - f\|^2 = O\left(\frac{1}{nh} + \frac{1}{nh^2} + \frac{1}{\sqrt{n}h} + ...
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Expected number of replacements during weighted reservoir sampling?

Consider the problem of taking a weighted sample of size $K$ from a stream of unknown but finite size $N$ in a single pass. Reservoir sampling solves this by assigning each item from the stream with ...
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260 views

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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436 views

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 ...
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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 ...
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Verify accuracy of asymptotic variance of estimator

The asymptotic variance of a maximum likelihood estimator can be obtained from the inverse of the Hessian of the log-likelihood function at the MLE, and the variance of derived quantities can be ...
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Prove that the MLE $\hat{p}(1-\hat{p})$ is a asymptotically efficient

Consider when $X_1, ..., X_n \sim $ Bernoulli($p$). We want to estimate $p(1-p)$. Suppose $\hat{p}=\frac{1}{n}\sum_{i=1}^nX_i$. Prove that the MLE $\hat{p}(1-\hat{p})$ is a asymptotically efficient ...
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70 views

Correctness of a proof for Hodges' estimator

We know the following is Hodges' estimator: $$ \delta_n = \begin{cases} \bar{X}_n & |X_n| \geq n^{-1/4} \\ a\bar{X}_n & |X_n| < n^{-1/4} \\ \end{cases} $$ where $X_1, ..., X_n \sim ...
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54 views

Asymptotic distribution for moments of gaussian distribution

Is there a way to find the asymptotic distribution for the moments of Gaussian distribution? More specifically, say you have $X_1, ..., X_n \sim N(\mu, \sigma^2)$. For a moment $m_{n, k} ...
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Is $X_{(1)} + X_{(n)}$ a good estimator for $\theta$?

Problem 8.7 From Van der Vaart's Asymptotic Statistics: Given a sample of size $n$ from the uniform distribution on $[0,\theta]$, the maximum $X_{(n)}$ of the observations is biased downwards. ...
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Asymptotic variance of GMM with efficient instrument

This question emerged from reading Wooldridge's Econometric Analysis of Cross Section and Panel Data, second edition, section 14.4.3, where the asymptotic distribution of the GMM (Generalized Method ...
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44 views

Convergence in probability for two statistics of Laplace random variables

Suppose $X_i$ are iid random variables. $X_i \sim \mathrm{Laplace}(\lambda)$. Also define: $$ U_n = \frac{1}{n-2}\sum_{i=1}^n{|X_i|} $$ $$ V_n = \sqrt{\frac{1}{n-1} \sum_{i=1}^n{X_i^2} } $$ Given ...
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Asymptotic distribution of t-ratio

I am looking at a problem where I have to calculate the asymptotic distribution of a t-ratio, after having run a OLS regression. I have re-written the expression so as to be t = z * sqrt ...