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

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Limiting variance of normal mean

In Casella's Statistical Inference´╝îin 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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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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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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112 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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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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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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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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51 views

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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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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39 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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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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33 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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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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166 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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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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41 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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242 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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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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47 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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43 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 ...
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What is the limiting distribution of the sample mean?

My question is relatively simple: what is the limiting distribution of the sample mean? But there are some technicalities I want to discuss. context: I was asked this problem in an exam, and I feel ...
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proving the asymptotic distribution of the mean

Let ${X_t} = \mu + \sum\limits_{j = - \infty }^{ + \infty } {{\psi _j}{\varepsilon _{t - j}}}$ with $\varepsilon$ is a white noise iid with variance $\sigma^2$ , $\sum\limits_{j = - \infty }^{ + ...
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Big O of inverse function plus constant

if I have a function $$f(i) = \frac{a}{1 + bi} + c,$$ where $a$, $b$, and $c$ are positive constants, and $i \geq 1$ is an integer. Can I say $$f(i) = O(1/i)$$ Wiki says that $f(i) = O(g(i))$ as ...
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asymptotic covariance between mean and standard deviation

I am trying to estimate the asymptotic covariance between mean and standard deviation. I know the following $$\sqrt n \hat \mu \xrightarrow{d}N\left( {\mu ,{\sigma ^2}} \right),\sqrt n \hat \sigma ...
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asymptotic distribution of joint random variables

I am trying to understand the asymptotic distribution of the following expression under normality $$ {\hat \sigma \hat S - \sigma S} $$ Where $\sigma$ and $S$ are the population standard deviation ...
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Asymptotic distribution of a recursive statistic

I have a (time series related) test statistic which is asymptotically normal. I would like to know what is the asymptotic distribution of its maximal value obtained by a recursive estimation. For ...
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asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
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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 ...
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71 views

Understanding $O_p$

One thing I feel like I have never mastered is the concept of $O_p$ convergence and how to use it. I understand the basic idea and what bounded in probability means, but I always have a hard time ...
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Intuition for uniform integrability as asymptotic light tails

I am trying to get an intuition to the concept of uniform integrability: $$ \lim_m \sup\lim_n \mathbb{E}[|X_n| I_{\{X_n \geq m \}}] \to 0 $$ Can this be seen as a form of "asymptotic light tails"? ...
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Proof of asymptotic variance

How do you prove that $X_n - E[X_n] = O_p(\sqrt{Var(X_n)})$ It's used in my textbook and I don't know where they get it from.
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How does Pearson's Chi Squared Statistic approximate a Chi Squared Distribution

So if Pearson's Chi Squared Statistic is given for a $1 \times N$ table, then its form is: $$\sum_{i=1}^n\frac{(O_i - E_i)^2}{E_i}$$ Then this approximates $\chi_{n-1}^2$, the Chi-Squared ...
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Express Multinomial as vector sum of bernoulli trials?

So we know we can think of the binomial as a sum of iid bernoulli. Can we similarly express the multinomial as a vector sum of dependant bernoulli's and get the asymptotic distribution that way? I ...
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Asymptotic distribution of uniform order statistics

It can be shown that for an iid sample from a Uniform(0, 1) distribution, \begin{equation} n(1-U_{(n)}) \rightarrow exp(1) \\ n(U_{(1)}) \rightarrow exp(1) \end{equation} To see this just try finding ...
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limiting distribution of $(n-1)S^2/\sigma^2$

I need to prove that the limiting distribution of $(n-1)S^2/\sigma^2$ is a Normal, where $S^2$ is the sample variance. However, I have no clue on how to do it. We know that $(n-1)S^2/\sigma^2 \sim ...
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47 views

Subtracting t-distributions

I have two linear regression parameters of interest, b1 and b2. Both parameters are from a linear model built from 14 datapoints and having 7 model parameters, including an intercept. Interest lies in ...
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Asymptotically unbiased estimator using MLE

I am learning Maximum likelihood estimators for a inference class. And this is a problem I came across. Let $X_1,X_2,X_3,\ldots, X_n$ be a random sample with p.m.f $$p(X)=\theta(1-\theta)^x; ...
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343 views

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 ...
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Simulating Convergence in Probability to a constant

Asymptotic results cannot be proven by computer simulation, because they are statements involving the concept of infinity. But we should be able to obtain a sense that things do indeed march the way ...
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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 ...
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Convergence of a sequence of random variables [duplicate]

Suppose $P(X=1)=P(X=-1)=1/2$ and define $$X_n=\begin{cases} X\ \text{with probability}\ 1-\frac{1}{n} \\ e^n\ \text{with probability}\ \frac{1}{n} \end{cases}$$ I then need to prove or disprove ...