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

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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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33 views

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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24 views

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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28 views

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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46 views

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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29 views

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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40 views

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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60 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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73 views

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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46 views

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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14 views

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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45 views

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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84 views

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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43 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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114 views

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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139 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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81 views

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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253 views

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 ...
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Limiting distribution of the first order statistic of a general distribution

Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i>0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)>0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting ...
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185 views

What are the regularity conditions for Likelihood Ratio test

Could anyone please tell me what the regularity conditions are for the asymptotic distribution of Likelihood Ratio test? Everywhere I look, it is written 'Under the regularity conditions' or 'under ...
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60 views

Cramer's theorem for a precise normal asymptotic distribution

I am working on a homework problem for my probability class: (Cramer Application) A. Let $X_1, X_2, ... X_n$ be a sample from a distribution with pdf $f(x;p) = q^xp$. Determine the MLE of $p$ ...
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Probability of ongoing experiment

Suppose, I do a experiment where I have an event 'a' true 1000 times in 1000 trials. So, the probability becomes 1000/1000 = 1. If I am going to do another trial, my prediction about event 'a's ...
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Empty asymptotic confidence interval?

I have a sample $x=(4, 3, 1, 2, 2, 2, 2, 5, 7, 3, 1, 2, 3, 4, 3, 2, 3, 3, 3, 4)$ of size $n=20$. from a binomial distribution with 10 trials and probability of success $p$. I am asked to construct the ...
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151 views

Can MCMC iterations after burn in be used for density estimation?

After burn-in, can we directly use the MCMC iterations for density estimation, such as by plotting a histogram, or kernel density estimation? My concern is that the MCMC iterations are not ...
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92 views

Obtain the asymptotic distribution of X/Y if X and Y are iid and independent of each other

. We just covered large sample theory and I'm going through the examples in our textbook. This one kinda confused me and I was hoping someone could help me understand why and how the book got, ...
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151 views

Normal Approximation of the sum of correlated Bernoulli Random Variables

Hi I am looking for a result (if it exists !!!) in the direction of Normal approximation for sum of correlated Bernoulli random variables (edit : with the same parameter $p$) where correlation between ...
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128 views

Example of CLT when moments do not exist

Consider $X_n = \begin{cases} 1 & w.p (1 - 2^{-n})/2\\ -1 &w.p~ (1 - 2^{-n})/2\\ 2^{k} &w.p~ 2^{-k} \text{ for } k > n\\ \end{cases}$ I need to show that even though this has ...
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Relationship between $\text{Cov}(x_i^2, e_i^2)$, the asymptotic variance of b under homoscedasticity and heteroscedasticity?

I am trying to figure out the relationship between $\text{Cov}(x_i^2, e_i^2), V$ and $V_0$, where: $V=$ asymptotic variance of $\sqrt{n(\hat{\beta}-β)}$ under heteroskedasticity, and $V_0=$ ...
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65 views

Deriving sampling distribution

Assume we have the following function: $$f(p) = \frac{1}{(1-p)d}\ln\left(\frac{1}{T}\sum_{t=1}^{T}\left[\frac{1+X_t}{1+Y_t} \right]^{1-p} \right)$$ where $d$ is a constant $T$ is a constant $X_t$ ...
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51 views

Asymptotic Least Squares question (with random regressors)

Consider the DGP $y_i=x_i+\epsilon_i$, where $\epsilon_i \sim Z$. We estimate $\beta=1$ by regression without a constant term, so in $y_i=\beta x_i + \epsilon_i$. Show that this DGP does ...
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67 views

Asymptotics of OLS coefficients for unequal variance RHS variables

This seems to be a very general question about the bias OLS produces for RHS variables with unequal variance, but I was not able to find an explicit solution anywhere. Suppose we have realizations of ...
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42 views

Dominant term in convergence in MSE

The speed of convergence in MSE is typically dominated by the variance. Consider the ML estimator of the variance: Its variance decays as $\mathcal{O}(1/n)$ and the bias term decays as ...
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61 views

Asymptotic distribution of Kernel density estimator

For my research I am looking for proof of the asymptotic distribution of the univariate Kernel density estimator as proposed by Rosenblatt 1956 and Parzen 1962. A proof is for example given here and ...
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423 views

How many clusters for linear mixed models and GEE?

I have a data set with repeated measurements on subjects. The total sample size is $n=118$ and the number of clusters (i.e. subjects) is $m=49$. The smallest cluster is of size 2 and the largest ...
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221 views

Proof that the log-likelihood is asymptotically quadratic

I was reading this article, where the author says that Maximum Likelihood (ML) estimates are asymptotically normal if the log-likelihood is asymptotically quadratic. I have heard or read other ...
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74 views

What does it mean to scale random variables?

We just started learning asymptotic theory, and to prove the lindberg-levy central limit theorem, weak law of large numbers etc, we 'scale and standardize' the RVs so it ends up having a standard ...
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218 views

Find the limiting distribution of $\sqrt{n} \left(\sqrt{\bar{X}} -1 \right) $ if $\sqrt{n} \left( \bar{X}-1 \right) \to N(0,1)$

Find the limiting distribution of $\sqrt{n} \left(\sqrt{\bar{X}} -1 \right) $ if $\sqrt{n} \left( \bar{X}-1 \right) \to N(0,1)$. Can you please check my work below? In principle, the Delta method ...
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95 views

Limiting Distribution of $W_n=\frac{Z_n}{n^2}$ , $Z_n \sim \chi ^2 (n)$

My try ended in an awkward result. I thought it best to use the moment generating function (MGF) technique. We can derive the MGF of $W_n$ as follows: $$ E \left[ e^{tZ /n^2} \right]= ...
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65 views

Limiting behavior of a martingale

This is a homework question: Suppose that $X_0=1$ and that for $n\geq 1$ $$X_n\sim \left\{ \begin{array}{l l} U(0,X_{n-1}) & \quad \text{with probability $1-X_{n-1}/2$}\\ U(X_{n-1},1) ...
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Cramer-von Mises using pdf instead of Cdf

Cramer-von Mises gives a nice way to compare distributions with: $$ \int [(\hat{F}_n(x) - F_0(x)]^2dF_0(x); $$ I'd like to use something like this to compare two empirical distributions, is there ...
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Should the mean of the bootstrapped distribution always be asymptotically equal to the sample estimate?

Suppose I bootstrap the distribution of the sample mean. Normally, one would use the mean of the bootstrapped distribution as point estimate of the parameter and the s.d. as its standard error. The ...
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67 views

Showing MGF of Poisson converges to MGF of N(0,1)

I'm trying to finish up a proof of the CLT for the Poisson distribution, but am having some trouble evaluating a limit. I've shown that the moment generating function for the standardized Poisson is ...
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35 views

Truncated Poisson Asymptotics

This is a homework problem. I have figured out part (a) but I need help with part (b). I include part (a) for completion. Suppose $X_1,\ldots,X_n$ are iid Poisson random variables. Furthermore, let ...
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108 views

Deriving the asymptotic distribution of a particular equation

Question: Assume we have the following equation: $$\widehat{\Theta}(\rho) = \frac{1}{(1-\rho)\Delta t} \ln\left(\frac{1}{T} \sum_{t=1}^T \left(\frac{1+r_t}{1+rf_t}\right)^{1-\rho} \right) \ \ \ \ ...
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182 views

How to get asymptotic covariance matrix when observed information matrix is singular

I'm fitting different models by Maximum Likelihood. To do this I'm using a stochastic version of Newton-Raphson algorithm, where both the gradient and the Hessian of the likelihood are estimated at ...
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310 views

Cauchy Distribution and Central Limit Theorem

In order for the CLT to hold we need the distribution we wish to approximate to have mean $\mu$ and finite variance $\sigma^2$. Would it be true to say that for the case of the Cauchy distribution, ...
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184 views

Why don't asymptotically consistent estimators have zero variance at infinity?

I know that the statement in question is wrong because estimators cannot have asymptotic variances that are lower than the Cramer-Rao bound. However, if asymptotic consistence means that an estimator ...