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

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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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15 views

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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109 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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26 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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1answer
57 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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58 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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28 views

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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54 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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44 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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1answer
54 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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18 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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1answer
40 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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1answer
136 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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1answer
84 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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1answer
68 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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148 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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70 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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55 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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36 views

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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67 views

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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1answer
44 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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1answer
28 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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95 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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102 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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1answer
179 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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125 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 ...
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56 views

Stationarity in OLS time series and asymptotic properties

I think I lack somewhat deeper understanding of this topic, but I thought stationarity is required in order for OLS to have asymptotic properties. "But stationarity is not at all critical for OLS to ...
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3answers
275 views

Asymptotic Theory in Economics

I am interested in deepening my Asymptotic Theory understanding. My current knowledge is that of a typical PhD student (from a decent University), say at the level of Green's textbook. Are there any ...
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63 views

Ratio of lengths of two confidence intervalls

I have two random variables: (1) With standard normal distribution. Confidence interval $I_1$, which is centered and has probability of $(1-\alpha)$ (2) With T-distribution. Confidence intervall ...
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136 views

Is there a statistical application that requires strong consistency?

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is ...
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84 views

Convergence in Probability of Empirical Median

I'm stuck with this one. Let $X_1,...X_n$ be an i.i.d. sequence of random variables with CDF F. The empirical CDF of $X_i$ is defined $$ \hat F_n(x) = \frac{1}{n} \sum_{1 \leq i \leq n} I\{X_I \leq x ...
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45 views

Asymptotic distribution of Pearson Chi Square?

Suppose under $H_1$, $X_i$'s are independent random variables with $\mathrm{Poisson}(\theta_i)$. Under the null hypothesis, $X_i$'s are IID from $\mathrm{Poisson}(\theta)$. Define pearson chi square ...
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76 views

Conditions for Poisson approximation of the superposition of non-Poisson processes

It is well known that the superposition of $N$ Poisson processes is itself a Poisson process with an intensity given by $\sum_{n=1}^{N} \lambda _{n}$. Conversely a superposition including any ...
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81 views

Asymptotic Deming regression line

The Deming model is given by independent paired observations $(x_i,y_i)$ with the following distributional assumption: $$({\cal D})\colon \large \begin{cases} x_i = x^*_i + \epsilon_i \\ y_i = y^*_i + ...
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127 views

Hypothesis test for the presence of a Gaussian signal in i.i.d additive Gaussian noise

Suppose there exists a sequence of $n$ numbers with two possible instantiations: The sequence contains all zeros; $n-1$ of the numbers are zeros, and one is a zero-mean Gaussian random variable ...
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160 views

Bounded in probability and finite expectation

Let $x_t = O_p(1)$, meaning that for all $\varepsilon > 0$ there exists $M_{\varepsilon} < \infty$ s.t. $P(|X_t| > M_{\varepsilon}) < \epsilon$ for all $t \in \mathbb{N}$. Does it imply ...
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58 views

Asymptotic vs. bootstrap test statistic, size and power properties

I am running a VAR based Granger non-causality tests. I've obtained asymptotic and bootstrap $p$ values for Wald joint test of 0 restriction on a set of lagged variables. It appears that bootstrap ...
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44 views

C.A.N. Estimator

I'm searching for the paper(s)/book(s) where the concept of a c.a.n. (consistent asymptotically normal) estimator was first defined and its basic properties proved. I'm particularly interested in the ...
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34 views

Classes of asymptotically normal statistics

Consider an iid sample $X_1, \dots, X_n$, from a "well behaved" distribution with pdf $f(X)$. My question is the following: what are the classes of statistics $S(X_1, \dots, X_n)$ that are ...
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60 views

Convergence rate of a non-linear function of the sample mean

We have a iid sequence of random variables $X_1, X_2, \dots, X_n$, where $E(X_i) = \mu$ and $var(X_i) = \sigma^2$. The sample mean $\bar{X}$ converges to $\mu$ at rate $\sqrt{n}$ thanks to the LLN. ...
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How is “asymptotic sufficiency” for a statistic defined?

From Wikipedia: The score plays an important role in several aspects of inference. For example: in formulating a test statistic for a locally most powerful test; in approximating the error in a ...
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71 views

Asymptotic normality and normalization wrt variance

Let $X_n, n \in \mathbb N$ be a sequence of random variables with finite variances. As $n \to \infty$, are the following two equivalent: $X_n \to N(0, \sigma^2)$ for some $\sigma^2 \in [0, \infty)$, ...
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66 views

How to code separate data for a regression parameter in nonlinear regression?

Say I have a bunch of measurements over a range of my independent variable (example data at the end) and I want to fit them to an equation such as a classic pharmacological inhibition curve: ...
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Implications of lower-bounded total variation distance on hypothesis testing

Let $\{X_i\}_n$ be a sequence of $n$ random variables independently and identically drawn from either $P$ or $Q$. Thus the sequence $\{X_i\}_n$ has a product distribution, which is either $P^n$ or ...
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198 views

Large sample asymptotic/theory - Why to care about?

I hope that this question does not get marked "as too general" and hope a discussion gets started that benefits all. In statistics, we spend a lot of time learning large sample theories. We are ...
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1answer
121 views

Confidence intervals based on the CLT: ever useful?

Suppose, for concreteness, that I am trying to estimate the mean of a population using a random sample of size $N$. Many elementary books discuss forming a confidence interval for the population mean ...
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80 views

How to determine the asymptotic variance of the following statistic?

Given $T_n = \sum_{i=1}^n c_i X_i$, and integer $m$ with $0\leq m\leq n$, where $X_1, \dots, X_n$ are $\{0,1\}$-valued random variables, and have a joint probability mass function which takes ${n ...
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49 views

Question about asymptotics of steepest descent method in the context of adaptive filtering

The model which will be used is defined as $e(n) = d(n) - y(n)$ with $y(n) = x(n)^Tw(n)$. where $e(n)$ is the error term of the n-th observation, $x(n)$ the input vector of the n-th ...
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96 views

Distribution/expected length of the shortest path in infinite random geometric graphs

Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
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151 views

Density of robots doing random walk in an infinite random geometric graph

Consider an infinite random geometric graph in which the node locations follow a Poisson point process with density $\rho$ and edges are placed between the nodes that are closer than $d$. Therefore, ...