The asymptotics tag has no wiki summary.
0
votes
0answers
12 views
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 ...
5
votes
0answers
69 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 ...
4
votes
1answer
97 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 ...
2
votes
0answers
35 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 ...
1
vote
0answers
42 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 ...
2
votes
0answers
56 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 ...
8
votes
1answer
126 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, ...
2
votes
1answer
136 views
Question about a derivative of the 2nd-step moments in a two-step estimator as a joint GMM-estimators approach
I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the Handbook of Econometrics, Volume 4, 1994, page 2176).
In the model I'm interestend in has some former ...
5
votes
1answer
76 views
Elementary approach to higher order asymptotics
I am trying to understand “higher order asymptotics”. I find several texts on Likelihood asymptotics, nothing’s easy to read... if you have any nice pointers on this direction, I’ll be interested; ...
0
votes
0answers
38 views
Distribution of transformation
Suppose $X_1,\ldots,X_n$ are i.i.d. $\mathcal U(0,1)$. I am looking for the asymptotic distribution of $$T_n = \prod_{i=1}^n [e{X_i}]^{1/\sqrt{n}} \>.$$
7
votes
2answers
215 views
Simple question about the asymptotics of estimators
Consider any arbitrary estimator called $\hat{M}$ (e.g., regression coefficient estimator or specific type of correlation estimator, etc) that satisfies the following asymptotic property:
...
1
vote
0answers
164 views
When are the asymptotic variance of OLS and 2SLS equal?
Assume the model $ \ y = X\beta + u \ $ with $\ W \ $ is a $ \ n\times l \ $ so called matrix of instruments.
The following assumptions hold. There is a law of large numbers (LLN) for 1.,2.,3. and ...
1
vote
1answer
92 views
Asymptotic assumptions in OLS
Let us assume the following data generating process:
(1) $ \ \ y = X\beta + u$
where $y$ and $u$ are $n\times 1$ vectors, $X$ is a $n\times k$-matrix with $rk(X)=k$ and $\beta$ is a $k\times ...
2
votes
0answers
72 views
Can we approximate the distribution of S?
I want to understand how the sampling distribution of the whole covariance matrix behaves for large $n$. I am trying to use the delta method and multivariate CLT. I am trying to show that when the ...
0
votes
0answers
91 views
Deriving asymptotic distribution
I'm working on a question and I appreciate if you could guide me on how to approach it. Here is the question:
Consider $Y_1, Y_2, \ldots, Y_n$ as iid with density $f(y;\theta)$ and assume that the ...
7
votes
1answer
239 views
Observed information matrix is a consistent estimator of the expected information matrix?
I am trying to prove that the observed information matrix evaluated at the weakly consistent maximum likelihood estimator (MLE), is a weakly consistent estimator of the expected information matrix. ...
2
votes
0answers
135 views
Asymptotic normality of MLE in exponential with higher-power x
Given the distribution:
$f(x;\theta) = \frac{3}{\theta}x^2e^{-x^3/\theta}$ if $x>0$
the MLE for $\theta$ is $\frac{1}{n}\sum_{i=1}^n x_i^3$. It's an unbiased estimator with variance $\theta^2/n$. ...
4
votes
0answers
34 views
Limiting distributions identified as functions of Brownian motion or stochastic integrals
I am teaching a stochastic processes course to MA stat students, and to stay on topic I would like some examples of limiting distributions in stat that are identified as functions of brownian motion ...
2
votes
0answers
138 views
Parameter estimation for the sum of two Independent (not necessarily i.d.) Gamma RVs
I'm a bit of a stats newbie so take it easy on me if this ends up being somehow trivial. I'm working on a problem that involves parameter estimation for the sum of two independent gamma distributions ...
3
votes
3answers
177 views
Justification for use of $\chi^2(1)$ in Wald and score test
In a recent exam, we were asked to justify the use of the $\chi^2(1)$ distribution in performing the Wald or Rao's score test. There was only 1 mark for this (approx. 2.5 mins worth of time). My ...
2
votes
0answers
94 views
Variations of Assouad’s lemma
I'm reading the paper Optimal Rates of Convergence for Sparse Covariance Matrix Estimation by T. Cai and H. Zhou.
They refer to a specific version of Assouad's Lemma (see lemma 2, p. 6 of the linked ...
2
votes
0answers
88 views
Asymptotics of 0-1 classification loss
I am interested in training a simple binary linear classifier. That is, I will find a vector of weights $\bf w$ such that I can predict the class of new example by the sign of
$f(x) = w^T x$.
I ...
1
vote
1answer
355 views
Variance and Asymptotic normality of sample variance of normal distribution
Let $X_1,X_2$,....,be a random sample from $N(q,w^2)$; $q,w$ are unknown. Let $S_n$ be the sample standard deviation.
i.e $S_n^2=\frac{1}{n-1}\sum(X_i-\bar{X})^2$
What is $Var(S_n^2)$? and how to ...
-1
votes
1answer
201 views
proof FGLS asymptotically efficient
Prove that FGLS is asymptotically efficient. Does one have to use Cramer Rao to do this?
1
vote
1answer
157 views
Asymptotic Efficiency of Two-Stage Least Squares
Apparently Wooldridge, Introductory Econometrics, 2002ed is the only book showing that two-stage least squares (2SLS) is asymptotically efficient. I cannot get a copy of the proof.
Is it correct to ...
2
votes
1answer
107 views
Sample size planning with required CI width and guessed propotion, using “Wilson” method
I refer to this link for calculating the sample size needed based on the width confidence interval needed, and the guessed proportion. I realize that it is use the "asymptotic" method when calculating ...
6
votes
1answer
200 views
How to compute asymptotic confidence intervals for differences in quantiles?
Can anyone give me advice on computing the asymptotic confidence intervals for a difference in quantiles of a distribution? For example, I have fit a log-normal distribution to doubly interval ...
10
votes
3answers
225 views
How many of the biggest terms in $\sum_{i=1}^N |X_i|$ add up to half the total?
Consider $\sum_{i=1}^N |X_i|$
where $X_1, \ldots, X_N$ are i.i.d. and the CLT holds.
How many of the biggest terms add up to half the total sum ?
For example, 10 + 9 + 8 $\approx$ (10 + 9 + 8 $\dots$ ...
4
votes
0answers
168 views
How does number of observations supporting alternate hypothesis on a test of a variance have to scale so that null is rejected?
Informal explanation: In the course of my research I've run into the following problem: I am observing a machine that outputs random numbers. Most (if not all) of these random numbers come from the ...
4
votes
1answer
68 views
Asymptotics of the survival function for Anderson Darling distribution?
I am using the ADinf procedure of Marsaglia & Marsaglia to compute the CDF of the Anderson Darling statistic. I am interested in the survival function, 1 minus ...
0
votes
0answers
28 views
Probing the existence of the asymptotic distribution of a time series
I am studying the properties of random walks with barriers and wondered what can be said, as generally as possible, about the existence of asymptotic distributions.
Consider a Gaussian process with a ...
2
votes
1answer
78 views
Convergence in distribution from ratio of the two densities
I want to prove $[X|y] \rightarrow \delta_0(X)$, as $y$ goes to $\infty$, i.e. the distribution of $[X|y]$ converges to degenerate distribution at zero for large enough $y$.
I know the density ...
4
votes
0answers
90 views
Factor models with small noises
The standard factor model formulation is
$y=W x+\epsilon$
where $x \sim \mathcal{N}(0, I)$, $\epsilon \sim\mathcal{N}(0, \Sigma)$. $W$ and $\Sigma$ are typically estimated from MLE. The solution can ...
10
votes
3answers
485 views
Can the empirical Hessian of an M-estimator be indefinite?
Jeffrey Wooldridge in his Econometric Analysis of Cross Section and Panel Data (page 357) says that the empirical Hessian "is not guaranteed to be positive definite, or even positive semidefinite, for ...
9
votes
4answers
247 views
Do third order asymptotics exist?
Most asymptotic results in statistics prove that as $n \rightarrow \infty$ an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood ...
5
votes
4answers
1k views
Asymptotic distribution of multinomial
I'm looking for the limiting distribution of multinomial distribution over d outcomes. IE, the distribution of the following
$$\lim_{n\to \infty} n^{-\frac{1}{2}} \mathbf{X_n}$$
Where $\mathbf{X_n}$ ...
5
votes
1answer
195 views
Lumping in Markov process with absorbing states
I have a four-state, discrete time Markov process with time-dependent transition matrices such that after a given time T the matrices become constant. The idea is people in a program leaving the ...
4
votes
2answers
325 views
How to compute efficiency?
Suppose instead of maximizing likelihood I maximize some other function g. Like likelihood, this function decomposes over x's (ie, g({x1,x2})=g({x1})g({x2}), and "maximum-g" estimator is consistent. ...
