All Questions
Tagged with estimators asymptotics
39
questions
0
votes
1
answer
23
views
Unbiased and consistent estimator with positive sampling variance as n approaches infinity? (Aronow & Miller) [duplicate]
In Aronow & Miller, "Foundations of Agnostic Statistics", the authors write on p105:
[A]lthough unbiased estimators are not necessarily consistent, any unbiased estimator $\widehat{\...
- 21
2
votes
1
answer
120
views
Sum of asymptotically independent random variables - Convergence
Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$.
If $X_i\sim Ber(\pi_i)$, I ...
5
votes
2
answers
523
views
Asymptotic unbiasedness + asymptotic zero variance = consistency?
Here, Ben shows that an unbiased estimator $\hat\theta$ of a parameter $\theta$ that has an asymptotic variance of zero converges in probability to $\theta$. That is, $\hat\theta$ is a consistent ...
- 65k
5
votes
2
answers
128
views
Sufficient conditions for asymptotic efficiency of MLE
Maximum-likelihood estimators are, according to Wikipedia, asymptotically efficient, that is they achieve the Cramér-Rao bound when sample size tends to infinity. But this seems to require some ...
- 1,099
1
vote
1
answer
218
views
Is convergence in probability implied by consistency of an estimator?
Every definition of consistency I see mentions something convergence in probability-like in its explanation.
From Wikipedia's definition of consistent estimators:
having the property that as the ...
2
votes
2
answers
682
views
What does the likelihood function converge to when sample size is infinite?
Let $\mathcal{L}(\theta\mid x_1,\ldots,x_n)$ be the likelihood function of parameters $\theta$ given i.i.d. samples $x_i$ with $i=1,\ldots,n$.
I know that under some regularity conditions the $\theta$ ...
- 956
1
vote
1
answer
59
views
How to find asymptotically normal estimator if I know probability density function [closed]
I have $X_1, X_2,\ldots,X_n$ be a random sample of size n from a distribution with probability density function:
$$p(x) = \theta^2xe^{-\theta x}I (x > 0).$$
How can I find an asymptotically normal ...
1
vote
1
answer
74
views
Asymptotic property of estimators
I'm studying the asymptotic properties of estimators.
Let $\{ \hat{\theta}_T : T=1,2,3... \}$ be a sequence of estimators of the $p \times1$ vector $\theta \in \Theta $, and $T$ is the sample size. ...
- 321
3
votes
1
answer
183
views
Is asymptotic unbiasedness different from unbiasedness in practice?
Given some estimator T for a parameter θ, by definition T is unbiased if its bias B(T) is 0. It is asymptotically unbiased if B(T) is not 0, but some value that tends to 0 as n goes to infinity. My ...
4
votes
2
answers
3k
views
What does asymptotic efficiency mean?
I read some comparison articles, and always find "asymptotic efficiency," "asymptotically less efficient," and "asymptotically normal."
I am really confused about the ...
- 650
2
votes
0
answers
61
views
How many samples does one need to perform polynomial regression of degree $m$?
Suppose $(X_i, Y_i)$, $i = 1,\dots, n$ are random variables such that
$$X_i\sim N(0,1)$$
$$Y_i = f(X_i) + \epsilon_i$$
where the $\epsilon_i$ are i.i.d. standard Gaussian and $f(x)=\sum_{k = 0}^\infty ...
- 21
3
votes
1
answer
237
views
Delta Method around zero is a N(0, 0)
I have this problem: $\sqrt N \hat{\theta} \sim N(0, V)$ where $E(\hat{\theta}) = \theta_{0} = 0$. I must find the asymthotic distribution of $\frac{N}{V}\hat{\theta}^{2}$ but if I use the Delta ...
- 33
7
votes
1
answer
563
views
Deriving the limiting distribution of the Hodges-Le Cam estimator in Bickel and Doksum (2015)
I am trying to better understand the Hodges-Le Cam estimator, and am having difficulty rendering explicit some of the asymptotic arguments in the derivation of the estimator's limiting distribution. I ...
- 2,550
1
vote
0
answers
166
views
Asymptotic efficiency of estimators of autoregressive models
Are OLS or MLE estimators of autoregressive model asymptotically efficient if errors are i.i.d?
Consider the case of an AR(1) model $$x_t=\alpha x_{t-1} + \epsilon_t$$
with $\epsilon_t$ ~ $i.i.d. N(0,\...
- 51
2
votes
0
answers
73
views
Different regularity conditions for finite population CLT
I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has.
...
- 83