All Questions
Tagged with estimators mathematical-statistics
101
questions
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
1
vote
0
answers
23
views
Degrees of freedom for estimation
In the context of estimators, why is it that in general dividing by the degrees of freedom(instead of the sample size) leads to unbiasedness? I see the value in substituting degrees of freedom for ...
1
vote
0
answers
58
views
Is there a good review on complete class theorems?
I'm trying to get an overview of the various results called "complete class theorems" and their relatives, especially the ones that say things along the lines of "every admissible ...
- 425
1
vote
0
answers
120
views
Distribution of $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ [closed]
Given $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$, find the distribution of the sample inter quartile range, $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ in terms of $F$ where, $F_n$ is the emperical distribution ...
- 385
4
votes
1
answer
109
views
Probability mass function of sample median (Bootstrap)
Consider a sample $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$. Let $T_n=F_n^{-1}(1/2)$ be the sample median where, $F^{-1}(x)=\inf\{t:F(t)\ge x\}$ and $F_n(y)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}(X_i\le ...
- 385
0
votes
0
answers
14
views
Optimality criterion for mean estimators
Assume a sample size of $n>5$, a given variance $\sigma^2 > 0$ and a $\delta \in (2e^{-n/4}, 1/2)$.
Proof that there exists a distribution with variance $\sigma^2$ such that for any mean ...
- 1
1
vote
1
answer
119
views
Difference between consistent and unbiased estimator [duplicate]
I have a problem where I have to think of an example to explain a practical example of consistency and unbiased. The example I thought of is the sample mean.
Consistency is when the estimator (sample ...
3
votes
1
answer
52
views
Properties of statistical estimators when data is a collection of estimates
Assume I have a statistical estimator $\theta$ that has nice properties (say, unbiased and consistent) when the data $Y=\{y_1,y_2,\dots,y_n\}$ is i.i.d. (possibly with additional assumptions). But now,...
- 213
3
votes
1
answer
166
views
How does Huber compute the $\operatorname{var}(s_n)/E[s_n]^2$ and $\operatorname{var}(d_n)/E[d_n]^2$?
(N.B. I am cross posting this question from math stackexchange since after
x days I have still not received any responses.)
How does Huber in book 'Robust statistical procedures' in chapter 1 compute ...
- 31
4
votes
1
answer
300
views
Cramer-Rao lower bound for the variance of unbiased estimators of $\theta = \frac{\mu}{\sigma}$
Let $X_1, \cdots, X_n$ be a sample from the $N(\mu, \sigma^2)$ density, where $\mu, \sigma^2$ are unknown.
I want to find a lower bound $L_n$ which is valid for all sample-sizes $n$ for the variance ...
- 63
1
vote
1
answer
137
views
Fisher Information for $\bar{X}^2 - \frac{\sigma^2}{n}$ with $X_1, \dots, X_n$ normally distributed
I need to find the Fisher Information for $T = \bar{X}^2 - \frac{\sigma^2}{n}$ with $X_1, \dots, X_n$ normally distributed sample with unknow mean $\mu$ and know variance $\sigma^2$. For this I'm ...
- 111
2
votes
0
answers
61
views
Is there a theory of M-Estimation for non-unique argmins?
Given some i.i.d. random variables $x_1,\ldots,x_n\in\mathbb R^d$, an M-estimator $\hat\theta_n\in\mathbb R^p$ is a parameter which minimizes
$$\hat\theta_n=\arg\min_{\theta\in\Theta} \sum_{i=1}^n\...
6
votes
3
answers
241
views
What 's the $(\Omega,\mathcal{F},P_{\theta} )$ those $T_{n}$ defined on?
Definition (Consistency)
Let $T_1,T_2,\cdots,T_{n},\cdots$ be a sequence of estimators for the parameter $g(\theta)$ where $T_{n}=T_{n}(X_1,X_2,\cdots,X_{n})$ is a function of $X_{1},X_{2},\cdots,X_{n}...
- 330
2
votes
1
answer
167
views
I need to prove that $\hat\theta=\max\{X_1,...,X_n\}$ is a mean square consistent estimator for $\theta$
Let $X_1,...,X_n$ a i.i.d from a population with distribution $U[0,\theta]$, i.e.,
$f_{X_i}(x)=\frac{1}{\theta}g_{[0,\theta]}(x)$, for $i=1, \ldots, n$
where
\begin{align}
g_{[0,\theta]}(x) =
\begin{...
1
vote
1
answer
59
views
How to find asymptotically normal estimator if I know probability density function [closed]
I have $X_1, X_2,\ldots,X_n$ be a random sample of size n from a distribution with probability density function:
$$p(x) = \theta^2xe^{-\theta x}I (x > 0).$$
How can I find an asymptotically normal ...
