All Questions
Tagged with estimators self-study
87
questions
2
votes
0
answers
46
views
No Existence of Efficient estimator
I need to prove that given $(X_1,...,X_n)$ from the density $$\frac{1}{\theta}x^{\frac{1}{\theta}-1}1_{(0,1)}$$ no efficient estimator exists for $g(\theta)$=$\frac{1}{{\theta}+1}$.
I have shown that ...
1
vote
1
answer
54
views
What is the distribution of the unbiased estimator of variance for normally distributed variables?
I must be making some mistake in my derivation of the distribution of the unbiased variance estimator for i.i.d. $X_{i} \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$.
We have $\bar{X} =\frac{1}{n}\sum\...
- 11
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
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 ...
1
vote
0
answers
42
views
Show that $T=\sum_{i=1}^n X_i$ is a sufficient statistic for $p$ [duplicate]
I try to use the definition of sufficient statistic to prove that
Suppose that $X_1,\dots, X_n$ is an iid random sample from $X\sim \mathrm{Bernoulli}(p)$. Show that $T=\sum_{i=1}^n X_i$ is a ...
- 747
0
votes
1
answer
50
views
Minimizing variance of sequence of independent but not identically distributed random variable
I tried to work on the problem
Let $(X_n)$ be a sequence of independent random variables with $E[X_n]=\mu$ and $Var[X_n]=n$ for every $n \in \mathbb{N}$. Find the statistic of the form $\sum_{i=1}^...
- 181
0
votes
0
answers
85
views
Estimator for the propensity for consumption c = C/Y
I've an exercise where it asks to propose an estimator for the propensity for consumption: $c = C/Y$ where $C$ is the consume and $Y$ is the income.
Since the consumption function $C = c_0 + c_1 Y$ is ...
- 83
9
votes
1
answer
421
views
Minimax estimator for geometric distribution
I'm trying to solve this problem:
Let $X$ be a single sample from Geo($p$) where $p ∈ (0, 1)$. Find a minimax estimator for $p$ under the loss $L(p, δ(x)) = (p−δ(x))^2/
p(1−p)$ .
I'm trying to put ...
- 231
3
votes
1
answer
88
views
Consistency of a simple estimator for $y_i = \beta_1 x_i + u_i$
Let $y_i = \beta_1 x_i + u_i$ for $i=1,2,..,n$. If I define $$\hat \beta_1 = \frac{y_1 + y_n}{x_1 + x_n}$$ then whether my $\hat \beta_1$ will be consistent or not in this setup?
For my estimator to ...
- 43
0
votes
0
answers
123
views
Unbiased estimators for uniform distribution
I have a given question that I just cannot figure out. From what I can understand, both estimators would be unbiased (since e_1 is the sample mean and e_2 is an unbiased estimator for uniform ...
5
votes
2
answers
350
views
What is Bayes estimator of $\theta$ when loss function is $L(\theta,a)=I(|\theta -a|>\delta)$?
Suppose $X$ given $\theta$ has pdf $f(x\mid \theta)=e^{-(x-\theta)}I(x>\theta)$ and there is a standard Cauchy prior on $\theta$. As part of an exercise, I am trying to find a Bayes estimator of $\...
- 11.5k
1
vote
0
answers
42
views
On the naming of two different median estimators
Assume that $X \sim \mathcal{E}(\lambda)$ is, for example, exponential with $\lambda > 0$. Given a data sample $X_1, \ldots, X_n$, assume that I want to estimate the median of $X$. Consider these ...
- 642
1
vote
1
answer
77
views
consistency of maximum likelihood estimator
For population with n size and following density function
$$f(y, a)= (1/6a^4)y^3e^{-y/a}$$
For that, I have found the maximum likelihood estimator of a which is $\hat{a}= \bar{y}/4$
I have also shon ...
- 1,140
0
votes
0
answers
31
views
Antithetic variate as control variate to find optimal constant [duplicate]
Problem:
If $\hat{θ}_1$ and $\hat{θ}_2$ are unbiased estimators of $θ$, and $\hat{θ}_1$ and $\hat{θ}_2$ are antithetic, we derived that $c^∗ = 1/2$ is the optimal constant that minimizes the variance ...
- 327