Questions tagged [umvue]
UMVUE stands for Uniform Minimum Variance Unbiased Estimation.
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What are the uniformly minimum variance unbiased estimators (UMVUE) for the minimum and maximum parameters of a PERT distribution?
I believe the answers to this question are the sample minimum and the sample maximum, but I have not been able to find a reference or proof of this.
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In linear models, why are we focused on BLUE rather than UMVUE?
It seems more natural to talk about UMVUE (following the basic statistical estimation theory). But when we turn to lm, we only care about BLUE, why? Are there any insurmountable difficulties here?
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Disproving the regularity condition of Cramer-Rao Lower bound
Let $X = (X_1,\cdots, X_n)$ where $X_1,\cdots,X_n$ be i.i.d from the uniform distribution $U(0,\theta)$ with $\theta>0$. I was asked to show the regularity condition of the Cramer-Rao lower bound:
$...
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UMVUE for a Uniform distribution [duplicate]
How did we derive the PDF and CDF highlighted in green?
Thanks
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Poisson: finding UMVUE for $\lambda + \lambda^2$ [closed]
Let $X_1,..., X_n$ be iid sample from the Poisson distribution with parameter $\lambda$. Find the UMVUE of $\lambda + \lambda^2$.
I know $T := \sum\limits_{i=1}^n X_i$ is complete and sufficient for $\...
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UMVUE of $\theta$ for $\mathrm{Uniform}(0,\theta) $ where $\theta \in[1, \infty)=\Theta$
Let $X_1, \ldots, X_n$ be a random sample from $\mathrm{U}(0, \theta)$, where $\theta \in[1, \infty)=\Theta$, say. Here I tried to find complete-sufficient statistics for $\theta$ as my main target is ...
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UMVUE for $g(\theta)=\theta^2$ of Poisson random variables
Let $X_1,...X_n$i.i.d.~$Pois(\theta)$ with unknown $\theta>0$, I want to find the UMVUE for $g(\theta)=\theta^2$.
I know that $T(x)=\Sigma_{i=1}^{n}x_i$ is complete and sufficient for $\theta>0$....
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Confused about UMVE and Linear Combination of Two Unbiased Estimators, Taking Unif (0, $\theta$) as an Example
A dumb question here ...
I am confused about UMVE and Linear Combination of Two Unbiased Estimator (that can potentially create a more efficient/lower-variance estimator).
Taking $\mathcal{Unif}(0, \...
DumbMLE
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Nonexistence of UMVUE for non-constant function?
I tried to prove the problem:
Suppose X $\sim \ U(\theta-1,\theta+1)$, $\theta \in \mathbb{R}$. Then there is no UMVUE for $g(\theta)$ unless $g$ is a constant function.
Here is my attempt:
Suppose $...
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Finding UMVUE of a parameter in form of probability of discrete random variables
We have $X$ and $Y$ as independent discrete random variables both in ${1, 2, ...}$.
Their pmf's are:
$f(x|\alpha)=P(X=x)=\alpha(1-\alpha)^{x-1}, x=1, 2, ...$
$f(y|\beta)=P(Y=y)=-\frac{1}{\log\beta}\...
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When is it better to have an unbiased estimator instead of one that has a smaller risk?
I just learned that for $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ i.i.d, the sample variance $\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2$ is unbiased, and it is in fact UMVUE.
However, it is not ...
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UMP Test and UMVUE when there are nuisance parameters
Consider $X_1,...,X_n \sim Weibull(\theta, c)$ where $c>0$ is unknown. Several textbook examples consider when $c$ is known, but here, we consider when $c$ is unknown. Suppose now we wanted to find ...
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UMVUE of $ \theta^2 (1- \theta ) $ X is random sample from bernoulli distribution
Let $ X_1, X_2 ..... X_n $ be a random sample from bernoulli distribution with parameter $ \theta $ , Obtain UMVUE of $ \theta^2 (1- \theta ) $
MY APPROACH
I calculated that T = $ \sum X_i $ is ...
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Finding UMVUE for exponential sample [duplicate]
Let $X_1,...,X_n$ be a random sample of i.i.d. exponential distribution with probability density function $$f(x|\theta)=\frac{1}{\theta}exp(-\frac{x}{\theta}), \ x\geq0$$
Let $S_n=\sum_{i=1}^nX_i$ and ...
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UMVUE for P(X > k) in exponential distribution [duplicate]
I have to find UMVUE for
$exp(-k*a)$ where X ~ Exponential(a); k is a positive real number.
I tried it using Lehmann-Scheffe theorem.
Since, T = $sum(xi) (i = 1,..,n)$ is complete sufficient statistic ...
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UMVUE of $\theta = P(X_1 \leq c)$ [duplicate]
I could use some help as im working through a practice/homework problem.
Let Let $X_1, X_2, ... X_n \overset{\text{iid}}\sim N(\mu,1)$, Find the UMVUE of $\theta = P(X_1 \leq c) $ where c is a known ...
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Can a minimum variance unbiased estimator be inconsistent?
Title says it all. I'm guessing it shouldn't be possible for an estimator to be UMVUE and yet not consistent. Is there a counter-example to this? Or is there a proof that it can't happen?
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Rao-Blackwell and unbiased estimators of zero
In Casella & Berger we are trying to prove that any estimator $\phi$ based on a complete sufficient statistic T is the unique best unbiased estimator of its expectation.
However, in the preceding ...
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Finding the M.V.U.E of n Bernoulli trials [duplicate]
Let $r$ be the observed number of successes in $n$ Bernoulli trials with probability $\pi$ of success. Then M.V.U.E (Minimum Variance Unbiased Estimator) of $\pi (1-\pi)$ is ?
$n$ Bernoulli trials ...
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Finding UMVUE of difference of exponentals
Let $X_1, \ldots, X_n$ be a sample from an exponential distribution with p.d.f. $f(x; \theta) = \theta e^{-\theta x}$ for $x > 0$ where $\theta > 0$ is an unknown parameter.
I would like to find ...
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Finding UMVUE of function of poisson parameter
I am to estimate $\exp(-\lambda)\lambda^2/2$ from the distribution $Exp(\lambda) \sim \frac{e^{-\lambda}\lambda^x}{x!}$
I used the indicator function $W=\mathbb I_{2}(X_1)$ as an initial unbiased ...
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Can we say that MSE of MLE is always at most equal to that of UMVUE?
The question is in the title. I was wondering if that is the case, because we are considering a more general class, rather than focusing on the class of unbiased estimators, as we do in case of UMVUE. ...
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Obtaining the UMVUE for $U(-\theta, \theta)$ for $\frac{\theta}{1+\theta}$
Let $X_1, X_2, .., X_n$ be $n$ random variables from $U(-\theta, \theta)$. We need to obtain the UMVUE of $\frac{\theta}{1+\theta}$.
I already derived that $Y = Max|X_i|$ is a complete and sufficient ...
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Finding the UMVUE of $e^{3\lambda}$ in Poi($\lambda$)
Let $ X = (X_1, ... , X_n)$ iid variables coming from Poisson
distribution with mean $\lambda$.
Find the UMVUE of $e^{3\lambda}$.
I tried understanding the solution below (in the possible duplicate ...
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UMVUE of two-parameter exponential family distribution
Suppose $\{X_{i}\}_{i=1}^n\overset{i.i.d}{\sim}X$, where $X$ has
density $$f_{X}(x)=\frac{1}{b}\exp\left\{\frac{x-a}{b}\right\},x>a$$
What is the UMVUE of $\mathbb{P}(X_1<u)$?
Here is what I've ...
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$X\sim\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. Find UMVUE of $\frac{a}{b}$
Suppose $\{X_i\}_{i=1}^n\overset{i.i.d}{\sim}X,$ and $X$ has density
$f(x)=\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. What is the UMVUE of
$\frac{a}{b}$?
Here is what I've done so far.
It can be ...
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Proving the nonexistence of UMVUE for $\text{Unif}\{\theta-1, \theta, \theta+1\}$
I am trying to prove that
There is no UMVUE for $\theta$ for the distribution
$\text{Unif}\{\theta-1, \theta, \theta+1\}$, $\theta$ is an integer.
Here is what I have attempted.
I am trying to use ...
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Completeness of a statistic - Open ball
I was studying the slides of the course in statistics, but there is a theorem that is not clear for me. This chapter was about finding a complete statistic, and it explains that it can be found with ...
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Bayes Estimate for Mean Squared Loss in Uniform Prior
Can some one please help me out in Verifying if my prior distribution is uniform then will my Bayes estimate will always be MLE or UMVUE?
If $X_i$ follow iid $N(\theta,1)$ and prior distribution of $\...
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Can a Bayesian estimator perform better than an MVUE?
According to wikipedia:
In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any ...
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How to show if $c=X'w$ has an solution, then there exists uniqu $w_0$ such that $var(w_0'Y)\leq var(w'Y)$?
The whole question is the following:
Consider the linear model $\mathbf{Y}=\mathbf{X}\mathbf{\beta}+\varepsilon,$ where $\mathbf{X}$ is a known $n \times p$ matrix, $\mathbf{\beta}$
is a $p$ -vector ...
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Find the UMVUE of $P\{{X_i < Y_i\}}$ for $X_i,..X_n \sim N(\mu, \sigma^2)$ and $Y_i,..Y_n \sim N(\nu, \sigma^2)$
I have to find the UMVUE of $P\{{X_i < Y_i\}}$ for $X_i,..X_n \sim N(\mu, \sigma^2)$ and $Y_i,..Y_n \sim N(v, \sigma^2$).
I know that I have to find the joint distribution of $X$ and $Y$ which I ...
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UMVUE of the probability of a conditional poisson probability $P_{\lambda} (X=r )$ [duplicate]
Consider a $X_1, ... X_n \sim~ Poisson(\lambda)$, I want to obtain the UMVUE of $P_{\lambda} (X=r)$.
This is my approach:
$\operatorname{\mathbb{E}}_{\theta}[h(t)] = P_{\lambda} (X=r)$.
The ...
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Let $X_1,X_2,\dots,X_n$ be random sample from Poisson($\theta$). Find MVUE of $e^{-2\theta}$
Question: Let $X_1,X_2,\dots,X_n$ be random sample from Poisson($\theta$). Find MVUE of $e^{-2\theta}$
My attempt has been by modifying the answer from this question:
The Poisson distribution is a one-...
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Techniques for finding UMVUEs
I'm learning about the different techniques available to find the UMVUE such as Rao-Blackwell and Lehmann-Scheffé theorems. My question is how to know when is better to use one method from the other ...
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UMVUE of Bernoulli random variables
Let $X_1, X_2..... X_n$ be a random sample from a Bernoulli population with parameter $p$.
A sufficient statistic is $\sum_{i=1}^{n}X_i$. If we define
$$
U(X_1,X_2,\ldots,X_n)= \begin{cases}1/2n &\...
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Find best unbiased estimator for $\theta$ when $X_i\sim U(-\theta,\theta)$
I am having an issue finding a best unbiased estimator for $\theta$. Any help is appreciated.
Let $X_1, ..., X_n$ be a random sample from a population with pdf:
$f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
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Find the MVUE of $1-e^{-2\theta}$ [closed]
I'm given the following question and I have to choose one of the four possible answer:
Let $X_1$ be a sample of size $n=1$ from distribution whose p.d.f. is:
$$ f(x;\theta)=\theta e^{- \theta x} \...
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How can I find the BUE of $\theta$ in the simple linear relationship $Y_i=\theta x_i^2+\epsilon_i$?
Let $Y_1,...,Y_n$ be described by the relationship $Y_i=\theta x_i^2+\epsilon_i$, where $x_1,...,x_n$ are fixed constants and $\epsilon_1,...,\epsilon_n$ are iid $N(0,\sigma^2)$. How can I find the ...
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How do I find the UMVUE of $\sqrt{\alpha}$ here?
new user here self-studying some mathematical statistics. I came across this problem and am stuck.
Problem: Suppose that for $i = 1, ... , n$, the positive random variables $X_i$ are independent and ...
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UMVUE of $\exp(2 [\mu + \sigma^2])$ for $N(\mu, \sigma^2)$
Suppose that $X_1, \ldots X_n$ are iid $N(\mu, \sigma^2)$ where $\mu$ and $\sigma$ are both unknown with $\mu \in \mathbb{R}$, $\sigma \in \mathbb{R}+$, $\boldsymbol{\theta} = (\mu, \sigma)$, $n \geq ...
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Finding good estimators for a function of bernoulli parameter [duplicate]
Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ l'm interested in finding estimator of $(1-\theta)^{1 / k},$ when $k$ is a positive integer. I am considering the following ...
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UMVUE of parameter from Zero Truncated Poission distribution
Let $x_1,...,x_n$ have the distribution
$$
P(X = x) = \frac{\theta^xe^{-\theta}}{x!(1-e^{-\theta})}, \ \ x = 1,2,3...
$$
Now we want to find UMVUE for $e^{-\theta}$. My first thought was to apply ...
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Finding UMVUE for a function of a Bernoulli parameter
Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ I'm interested in finding the UMVUE of $(1-\theta)^{1/k}$, when $k$ is a positive integer. .
I know $\sum X_{i}$ is a ...
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How can I find an unbiased estimator for $\frac{1-\theta}{\theta}$ to obtain this quantity's UMVUE?
Let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=\theta(1-\theta)^x,x=0,1,\cdots; 0 < \theta <1$ is unknown. Find the UMVUE of $\frac{1-\theta}{\theta}$.
My work:
I know that I should ...
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explanation of why an UMVUE doesn't necessarily have to achieve the CRLB?
I'm studying uniformly minimum variance unbiased estimator(UMVUE). I have seen question on this site asking why the UMVUE doesn't achieve the CRLB(Cramer Rao lower bound), and all of the answers have ...
user255658
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Rao-Blackwell part of the Lehmann-Scheffe theorem
I'm trying to understand the proof of this theorem.
An unbiased estimator $T$, that is a function of a complete statistic $S$, is unique, i.e. there can't be other unbiased estimators that are ...
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Show that $E[u(X_1)\mid\sum^n_{i=1}X_i=z]=\Phi \left( \frac{\sqrt n(c-z/n)}{\sqrt{n-1}} \right)$ [duplicate]
Let $X_1, ..., X_n$ be a random sample from the $N(\mu,1)$ distribution. First show that $$E \left[ u(X_1)\, \Big|\, \sum^n_{i=1}X_i=z \right] = \Phi \left( \frac{\sqrt n(c-z/n)}{\sqrt{n-1}} \right)\,,...
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Show that a linear combination of UMVU estimators is also a UMVU estimator
Suppose I have two estimators $\delta_1$ and $\delta_2$ with finite second moments, and they are UMVU estimators of $f_1(\theta)$ and $f_2(\theta)$, respectively.
Now, for some real numbers $n_1$ and ...
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Proving $X$ is a complete statistic to find a UMVUE
I'm learning about Stein's phenomenon. This standard problem is considered:
Let $X_1, \dots, X_p$ be independent random variables with $X_i \sim N(\theta_i, 1)$ for $i = 1, \dots, p$. Let $\theta = (\...
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