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Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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MLE and UMVUE from an ordered sample of the exponential distribution

I am having a lot of trouble with every part of the problem below. Now, finding MLE's is simple in principle. I just find the distribution for $Y$ and then use calculus to find the value of $\sigma$ ...
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Proving an Estimator of the sample variance to be MVUE

Question: Prove that $\hat{\sigma}_x^2=\displaystyle\frac{1}{N-1}\sum_{i=1}^N(X_i-\overline{X})^2$, with $\overline{X}=\frac{1}{N}\sum_{i=1}^N X_i$ is an unbiased, minimum variance estimator of the ...
Subhasis Biswas's user avatar
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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.
Nick Stats's user avatar
2 votes
1 answer
90 views

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?
Voyager's user avatar
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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: $...
Nothing's user avatar
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216 views

UMVUE for a Uniform distribution [duplicate]

How did we derive the PDF and CDF highlighted in green? Thanks
learn_to_code1's user avatar
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1 answer
250 views

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 $\...
Pramesh Pudasaini's user avatar
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249 views

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 ...
Debarghya Jana's user avatar
3 votes
1 answer
498 views

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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75 views

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, \...
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1 vote
1 answer
122 views

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}\...
AlgoManiac's user avatar
3 votes
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68 views

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 ...
Phil's user avatar
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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 ...
s l's user avatar
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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 ...
simran's user avatar
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0 answers
83 views

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 ...
Alex He's user avatar
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1 answer
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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 ...
Kcd's user avatar
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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 ...
John's user avatar
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6 votes
1 answer
592 views

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?
ryu576's user avatar
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1 vote
3 answers
680 views

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 ...
Artem Moskalev's user avatar
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107 views

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 ...
Kalvin's user avatar
  • 423
2 votes
1 answer
272 views

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 ...
user7423043's user avatar
0 votes
1 answer
246 views

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 ...
smaillis's user avatar
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3 votes
0 answers
114 views

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. ...
Martund's user avatar
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263 views

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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3 votes
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196 views

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 ...
Faithhhhhh's user avatar
2 votes
0 answers
629 views

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 ...
Tan's user avatar
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6 votes
1 answer
170 views

$X\sim\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. 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 ...
Tan's user avatar
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6 votes
2 answers
574 views

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 ...
Tan's user avatar
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2 votes
2 answers
382 views

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 ...
Javier Moreno Sepena's user avatar
0 votes
1 answer
378 views

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 $\...
StatQuestioner's user avatar
1 vote
2 answers
572 views

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 ...
helperFunction's user avatar
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0 answers
31 views

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 ...
JoZ's user avatar
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0 answers
131 views

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 ...
user1916067's user avatar
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0 answers
89 views

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 ...
user1916067's user avatar
6 votes
0 answers
947 views

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-...
abcde's user avatar
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1 vote
0 answers
65 views

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 ...
Daniel's user avatar
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0 votes
0 answers
338 views

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 &\...
Muskaan Madan's user avatar
0 votes
1 answer
2k views

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: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
Student's user avatar
  • 33
1 vote
1 answer
166 views

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} \...
matmin's user avatar
  • 23
1 vote
1 answer
188 views

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 ...
Ron Snow's user avatar
  • 2,103
7 votes
1 answer
625 views

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 ...
BonnieKlein's user avatar
0 votes
0 answers
537 views

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 ...
Jackson's user avatar
  • 133
0 votes
0 answers
46 views

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 ...
wanderer's user avatar
  • 214
0 votes
1 answer
754 views

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 ...
FAHRB's user avatar
  • 47
5 votes
2 answers
2k views

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 ...
wanderer's user avatar
  • 214
2 votes
1 answer
495 views

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 ...
Ron Snow's user avatar
  • 2,103
5 votes
1 answer
3k views

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 ...
user avatar
3 votes
1 answer
406 views

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 ...
Maverick Meerkat's user avatar
1 vote
0 answers
81 views

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)\,,...
StatisticsPersonInTraining's user avatar