Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

Filter by
Sorted by
Tagged with
0
votes
0answers
29 views

Umvue of probability cutoff when variance is unknown for normal distribution

Let $X_1,X_2,..,X_n$ be a random sample from $N(a,b^2)$ where $b$ is unknown and $a$ is known. Calculate the umvue of $P(X\le c)$ for a constant $c.$ I know a complete sufficient statistic is the ...
-1
votes
0answers
32 views

Finding the UMVUE of $P(X_1\ge k)$ [duplicate]

Let $X_1,X_1,...,X_n$ be a random sample of size $n$ from $N(\mu, \sigma^2)$ where both $\mu$ and $\sigma$ are unknown. It is required to find the UMVUE of $P(X_1\ge k)$. In this problem I am able to ...
0
votes
1answer
33 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 ...
2
votes
0answers
14 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. ...
0
votes
0answers
19 views

MVU for random deviation in the samples of the random variable normal with mean 0 and variance unknow

I am trying to calculate the MVU for $ \sigma $, when we have a random sample of a distribution $ N (0, \sigma^ {2}) $, using the Lehman-Scheffe theorem. We have to $\sum^{n}_{i=1}X^{2}_{i}$ is ...
0
votes
1answer
69 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 ...
3
votes
0answers
47 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 ...
2
votes
0answers
64 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 ...
4
votes
1answer
81 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 ...
4
votes
1answer
98 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 ...
2
votes
2answers
69 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 ...
0
votes
1answer
55 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 $\...
0
votes
2answers
109 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 ...
0
votes
0answers
27 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 ...
0
votes
0answers
24 views

UMVUE of $\exp(t \mu)$ for $N(\mu, \sigma^2)$ where $\mu$ is not known but $\sigma^2$ is known [duplicate]

I am new to UMVUE calculations and struggling a bit with this question. I want a hint about how to approach the derivation of the UMVUE for $\exp(t [\mu)$ for $N(\mu, \sigma^2)$ where (\mu is not ...
0
votes
0answers
92 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 ...
0
votes
0answers
47 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 ...
5
votes
0answers
200 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-...
1
vote
0answers
57 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 ...
0
votes
0answers
93 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 &\...
0
votes
1answer
308 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: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
0
votes
0answers
203 views

Difference between MVUE and efficient estimator

Is there any difference between an efficient estimator and MVUE? An efficient estimator has to be unbiased and its variance has to be equal to this obtained from Cramer-Rao theorem. MVUE also needs to ...
1
vote
1answer
73 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} \...
0
votes
1answer
60 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 ...
7
votes
1answer
169 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 ...
0
votes
0answers
144 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 ...
0
votes
0answers
37 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 ...
4
votes
2answers
579 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 ...
1
vote
1answer
173 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 ...
3
votes
1answer
434 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 ...
3
votes
1answer
113 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 ...
1
vote
0answers
67 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)\,,...
3
votes
2answers
379 views

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 ...
3
votes
1answer
189 views

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 = (\...
0
votes
0answers
45 views

How many classmates does a freshman have?

The freshmen at East China Normal University has just received their student ID. Let the last three digits of a student ID be ABC, then A is the class he is in, whereas BC is his number in the class. ...
2
votes
1answer
359 views

Conditional Expectation (Poisson) UMVUE

Suppose $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson distribution with mean $λ$. How can I find the conditional expectation $E \left( X_1\times X_2\times X_3 \mid \sum_{i=1}^n X_i= z \right)$...
3
votes
2answers
300 views

UMVUE for probability of cutoff

Let $X_i \sim N(\mu,1)$, i.i.d. We aim to find UMVUE for $p(\mu) = P_{\mu}(X_1 \leq u)$ for some fixed $u$. I have shown that $\bar{X}$ and $X_1 - \bar{X}$ are independent. ($\bar{X}$: sample mean). ...
3
votes
2answers
515 views

Estimator with variance equal to Cramér-Rao lower bound in $N(x_i\theta,1)$-distribution

Let $Y_1,\ldots, Y_n$ be independent and $N(x_i\theta,1)$ distributed, with for each $Y_i$ a mean of $x_i\theta$ for known $x_1,\ldots,x_n$. In a previous section of this exercise I found that the ...
4
votes
1answer
271 views

Showing a UMVUE does not exist for the center of symmetry for a family $f(x-\theta)$

I am unsure how to finish this problem in Lehmann's book. The problem asks to prove that among the class of all symmetric distributions $\mathcal{F}$, no UMVUE exists for the center of symmetry $\...
1
vote
1answer
313 views

Use the Lehmann-Scheffé theorem to deduce that $\overline{X}$ is an UMVUE estimator for $\theta$

Let $X_1,X_2,\ldots,X_n$ be a random sample whose distribution is $X\sim\operatorname{Bernoulli}(\theta)$. (a) Prove that $\sum_{i=1}^n X_i$ is complete. (b) Use the Lehmann-Scheffé to deduce that $\...
5
votes
0answers
64 views

How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
2
votes
1answer
159 views

UMVUE of the probability a Poisson R.V is odd?

Problem: Let $X_i \sim Pois(\lambda)$. Find the UMVUE of the probability that $X_1$ is odd. My attempt: I don't think there's any obvious unbiased estimator to use conditioning. So instead I write $...
2
votes
0answers
148 views

UMVUE of $P(X=0)^2=e^{-2\lambda}$ for $X\sim Pois(\lambda)$ with single observation

I found a similar question asked previously, however that question involved a random sample $X_1,X_2,\dots, X_n$ where as this problem only has a single observation, so their method of finding $P(X_1=...
1
vote
1answer
127 views

Unbiased Estimator based on Sufficient Statistic

suppose $X_1, ... , X_n$ are iid with pdf $f(x|\beta) = e^{-(x-\beta))}I_{(\beta, \infty)}(x)$ and the pdf of ( the smallest order statistic) $X_{(1)}$ is given by $f_{X_1}(x)$ = n $ *$ $e^{n(\...
2
votes
1answer
154 views

Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
0
votes
1answer
138 views

UMVUE of $\cos\theta$ when $X_i\sim U(0,\theta)$

$X\sim U(0,\theta)$. To find the umvue of $\cos\theta$ is it enough to find the umvue of theta and substitute for it. Umvue of $\theta$ being $(n+1)X_{(n)}/n$, is the answer $\cos (n+1)X_{(n)}/n$?
2
votes
1answer
181 views

Find UMVUE of $p^3$

Let $X_1, X_2, ..., X_n$ be a random sample from $Binom(1, p)$. I'm trying to find the UMVUE of $p^3$. Some thoughts: Apparently, $\bar{X}^3$ is not the answer, although it's the MLE of $p^3$. For ...
2
votes
1answer
1k views

UMVUE- geometric distribution where $X$ is the number of failures preceding the first success

$X_1, \dots, X_n$ iis geometric: $P(X=x) = (1-p)^{x}p$, $x=0,1,2, \dots$ My Attempt: $T=\sum_{i=1}^n X_i$ is a sufficient statistic $W= \begin{cases}1 & X_1= 0,\\ 0 & X_1\neq 0\end{cases}$ ...
3
votes
0answers
342 views

UMVUE for $g(p) = \mathbb{E}_p[X^2]$, where X follows a geometric distribution

I have a random variable X with pmf $$p_\lambda(x) = (1-p)^{x-1}p, \ \ x = 1,2,3,\ldots, \ \ p \in (0,1)$$ and I am trying to find a UMVUE for $$g(p) = \mathbb{E}_p[X^2]$$. Here is my attempt so ...
2
votes
0answers
153 views

Finding UMVUE for $g(\theta)$ that satisfy $g(0)=0$ in discrete uniform $f(x\mid\theta)=\frac{1}{\theta} I_{1,…,\theta}(x)$

Let $x_1, \ldots x_n, \overset{\text{i.i.d}}{\sim}f(x\mid \theta)=\frac{1}{\theta} I_{1,...,\theta}(x)$. I know that $T=X_{(n)}$ is complete and sufficient statistic for $\theta$ and $$f_T(t\mid\...