Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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21 views

UMVUE of the following parameter

Suppose I have $\{X_i : 1\le i \le m\}$ which are i.i.d random variables having Poisson distribution with parameter $\lambda$ and let $N_i = \min\{k : X_k > p \text{ and } k \ge i\}$ where $p<\...
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68 views

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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43 views

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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62 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?
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78 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 ...
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59 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 ...
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83 views

Minimum variance unbiased estimator for the parameter, $\lambda$ of a Poisson process

Consider a Poisson process. We start observing it at a time, $u_1$ in its time-line, until the time $u_2 = u_1 + u$. There are $m$ of these processes operating independently of each other. This is ...
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127 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 ...
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56 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 ...
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25 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. ...
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27 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 ...
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115 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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78 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 ...
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138 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 ...
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108 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 ...
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1answer
150 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 ...
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94 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 ...
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118 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 $\...
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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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30 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 ...
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106 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 ...
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54 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 ...
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311 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-...
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62 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 ...
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146 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 &\...
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1answer
592 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: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
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97 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} \...
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92 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 ...
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265 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 ...
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211 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 ...
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40 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 ...
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862 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 ...
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225 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 ...
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1answer
804 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 ...
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1answer
205 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 ...
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72 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)\,,...
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533 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 ...
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1answer
254 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 = (\...
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50 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. ...
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1answer
449 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 $E \left( X_1\times X_2\times X_3 \mid \sum_{i=1}^n X_i= z \right)$...
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492 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). ...
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2answers
724 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 ...
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1answer
412 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 $\...
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1answer
446 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 $\...
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70 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 ...
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1answer
184 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 $...
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160 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=...
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1answer
161 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
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1answer
164 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 $b(\theta)$...
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1answer
165 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 $\cos (n+1)X_{(n)}/n$?