# Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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### UMVUE of a parametric function [closed]

Assume $(X_1, . . . , X_n)$ is an i.i.d. sample from $P = \{f : f$ is a pdf and $E_f|X|< ∞\}$. Use the conditioning approach to find an UMVUE of $τ (f) = (E_f (X))^2$. Can someone provide a hint ...
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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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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 $\... 0 votes 0 answers 168 views ### UMVUE of$\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$as my main target is ... 2 votes 1 answer 215 views ### UMVUE for$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$.... 0 votes 0 answers 49 views ### Confused about UMVE and Linear Combination of Two Unbiased Estimators, Taking Unif (0,$\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, \... 1 vote
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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 ...
1 vote
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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?
1 vote
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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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### 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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### 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 ... 334 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
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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)\,,...
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 ...