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Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

2
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2answers
55 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). ...
2
votes
2answers
33 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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votes
1answer
41 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 $\...
4
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0answers
36 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 ...
0
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1answer
33 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
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0answers
27 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
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1answer
45 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
109 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
54 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 $\cos (n+1)X_{(n)}/n$?
1
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1answer
60 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 ...
1
vote
1answer
77 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
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0answers
119 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 ...
1
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0answers
48 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\...
3
votes
1answer
191 views

Does UMVUE of $\frac{\theta_{x}}{\theta_{y}}$ exist? X $\sim$ exp($\theta_{x}$), Y $\sim$ exp($\theta_{y}$)

I have got the following variables. $$ X\sim exp(\theta_{x_{i}}), \ Y \sim exp(\theta_{y_{i}}) $$ and want to find the UMVUE of $$ \frac{\theta_{x}}{\theta_{y}}. $$ As the complete statistics ...
0
votes
1answer
28 views

approximating gamma function to calculate variance of sigma UMVUE

I'm trying to show that variance of $\hat{\sigma}^{UMVUE}$(which is estimator of $\sigma$ in $N(\mu, \sigma^2)$) is larger than cramer-rao lower bound, which I have found to be $\frac{\sigma^2}{2n}$. ...
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0answers
140 views

Existence of UMVUE of $\theta$ for sample from $\small{\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}}$?

Suppose $X_1,X_2,\ldots,X_n$ is a random sample drawn from a distribution with pdf $$f_{\theta}(x)=\small\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}\quad,\,\theta>1$$ Does there ...
1
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1answer
61 views

Usefulness of Point Estimators: MVU vs. MLE

In a past class, two types of point estimators were introduced: minimum variance unbiased estimators (MVUs) and maximum likelihood estimators (MLEs). Supposedly, the MVU is optimal, unless an unbiased ...
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1answer
132 views

The UMVUE of ratio of parameters for two uniform distributions,

Let $X_1,\ldots,X_m$ be i.i.d. having the uniform distribution $U(0, \theta_x)$ and $Y_1,\ldots, Y_n$ be i.i.d. having the uniform distribution $U(0, \theta_y)$. Suppose that $X_i$’s and $Y_j$’s are ...
3
votes
0answers
112 views

Finding the UMVUE of $\theta^2$ where $f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of ${\theta^2}$ I ...
2
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1answer
52 views

Finding a UMVU estimator for the mean

I am struggling with the following problem. We are given an i.i.d sample of size $n,$ with the form $X_{i}=\mu+n_{i}$, where $\mu$ is a deterministic unknown constant, and $n_{i}$ is a noise with a ...
2
votes
2answers
504 views

Find the UMVUE of $\frac{\mu^2}{\sigma}$ where $X_i\sim\mathsf N(\mu,\sigma^2)$

Suppose $X_1, ..., X_4$ are i.i.d $\mathsf N(\mu, \sigma^2)$ random variables. Give the UMVUE of $\frac{\mu^2}{\sigma}$ expressed in terms of $\bar{X}$, $S$, integers, and $\pi$. Here is a ...
5
votes
1answer
613 views

Find UMVUE of $\theta$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

As a slight modification of my previous problem: Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\...
6
votes
2answers
617 views

Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$

Suppose $X_1, X_2, . . . , X_n$ are i.i.d Poisson ($\theta$) random variables, where $\theta\in(0,\infty)$. Give the UMVUE of $\theta e^{-\theta}$ I found a similar problem here. I have that the ...
9
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1answer
247 views

Find UMVUE of $\frac{1}{\theta}$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of $\frac{1}{\theta}$ ...
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0answers
33 views

Sufficient statistic for the mean of a generic distribution?

Is there such thing as a "sufficient statistics for the expectation?" From what I understand, a sufficient statistic is defined only when there is a family of distributions parametrized by $\theta$ ...
3
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1answer
193 views

UMVUE of distribution function $F$ when $X_i\sim F$ are i.i.d random variables

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from a population with distribution function $F$. Is the empirical distribution function $F_n$ the UMVUE of $F$? ( $F$ itself is the parameter of ...
3
votes
2answers
205 views

UMVU estimator for non-linear transformation of a parameter

Let $X_1, ..., X_n$ be iid. and $X_1\sim N(\mu,1)$. $\gamma(\mu)=e^{t\mu}$ for $t\neq 0$ My question is how to find an UMVU estimator for $\gamma(\mu)$ My concern is not so much about the specific ...
10
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1answer
587 views

On the existence of UMVUE and choice of estimator of $\theta$ in $\mathcal N(\theta,\theta^2)$ population

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from $\mathcal N(\theta,\theta^2)$ population where $\theta\in\mathbb R$. I am looking for the UMVUE of $\theta$. Joint density of $(X_1,X_2,\...
3
votes
1answer
845 views

UMVUE of $e^{-\lambda}$ from poisson distribution

Let $X_1,\ldots,X_n \sim$ Poisson$(\lambda)$. I wish to find UMVUE of $e^{-\lambda}$. Here $\overline{X}_n$ is complete and sufficient for $\lambda$ (hence for $e^{-\lambda}$??). Define $Y_i=1$ if $...
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0answers
319 views

The relationship between UMVUE and complete sufficient statistic

Let $X_1,...X_n$ $U(-\theta , \theta)$ I want to find the UMVUE of $\theta$ if it is exists. My answer is , there is no UMVUE in this case. Because there is no complete sufficient statistic that ...
2
votes
2answers
368 views

UMVUE for Bernoulli

Let $X_1,..,X_n$ be independent and $Bin(1,\theta)$ distributed. I would like to find the UMVUE for $\phi(\theta)=\theta^3$. I have a complete and sufficient statistic in $T=\sum_iX_i$, and a unbiased ...
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0answers
438 views

Is the OLS estimator the UMVUE (assuming Normality)?

Suppose $$ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{e} \, , \\ \mathbf{e} \sim \mathcal{N}(0,\mathbf{I}_P) \, . $$ We know that $\mathbf{\hat{b}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \...
4
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2answers
151 views

UMVUE estimates of uniform distribution mean and width

Given are the uniformly distributed samples $$x_n \overset{\text{iid}}{\sim} \mathcal{U}\left(\mu-\frac{w}{2}, \mu+\frac{w}{2}\right)$$ for $n = 1 \ldots N$.Then the UMVUE estimates of $\mu$ and $w$ ...
1
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2answers
394 views

UMVUE for $\theta$ where $X \sim unif\{1 , …, \theta\}$

Say we have $X \sim unif\{1, ... , \theta\}$ and we want to find the uniformly minimum variance unbiased estimator for $\theta$ My first assumption was $X_{(n)}$. Which I managed to show is complete ...
0
votes
1answer
69 views

Is there no UMVUE for this case?

Let $X_1,X_2,... X_n$ be $Normal(\mu,\sigma^2)$ I seem to recall it said that there is no UMVUE for $\mu$ if $\sigma^2$ is also unknown but cannot find why this is so. Is this true?
2
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1answer
158 views

Variance of OLS estimator of $\theta$ in $y_n = \theta x_n + \eta_n$ compared to Cramer-Rao

From Theodoridis' Machine Learning, problem 3.7: Derive the Cramer-Rao bound for the LS estimator, where the training data result from the model $$y_n = \theta x_n + \eta_n\text{, } \qquad n = 1, ...
0
votes
1answer
317 views

Difficulty in obtaining the uniformly minimum variance unbiased estimator of Poisson distribution

I have this problem I know in a Poisson distribution with the parameter theta, the UMVUE is Sample Mean. But I am not sure how do we obtain the UMVUE of theta square. Any help will be appreciated. ...
1
vote
1answer
780 views

Finding sufficient statistic for Weibull density function

I am given the follow problem and am having trouble finding the sufficient statistic. Suppose that Y$_1$, Y$_2$, ..., Y$_n$ denote a Weibull density function, given by: f ( y | $\theta$ ) = Let $...
3
votes
1answer
426 views

UMVUE of location parameter (shifted exponential)

Let $X_1,...,X_n$ be a sample from a distribution with pdf, $f_X(x) = e^{-x + \theta}, x \geq \theta$. Let $x_0 \geq \theta$ be given. I'm trying to find the UMVUE of $f_X(x_0) = e^{-x_0 + \theta}$. I ...
1
vote
0answers
385 views

Compute UMVUE of $\frac{\mu}{\sigma^2}$

Given $X_1,\ldots, X_7$ are i.i.d r.vs which follow $N(\mu, \sigma^2)$. Find the UMVUE of $\frac{\mu}{\sigma^2}$. My thought: First, we realize that $E(S^2) = \sigma^2$ where $S^2 = \frac{\sum_{i=1}^{...
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votes
0answers
462 views

Find the UMVUE of $6\theta^2$ given $f(x\mid\theta) = \frac{1}{2\theta^2} e^{\frac{-\sqrt{x}}{\theta}} I_{(0,\infty)}(x)$

Given $X_1, X_2,\ldots, X_n$ are i.i.d rvs with pdf $f(x\mid\theta) = \frac{1}{2\theta^2} e^{\frac{-\sqrt{x}}{\theta}} I_{(0,\infty)}(x)$ for $\ \theta > 0$. Find the UMVUE of $\ 6\theta^2$, and ...
5
votes
2answers
896 views

Finding UMVUE of a function of parameter belonging to Poisson distribution

Let $X_1, ..., X_n$ be iid from the Poisson ($\theta$) distribution. I have proven that $T = \sum_{i=1}^{n} x_i$ is the complete and sufficient statistic and it has a Poisson($n\theta$) distribution....
0
votes
1answer
259 views

Meaning of unique umvue

For a random sample of size $n$ from Bernoulli distribution with parameter $p$, find the unique UMVUE of $1-p(1-p)$. I found that $1-X_1$ is a UMVUE of $1-p(1-p)$. But can't this $X_1$ be replaced by ...
2
votes
1answer
727 views

Unbiased Estimator for the CDF of a Normal Distribution

Problem Statement Let $X_1, X_2, ..., X_n$ be i.i.d. random variables from a normal distribution with mean $\mu$ and variance $1$. Find an unbiased estimator for $\tau(\mu):=P(X_1>0)$. Attempt at ...
0
votes
1answer
68 views

Find the UMVUE of $U(n_1,θ)$ where $θ>n_1$

Let $X_1,X_2,\dots ,X_n$ follows $U(n_1,θ)$ where $θ>n_1$ and $n_1\le n$. Find the UMVUE of $θ$. My answer is: $$\frac{(n+1)X_n}n - \frac{n_1}n$$ Is that correct?
8
votes
2answers
4k views

Complete statistic for $\sigma^2$ in a $N(\mu,\sigma^2)$

I would like to know if the statistic $$T(X_1,\ldots,X_n)=\frac{\sum_{i=1}^n (X_i-\bar{X}_n)^2}{n-1}$$ is complete for $\sigma^2$ in a $N(\mu,\sigma^2)$ setting. Does this depend on whether $\mu$ is ...
20
votes
5answers
4k views

Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution?

It came as a bit of a shock to me the first time I did a normal distribution Monte Carlo simulation and discovered that the mean of $100$ standard deviations from $100$ samples, all having a sample ...
2
votes
1answer
69 views

How is asymptotic consistency related to mean squared errors?

Let $ X_1 $ , $X_2$ . . . , $X_n$ be a random sample from $ N (\mu , \sigma^2 )$. Then the UMVUE of $\sigma^2$ is $$ \frac{n}{n-1} \frac1 n \sum_i(X_i - \bar X)^2 $$ and its MLE is $$ \frac1 n\sum_i( ...
3
votes
0answers
235 views

Conceptual question - Is it impossible to get a UMVUE for an estimator if another, unknown parameter is required to reach the Cramer-Rao Lower Bound?

Say I'm interested in estimating the $\sigma^2$ of a normal population, which has a Cramer-Rao Lower Bound of $\frac{2\sigma^4}{n}$ if my calculations are correct. I found that the uniformly minimum ...
4
votes
2answers
568 views

UMVUE $g(\lambda)$ = $e^\lambda$ when $x_i \sim Pois(\lambda)$

Let $x_1 ... x_n$ be $Pois(\lambda)$ Find UMVUE of $e^\lambda$ From a previous question, I found the UMVUE of $e^{-\lambda}$ to be $(\frac{n-1}{n})^{t}$ where $t = \sum_{i=0}^n(x_i)$. $\sum_{i=0}^n(...