Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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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 size of only $n=2$, proved to be much less than,…
Carl
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The pdf of $\frac{X_1-\bar{X}}{S}$

Suppose $X_1, X_2,...,X_n$ be i.i.d from $N(\mu,\sigma^2)$ with unknown $\mu \in \mathcal R$ and $\sigma^2>0$ Let $Z=\frac{X_1-\bar{X}}{S}$, where $S$ is the standard deviation here. It can be shown that $Z$ has the Lebesgue…
Deep North
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How do I know which method of parameter estimation to choose?

There are quite a few methods for parameter estimation out there. MLE, UMVUE, MoM, decision-theoretic, and others all seem like they have a fairly logical case for why they are useful for parameter estimation. Is any one method better than the…
Christopher Aden
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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 $\frac{1}{\theta}$ and compute its variance I have learned about two such…
Remy
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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 $(X_1,X_2,\cdots,X_n)$ is…
StubbornAtom
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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 previously known or not? If $T$ is complete for…
user39756
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Find the joint distribution of $X_1$ and $\sum_{i=1}^n X_i$

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th version question 7.6.7. The problem is : Let a random sample of size $n$ be taken from a distribution with the pdf…
Deep North
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Find the unique MVUE

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th Version problem 7.4.9 at page 388. Let $X_1,...,X_n$ be iid with pdf $f(x;\theta)=1/3\theta,-\theta0$. (a) Find the mle…
Deep North
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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 Poisson distribution is an exponential family…
Remy
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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 $\mathbf{\hat{b}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}$ is the BLUE. Is it also the UMVUE? I can…
Patrick
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What is the necessary condition for a unbiased estimator to be UMVUE?

According to the Rao-Blackwell theorem, if statistic $T$ is a sufficient and complete for $\theta$, and $E(T)=\theta$, then $T$ is a uniformly minimum-variance unbiased estimator (UMVUE). I am wondering how to justify that an unbiased estimator is…
Alex Brown
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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 each have the cumulative distribution function…
BonnieKlein
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Sufficiency in Lehmann Scheffe

We are wondering what sufficiency in the Lehmann Scheffe Theorem is needed for. Our reasoning was: If an unbiased estimator is uncorrelated with all unbiased estimators of 0, it is UMVUE If the estimator is from a complete family, it is…
user1587692
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UMVUE for normal distribution $\sigma$

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where $\bar X$ is the sample mean and $S^2$ is the…
clarkson
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Need help finding UMVUE for a Poisson Distribution

My fellow classmates and I are stuck on a homework problem that is a three part problem to find the UMVUE of a Poisson distribution. The problem goes like this: Let X ~ Pois$(\lambda$), and we want to estimate $\theta=e^{-\lambda}$. a) Find the…
Perdue
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