Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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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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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 $*$ $e^{n(\... 1answer 119 views Proving the MVUE is the following I am stuck on the following question and I was wondering if can get some help. Let$f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$with$a(\theta)$decreases and$b(\theta)$... 1answer 66 views UMVUE of$\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$? 1answer 72 views Find UMVUE of$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$. For ... 1answer 103 views UMVUE- geometric distribution where$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}$... 0answers 136 views UMVUE for$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 ... 0answers 57 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 $$f_T(t\mid\... 1answer 206 views Does UMVUE of \frac{\theta_{x}}{\theta_{y}} exist? X \sim exp(\theta_{x}), Y \sim exp(\theta_{y}) I have got the following variables.$$ X\sim exp(\theta_{x_{i}}), \ Y \sim exp(\theta_{y_{i}}) $$and want to find the UMVUE of$$ \frac{\theta_{x}}{\theta_{y}}. $$As the complete statistics ... 1answer 30 views approximating gamma function to calculate variance of sigma UMVUE I'm trying to show that variance of \hat{\sigma}^{UMVUE}(which is estimator of \sigma in N(\mu, \sigma^2)) is larger than cramer-rao lower bound, which I have found to be \frac{\sigma^2}{2n}. ... 0answers 150 views Existence of UMVUE of \theta for sample from \small{\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}}? Suppose X_1,X_2,\ldots,X_n is a random sample drawn from a distribution with pdf$$f_{\theta}(x)=\small\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}\quad,\,\theta>1$$Does there ... 1answer 79 views Usefulness of Point Estimators: MVU vs. MLE In a past class, two types of point estimators were introduced: minimum variance unbiased estimators (MVUs) and maximum likelihood estimators (MLEs). Supposedly, the MVU is optimal, unless an unbiased ... 1answer 178 views The UMVUE of ratio of parameters for two uniform distributions, Let X_1,\ldots,X_m be i.i.d. having the uniform distribution U(0, \theta_x) and Y_1,\ldots, Y_n be i.i.d. having the uniform distribution U(0, \theta_y). Suppose that X_i’s and Y_j’s are ... 0answers 123 views Finding the UMVUE of \theta^2 where f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x) Let X_1, X_2, . . . , X_n be iid random variables having pdf$$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$where \theta >0. Give the UMVUE of {\theta^2} I ... 1answer 57 views Finding a UMVU estimator for the mean I am struggling with the following problem. We are given an i.i.d sample of size n, with the form X_{i}=\mu+n_{i}, where \mu is a deterministic unknown constant, and n_{i} is a noise with a ... 2answers 677 views Find the UMVUE of \frac{\mu^2}{\sigma} where X_i\sim\mathsf N(\mu,\sigma^2) Suppose X_1, ..., X_4 are i.i.d \mathsf N(\mu, \sigma^2) random variables. Give the UMVUE of \frac{\mu^2}{\sigma} expressed in terms of \bar{X}, S, integers, and \pi. Here is a ... 1answer 724 views Find UMVUE of \theta where f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x) As a slight modification of my previous problem: Let X_1, X_2, . . . , X_n be iid random variables having pdf$$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$where \... 2answers 808 views Finding UMVUE of \theta e^{-\theta} where X_i\sim\text{Pois}(\theta) Suppose X_1, X_2, . . . , X_n are i.i.d Poisson (\theta) random variables, where \theta\in(0,\infty). Give the UMVUE of \theta e^{-\theta} I found a similar problem here. I have that the ... 1answer 276 views Find UMVUE of \frac{1}{\theta} where f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x) Let X_1, X_2, . . . , X_n be iid random variables having pdf$$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$where \theta >0. Give the UMVUE of \frac{1}{\theta} ... 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 ... 1answer 217 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 ... 2answers 262 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 ... 1answer 673 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,\... 1answer 1k 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 ... 0answers 365 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 ... 2answers 422 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 ... 0answers 587 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) \, . $$We know that \mathbf{\hat{b}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \... 2answers 202 views UMVUE estimates of uniform distribution mean and width Given are the uniformly distributed samples$$x_n \overset{\text{iid}}{\sim} \mathcal{U}\left(\mu-\frac{w}{2}, \mu+\frac{w}{2}\right)$$for n = 1 \ldots N.Then the UMVUE estimates of \mu and w ... 2answers 463 views UMVUE for \theta where X \sim Unif\{1 ,\ldots, \theta\} Say we have X \sim Unif\{1, \ldots , \theta\} and we want to find the uniformly minimum variance unbiased estimator for \theta. My first assumption was X_{(n)}. Which I managed to show is ... 1answer 81 views Is there no UMVUE for this case? Let X_1,X_2,\ldots, X_n be \operatorname{Normal}(\mu,\sigma^2) I seem to recall it said that there is no UMVUE for \mu if \sigma^2 is also unknown but cannot find why this is so. Is this true?... 1answer 191 views Variance of OLS estimator of \theta in y_n = \theta x_n + \eta_n compared to Cramer-Rao From Theodoridis' Machine Learning, problem 3.7: Derive the Cramer-Rao bound for the LS estimator, where the training data result from the model$$y_n = \theta x_n + \eta_n\text{, } \qquad n = 1, ... 1answer 350 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. ... 1answer 859 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$...
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 ...