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### Exponential family where set of natural parameters has empty interior

In my math-stat class we have a theorem that goes: Let $\{P_\theta : \theta \in \Theta\}$ be a $k$ parameter exponential family (i.e. the density of a member of this family can be written as ...
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### Function of an efficient estimator?

Let $T$ be an efficient estimator for a parameter $\theta$. Is $g(T)$ (with $g(·)$ not linear) an efficient estimator for $g(\theta)$? Is $g(T)$ an minimum-variance unbiased estimator (UMVUE) for ...
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### Improving an unbiased estimator using the Rao-Blackwell theorem [closed]

Given data following a Bernoulli distribution $Y_1,...Y_n \sim B(1,p)$, we want to measure $\theta = \text{Var}(Y_1)$ I found this unbiased estimator $S^2 = \frac{\sum{(y_i-\bar{y}})^2}{n-1}$ I need ...
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### $X \sim$ Geometric$(p)$, $Y \sim$ NegativeBinomial$(2,p)$: find the MVUE for $p(1-p)$

Let $X$ and $Y$ be independent random variables with pdf \begin{align} f(x)&=p(1-p)^x,x=0,1,2,\ldots \\ f(y)&=(1+y)p^2(1-p)^y, y=0,1,2,\ldots \end{align} Find the MVUE for $$p(1-p).$$ I ...
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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 ...
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If x$_{1}$, x$_{2}$,...,x$_{n}$ is sampled from N($\theta$$_{1}, \theta$$_{2}$), how can I prove that E [(x$_{1}$ - $\theta$$_{1})^{4}] = 3\theta$$_{2}$$^{2}? I started off this question ... 1answer 85 views ### Finding UMVUE of parameter \displaystyle a^{-\theta}, (0<a<1) Suppose X_1,X_2,\ldots,X_n is a random sample of \operatorname{Beta}(\theta,1). How can I find UMVUE of parameter \displaystyle a^{\theta}, (0<a<1)? 0answers 85 views ### Function of an efficient estimator Let T be an efficient estimator for a parameter \theta. Let g(\cdot) be a (real valued) non-linear function. Is g(T) an UMVUE for g(\theta)? 2answers 194 views ### Sufficiency in Lehmann Scheffe We are wondering what sufficiency in the Lehmann Scheffe Theorem is needed for. Our reasoning was: If an unbiased estimator is uncorrelated with all unbiased estimators of 0, it is UMVUE If the ... 1answer 257 views ### Obtain the minimum variance unbiased estimators Let X_1, X_2,\cdots,X_n be a random sample from the distribution with the p.d.f$$f(x)=\frac{1}{\beta -\alpha},\alpha<x<\beta$$where$0<\alpha<\beta<\infty$. Obtain the minimum ... 0answers 416 views ### Uniform minimum variance unbiased estimator Let$X_1, X_2, ..., X_n$be an iid random sample from a Poisson$(\lambda)$distribution: a) find the UMVUE of$\theta$=$\lambda^k$for$k > 0$a known integer b) find the UMVUE of$\tau = ...
Here is a basic question that perhaps has a simple answer, but one that I was not able to find by quickly scanning the literature. Suppose that I have a collection of $n$ unopened boxes. Each box ...