Questions tagged [lehmann-scheffe]

I am currently trying to learn the two related concepts of the Rao-Blackwell theorem and the Lehmann-Scheffé theorem. Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. We have that $E[S^2] =…

Take the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. Now assume the $X_i$ are Poisson random variables with parameter $\lambda$. I am told that the Lehmann-Scheffè theorem directly implies the identity $E[S^2…

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 $X_{(1)}=\text{min}_{1\leq i\leq n}X_i$. Find…

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 for a, we need to find g(T) such that, E(g(T)) =…