Let $X_1,\ldots,X_n$ be iid with pdf
$$f(x\mid\theta) = \exp(\theta -x) I(x)_{(\theta, \infty)}$$
It is asked to find an unbiased estimator of $\theta$ that is a function of a sufficient statistic for $\theta$.
By factorization theorem, we show that $X_{(1)}$ is a sufficient statistical for $\theta.$ And, since
$$E(X) = \theta +1$$
the estimator $ \bar{X} -1$ is unbiased. So, by the Rao Blackell theorem,
$$W=E(\bar{X}-1\mid X_{(1)})$$ is an unbiased estimator that is function of the sufficient statistical. But it seems very complicated to evaluate the distribution of $\bar{X}-1\mid X_{(1)}$. How can I find this distribution? Is there a better unbiased estimador that I can use, in this case?
Thanks in advance!
Best Answer
We have $$F_X(x) = \int_{\theta}^{x}e^{\theta -t} dt = -e^{\theta}e^{-t}\Big|^{x}_{\theta} = 1 - e^{\theta -x} $$
Since $F_{X_{(1)}}(x_{(1)}) = 1 -[1-F_X(x_{(1)})]^{n}$, the density function of the minimum order statistic is
$$f_{X_{(1)}}(x_{(1)}) = nf_X(x_{(1)})[1-F_X(x_{(1)})]^{n-1}I(x)_{(\theta, \infty)} = ne^{\theta -x_{(1)}}[e^{\theta -x_{(1)}}]^{n-1}I(x)_{(\theta, \infty)}$$
$$\Rightarrow f_{X_{(1)}}(x_{(1)}) =ne^{n(\theta -x_{(1)})}I(x)_{(\theta, \infty)}$$
Then
and so
is an unbiased estimator based on the sufficient statistic.