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Suppose we have a random sample $\textbf{X}=(X_1,...,X_n)$ from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \end{matrix}\right.$

How does one find the linear estimator of the form $\hat{\theta}=a_0+\sum_{i=1}^{n}a_iX_i$ with the smallest mean square error? Is it right to assume that this estimator is unbiased?

I have calculated $E[X_i]=\theta+1$ and $Var[X_i]=1$. If we are assuming this estimator to be unbiased, then we would have $$MSE[\hat{\theta}]=Var[\hat{\theta}]=\sum_{i=1}^{n}a_i^{2}$$

I am trying to compute the MSE so this is the equation I would need to minimize. With the unbiased assumption, then we can say that $$E[\hat{\theta}]=a_0+(\theta+1)\sum_{i=1}^{n}a_i=\theta$$

Am I right to use this above equation as the constraint of a Lagrange multiplier problem? Should the constraint be $\sum_{i=1}^{n}a_i =\frac{\theta-a_0}{\theta+1}$?

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  • $\begingroup$ Do you mean an unbiased estimator of $1+\theta$ or of $\theta$? As one is unbiased and the other not. More generally, I do not understand the question since there exists a minimal sufficient statistic in that case and hence a UMVUE. $\endgroup$ – Xi'an May 28 at 12:03
  • $\begingroup$ An estimator of the location parameter $\theta$ with the smallest mean square error. $\endgroup$ – dsakiocxla May 28 at 12:14
  • $\begingroup$ How do I know if that is in the form I have specified? I need the estimator of that form with the smallest MSE. $\endgroup$ – dsakiocxla May 28 at 12:44
  • $\begingroup$ This is a self-study question, so please add the tag and develop further what you have tried to solve this question. $\endgroup$ – Xi'an May 28 at 13:03

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