# Minimizing Mean Square Error

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 $$\sum_{i=1}^{n}a_i =\frac{\theta-a_0}{\theta+1}$$?

• 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. – Xi'an May 28 at 12:03
• An estimator of the location parameter $\theta$ with the smallest mean square error. – dsakiocxla May 28 at 12:14
• How do I know if that is in the form I have specified? I need the estimator of that form with the smallest MSE. – dsakiocxla May 28 at 12:44
• This is a self-study question, so please add the tag and develop further what you have tried to solve this question. – Xi'an May 28 at 13:03