Question: Let \({{Y}_{1}},....,{{Y}_{n}}\sim Exp\left( \theta \right)\), so

1. Find the distribution of the smallest order statistics \(Y\left( 1 \right)=\underset{1\le i\le n}{\mathop{\min }}\,{{Y}_{i}}\).
2. Calculate \(E\left( Y\left( 1 \right) \right)\) and \(\operatorname{var}\left( Y\left( 1 \right) \right)\)
3. Let \({{\hat{\theta }}_{1}}=cY\left( 1 \right)\), for some constant c . Find a value of c such that \(E\left( {{{\hat{\theta }}}_{1}} \right)=\theta \)
4. Let \({{\hat{\theta }}_{2}}=\bar{Y}\), where \(\bar{Y}=\frac{1}{n}\left( {{Y}_{1}}+...+{{Y}_{n}} \right)\). Calculate \(E\left( {{{\hat{\theta }}}_{2}} \right)\) and \(\operatorname{var}\left( {{{\hat{\theta }}}_{2}} \right)\)
5. The goal is to find the estimator with the minimum mean square error (MSE). Calculate and compare the MSE of the two estimators \({{\hat{\theta }}_{1}}\) and \({{\hat{\theta }}_{2}}\) from parts (c) and (d).

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