# Suppose that Y_1,Y_2,Y_3 denote a random sample from anexponential distribution with density function f(y)=(\frac{1}{\lambda})\cdot e^{\frac{-y}{\theta}},y>0 =0 elsewhere Consider the following five estimators of \theta: \hat{\theta}_1=Y_1,\hat{\theta}_2=\frac{Y_1+Y_2}{2},\hat{\theta}_3=\frac{Y_1+2Y_2}{3},\hat{\theta}_4=\overline{Y} a) Which of these estimators are unbiased? b) Among the unbiased estimators, which has the smallest variance?

Question
Sampling distributions
Suppose that $$\displaystyle{Y}_{{1}},{Y}_{{2}},{Y}_{{3}}$$ denote a random sample from anexponential distribution with density function $$\displaystyle{f{{\left({y}\right)}}}={\left({\frac{{{1}}}{{\lambda}}}\right)}\cdot{e}^{{{\frac{{-{y}}}{{\theta}}}}},{y}{>}{0}$$
=0 elsewhere
Consider the following five estimators of $$\displaystyle\theta$$:
$$\displaystyle\hat{{\theta}}_{{1}}={Y}_{{1}},\hat{{\theta}}_{{2}}={\frac{{{Y}_{{1}}+{Y}_{{2}}}}{{{2}}}},\hat{{\theta}}_{{3}}={\frac{{{Y}_{{1}}+{2}{Y}_{{2}}}}{{{3}}}},\hat{{\theta}}_{{4}}=\overline{{{Y}}}$$
a) Which of these estimators are unbiased?
b) Among the unbiased estimators, which has the smallest variance?

2021-02-03

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### Relevant Questions

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Part B:
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