Suppose that $T$, the time to failure of a component is exponentially distributed. That is, the pdf for $T$ is $${f(x; θ) = \frac{1} {θ} e ^{\textbf{-}x/θ} , 0 ≤ x < ∞}$$. Suppose we test n components and record the failure times $T_1, . . . , T_n$.
(a) Show that $\hatθ = \overline{T} = n^{-1}\sum_{i=1}^{\infty} T_i$ is an unbiased estimator of $θ$.
(b) What is the variance of $\hatθ$?
(c) It can be shown that $Z = \min(T_1, . . . , T_n)$ also has an exponential distribution with parameter $θ/n$. That is $$g(z; θ) = \frac{1} {θ/n}e^ {−z/(θ/n)}$$ (Do not show this). Use this to find another unbiased estimator $\tilde{θ}$ of $θ$ and determine its variance.
For part (a): I showed $E (\hatθ) = E(\theta)$ through rules regarding addition of expected values.
For part (b): Through independence, I showed that $var(\hatθ) = \frac{θ^2}{n}$ (Here I assumed independence which I hope is correct. I'm also not entirely sure that I've done this question correctly).
But I'm having trouble with part (c). I know how to show something is an unbiased estimator but I'm unsure how to find an unbiased estimator. Would appreciate any help.
For part (c), you can apply Maximum Likelihood Method to the only data , namely $z=\min(t_1,\cdots,t_n)$. Then you will get $\hat{\theta}=nZ$, which is unbiased.