Let $(X_1;X_2;...;X_n)$ be a sample from exponential distribution with the unknown parameter $\lambda$ > 0. Show that the statistic $T_n(X_1;X_2;...;X_n) = nX_{1:n}$ is not consistent estimator of $\frac{1}{\lambda}$.

I see how it may work, because

$\mathbb{P}(nX_{1:n} \le t) = 1-e^{-\lambda t}$, so $T_n \to exp(\lambda)$ in distribution and nothing changes when $n$ is increasing, but I dont know how to show it formally.

  • $\begingroup$ In the last equality the left side does not involve t. It seems that there is some thing wrong in the calculation. $\endgroup$ – Janak Nov 15 '14 at 19:02
  • $\begingroup$ To show it formally we need to find an $\epsilon>0$, such that the following holds $P(|nX_{1:n}-\frac{1}{\lambda}|\geq\epsilon)\nrightarrow0$. $\endgroup$ – Janak Nov 15 '14 at 19:19

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