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Assume $X_1, X_2, X_3,\ldots,X_n$ are i.i.d. samples from Exp($\lambda$). Assume that the integer $k<n$, is it possible to find a an unbiased estimator for $\lambda$ from the k-th smallest ordered samples? I know it is possible to derive the an expression for the distribution of the order statistics, but I need an estimator for the original distribution parameter.

Anas Alhashimi
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  • By "k-th smallest ordered samples" would you mean all $k$ of the smallest values among the $X_i,$ the (single) $k^\text{th}$ order statistic, or something else? – whuber May 21 '21 at 12:53
  • yes from all samples that are smaller than $X_k$ – Anas Alhashimi May 21 '21 at 13:00
  • That's not the same as any of the interpretations I proposed, so let's clarify: by "$X_k$" do you mean the $k^\text{th}$ smallest value and by "smaller than" do you mean "less than or equal to"? – whuber May 21 '21 at 13:35
  • Yes exactly. It is my bad, it should be "less than or equal to". thanks – Anas Alhashimi May 21 '21 at 13:43
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    This observation scheme is called Type II censoring in lifetime analysis. – Yves May 21 '21 at 13:48
  • So I believe that this is nearly the same as this question. – Yves May 21 '21 at 14:00
  • Thanks Yves, it is an interesting answer! though, from the simulations, I was expecting 2^0.5 to appear in the answer. any idea how we obtained the the joint probability density? – Anas Alhashimi May 21 '21 at 14:10
  • The density is the product of the density for the observed values and of the Survival function for the censored values. Of related interest is the Rényi's representation for the order statistics of a sample from the exponential. – Yves May 21 '21 at 14:37

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