Xi'an's Og

Another intriguing question on X validated (about an exercise in Jun Shao’s book) that made me realise a basic fact about exponential distributions. When considering two Exponential random variables X and Y with possibly different parameters λ and μ,  Z⁺=max{X,Y} is dependent on the event X>Y while Z⁻=min{X,Y} is not (and distributed as an Exponential variate with parameter λ+μ.) Furthermore, Z⁺ is distributed from a signed mixture

conditionally on the event X>Y, meaning that there is no sufficient statistic of fixed dimension when given a sample of n realisations of Z⁺’s along with the indicators of the events X>Y…. This may explain why there exists an unbiased estimator of λ⁻¹-μ⁻¹ in this case and (apparently) not when replacing Z⁺ by Z⁻. (Even though the exercise asks for the UMVUE.)

This entry was posted on March 26, 2022 at 12:22 am and is filed under Books, Kids, Statistics with tags cross validated, exercises, exponential distribution, exponential family, homework, mathematical statistics, order statistics, textbook, unbiasedness. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.