on completeness

Another X validated question that proved a bit of a challenge, enough for my returning to its resolution on consecutive days. The question was about the completeness of the natural sufficient statistic associated with a sample from the shifted exponential distribution

f(x;\theta) = \frac{1}{\theta^2}\exp\{-\theta^{-2}(x-\theta)\}\mathbb{I}_{x>\theta}

[weirdly called negative exponential in the question] meaning the (minimal) sufficient statistic is made of the first order statistic and of the sample sum (or average), or equivalently

T=(X_{(1)},\sum_{i=2}^n \{X_{(i)}-X_{(1)}\})

Finding the joint distribution of T is rather straightforward as the first component is a drifted Exponential again and the second a Gamma variate with n-2 degrees of freedom and the scale θ². (Devroye’s Bible can be invoked since the Gamma distribution follows from his section on Exponential spacings, p.211.) While the derivation of a function with constant expectation is straightforward for the alternate exponential distribution

f(x;\theta) = \frac{1}{\theta}\exp\{-\theta^{-1}(x-\theta)\}\mathbb{I}_{x>\theta}

since the ratio of the components of T has a fixed distribution, it proved harder for the current case as I was seeking a parameter free transform. When attempting to explain the difficulty on my office board, I realised I was seeking the wrong property since an expectation was enough. Removing the dependence on θ was simpler and led to

\mathbb E_\theta\left[\frac{X_{(1)}}{Y}-\frac{\Gamma(n-2)}{\Gamma(n-3/2)}Y^\frac{-1}{2}\right]=\frac{\Gamma(n-2)}{n\Gamma(n-1)}

but one version of a transform with fixed expectation. This also led me to wonder at the range of possible functions of θ one could use as scale and still retrieve incompleteness of T. Any power of θ should work but what about exp(θ²) or sin²(θ³), i.e. functions for which there exists no unbiased estimator..?

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