## on completeness

Posted in Books, Kids, Statistics with tags , , , , , , on November 19, 2020 by xi'an

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..?

## absurdly unbiased estimators

Posted in Books, Kids, Statistics with tags , , , , , , , on November 8, 2018 by xi'an

“…there are important classes of problems for which the mathematics forces the existence of such estimators.”

Recently I came through a short paper written by Erich Lehmann for The American Statistician, Estimation with Inadequate Information. He analyses the apparent absurdity of using unbiased estimators or even best unbiased estimators in settings like the Poisson P(λ) observation X producing the (unique) unbiased estimator of exp(-bλ) equal to

$(1-b)^x$

which is indeed absurd when b>1. My first reaction to this example is that the question of what is “best” for a single observation is not very meaningful and that adding n independent Poisson observations replaces b with b/n, which gets eventually less than one. But Lehmann argues that the paradox stems from a case of missing information, as for instance in the Poisson example where the above quantity is the probability P(T=0) that T=0, when T=X+Y, Y being another unobserved Poisson with parameter (b-1)λ. In a lot of such cases, there is no unbiased estimator at all. When there is any, it must take values outside the (0,1) range, thanks to a lemma shown by Lehmann that the conditional expectation of this estimator given T is either zero or one.

I find the short paper quite interesting in exposing some reasons why the estimators cannot find enough information within the data (often a single point) to achieve an efficient estimation of the targeted function of the parameter, even though the setting may appear rather artificial.