Editing Lehmann–Scheffé theorem (section)

== Example for when using a non-complete minimal sufficient statistic ==

An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is '''not complete''', was provided by Galili and Meilijson in 2016.<ref>{{cite journal|title= An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator |author1=Tal Galili |author2=Isaac Meilijson | date = 31 Mar 2016 | journal = The American Statistician | volume = 70 | issue = 1 | pages = 108–113 |doi=10.1080/00031305.2015.1100683| pmc = 4960505 | pmid=27499547}}</ref> Let <math>X_1, \ldots, X_n</math> be a random sample from a scale-uniform distribution <math>X \sim U ( (1-k) \theta, (1+k) \theta),</math> with unknown mean <math>\operatorname{E}[X]=\theta</math> and known design parameter <math>k \in (0,1)</math>. In the search for "best" possible unbiased estimators for <math>\theta</math>, it is natural to consider <math>X_1</math> as an initial (crude) unbiased estimator for <math>\theta</math> and then try to improve it. Since <math>X_1</math> is not a function of <math>T = \left( X_{(1)}, X_{(n)}  \right)</math>, the minimal sufficient statistic for <math>\theta</math> (where <math>X_{(1)} = \min_i X_i </math> and <math>X_{(n)} = \max_i X_i </math>), it may be improved using the Rao–Blackwell theorem as follows: 

:<math>\hat{\theta}_{RB} =\operatorname{E}_\theta[X_1\mid X_{(1)}, X_{( n)}] = \frac{X_{(1)}+X_{(n)}} 2.</math>

However, the following unbiased estimator can be shown to have lower variance: 

:<math>\hat{\theta}_{LV} = \frac 1 {k^2\frac{n-1}{n+1}+1} \cdot \frac{(1-k)X_{(1)} + (1+k) X_{(n)}} 2.</math> 

And in fact, it could be even further improved when using the following estimator:  

:<math>\hat{\theta}_\text{BAYES}=\frac{n+1} n \left[1- \frac{\frac{X_{(1)} (1+k)}{X_{(n)} (1-k)}-1}{ \left (\frac{X_{(1)} (1+k)}{X_{(n)} (1-k)}\right )^{n+1} -1} \right] \frac{X_{(n)}}{1+k}</math>

The model is a [[Scale parameter|scale model]]. Optimal [[Equivariant Estimator|equivariant estimators]] can then be derived for [[loss function]]s that are invariant.<ref>{{Cite journal|last=Taraldsen|first=Gunnar|date=2020|title=Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comment|url=https://doi.org/10.1080/00031305.2020.1769727|journal=The American Statistician|volume=74|issue=3|pages=315|doi=10.1080/00031305.2020.1769727|s2cid=219493070 |issn=|url-access=subscription}}</ref>