Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem ties together completeness, sufficiency, uniqueness, and best unbiased estimation. 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 model. Optimal equivariant estimators can then be derived for loss functions that are invariant.

References

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