For > 0 consider a random sample from a uniform distribution on the interval from to 2, and use the factorization theorem to

For θ > 0 consider a random sample from a uniform distribution on the interval from θ to 2θ, and use the factorization theorem to determine a sufficient statistic for θ.

The Factorization Theorem How can a sufficient statistic be identified? It may seem as though one would have to select a statistic, determine the conditional distribution of the Xi's given any particular value of the statistic (no easy task look at the last example!), and keep doing this until hitting paydirt by finding one that satisfies the defining condition. This would be terribly time-consuming, and when the Xi's are continuous there are additional technical difficulties in obtaining the relevant conditional distribution. Fortu- nately, the next result provides a relatively straightforward way of proceeding.