Some puzzles appearing in statistical inference

Rao-Blackwell theorem is widely known to be a mathematically powerful technique that can be used to improve the precision of an estimator. The procedure entails exploiting a sufficient statistic to obtain an improved estimator or a uniformly minimum variance unbiased estimator. A modification of sufficient statistics is introduced here which can be applied for Rao-Blackwell theorem along with some fruitful applications that illustrate its properties. Also some theorems have been rewritten in statistical inference.