- Basu's theorem
It is often used in statistics as a tool to prove independence of two statistics, by first demonstrating one is complete sufficient and the other is ancillary, then appealing to the theorem.
Independence of sample mean and sample variance
Let "X"1, "X"2, ..., "X""n" be independent, identically distributed normal
random variables with mean"μ" and variance"σ"2.
Then with respect to the parameter "μ", one can show that
the sample mean, is a complete sufficient statistic, and
the sample variance, is an ancillary statistic.
Therefore, from Basu's theorem it follows that these statistics are independent.
*Basu, D., "On Statistics Independent of a Complete Sufficient Statistic," "Sankhya", Ser. A, 15 (1955), 377-380
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