Basu's theorem
In statistics, Basu's theorem states that any boundedly complete sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu.[1]
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.[2] An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the Example section below. This property (independence of sample mean and sample variance) characterizes normal distributions.
Statement
Let be a family of distributions on a measurable space and measurable maps from to some measurable space . (Such maps are called a statistic.) If is a boundedly complete sufficient statistic for , and is ancillary to , then is independent of .
Proof
Let and be the marginal distributions of and respectively.
Denote by the preimage of a set under the map . For any measurable set we have
The distribution does not depend on because is ancillary. Likewise, does not depend on because is sufficient. Therefore
Note the integrand (the function inside the integral) is a function of and not . Therefore, since is boundedly complete the function
is zero for almost all values of and thus
for almost all . Therefore, is independent of .
Example
Independence of sample mean and sample variance of a normal distribution (known variance)
Let X1, X2, ..., Xn be independent, identically distributed normal random variables with mean μ and variance σ2.
Then with respect to the parameter μ, one can show that
the sample mean, is a complete sufficient statistic – it is all the information one can derive to estimate μ, and no more – and
the sample variance, is an ancillary statistic – its distribution does not depend on μ.
Therefore, from Basu's theorem it follows that these statistics are independent.
This independence result can also be proven by Cochran's theorem.
Further, this property (that the sample mean and sample variance of the normal distribution are independent) characterizes the normal distribution – no other distribution has this property.[3]
Notes
- ^ Basu (1955)
- ^ Ghosh, Malay; Mukhopadhyay, Nitis; Sen, Pranab Kumar (2011), Sequential Estimation, Wiley Series in Probability and Statistics, vol. 904, John Wiley & Sons, p. 80, ISBN 9781118165911,
The following theorem, due to Basu ... helps us in proving independence between certain types of statistics, without actually deriving the joint and marginal distributions of the statistics involved. This is a very powerful tool and it is often used ...
- ^ Geary, R.C. (1936). "The Distribution of the "Student's" Ratio for the Non-Normal Samples". Supplement to the Journal of the Royal Statistical Society. 3 (2): 178–184. doi:10.2307/2983669. JFM 63.1090.03. JSTOR 2983669.
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References
- Basu, D. (1955). "On Statistics Independent of a Complete Sufficient Statistic". Sankhyā. 15 (4): 377–380. JSTOR 25048259. MR 0074745. Zbl 0068.13401.
- Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. Statistics: A Series of Textbooks and Monographs. 162. Florida: CRC Press USA. ISBN 0-8247-0379-0.
- Boos, Dennis D.; Oliver, Jacqueline M. Hughes (Aug 1998). "Applications of Basu's Theorem". The American Statistician. 52 (3). Boston: American Statistical Association: 218–221. doi:10.2307/2685927. JSTOR 2685927. MR 1650407.
- Ghosh, Malay (October 2002). "Basu's Theorem with Applications: A Personalistic Review". Sankhyā: the Indian Journal of Statistics, Series A. 64 (3): 509–531. JSTOR 25051412. MR 1985397.