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 Examples section below. This property (independence of sample mean and sample variance) characterizes normal distributions.

Statement

Let P_θ be a family of distributions on a measurable space (X, Σ). Then if T is a boundedly complete sufficient statistic for θ, and A is ancillary to θ, then T is independent of A.

Proof

Let P_θ^T and P_θ^A be the marginal distributions of T and A respectively.

Define B here and the meaning of A^{-1} B .....

$P_\theta^A(B) = P_\theta (A^{-1} B) = \int_{T(X)} P_\theta(A^{-1}B | T=t) \ P_\theta^T (dt) \,$

The P_θ^A does not depend on θ because A is ancillary. Likewise, P_θ(·|T = t) does not depend on θ because T is sufficient. Therefore:

$\int_{T(X)} \big[ P(A^{-1}B | T=t) - P^A(B) \big] \ P_\theta^T (dt) = 0 \,$

Note the integrand (the function inside the integral) is a function of t and not θ. Therefore, since T is boundedly complete:

$P(A^{-1}B | T=t) = P^A(B) \quad \text{for all }t\,$

Therefore, A is independent of T.

Example

Independence of sample mean and sample variance of a normal distribution

Let X₁, X₂, ..., X_n be independent, identically distributed normal random variables with mean μ and variance σ².

Then with respect to the parameter μ, one can show that

$\widehat{\mu}=\frac{\sum X_i}{n},\,$

the sample mean, is a complete sufficient statistic – it is all the information one can derive to estimate μ, and no more – and

$\widehat{\sigma}^2=\frac{\sum \left(X_i-\bar{X}\right)^2}{n-1},\,$

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

1. Basu (1955)
2. Ghosh, Malay; Mukhopadhyay, Nitis; Sen, Pranab Kumar (2011), Sequential Estimation, Wiley Series in Probability and Statistics 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 ...
3. 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.

References

• Basu, D. (1955). "On Statistics Independent of a Complete Sufficient Statistic". Sankhyā 15 (4): 377–380. JSTOR 25048259. MR 74745. 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 (Boston: American Statistical Association) 52 (3): 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.