The Basu's theorem are three statements of mathematical statistics that a connection between the sufficiency , the completeness and the distribution of free produce.
They were established and proven in 1955 by Debabrata Basu .
sentences
For all sentences there is always a statistical model with a basic set , σ-algebra and distribution class . In addition, let sub- σ-algebras of .
![(\ Omega, \ mathcal A, \ mathcal P)](https://wikimedia.org/api/rest_v1/media/math/render/svg/38cfacb98ecdaf3e55a8d5f3060473578107ce91)
![\ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![{\ mathcal P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6)
![\ mathcal C, \ mathcal D](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f481f9a18d87845978613e8af508a6d753d40b)
![\ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
Relationship sufficiency, completeness and freedom of distribution
Is
![f: (\ Omega, \ mathcal A) \ to (X, \ mathcal X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf656e96da65d1ce6ab2cc0a6953490e70a560c)
a non-distribution statistic and is
![g: (\ Omega, \ mathcal A) \ to (Y, \ mathcal Y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4fa3bcc4faace6224ceea31fc84a5f6ed9d081)
a sufficient and limited complete statistic , then and for all are stochastically independent random variables .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
Freedom of distribution and independent sufficient σ-algebras
A lot of exist for everyone , so that
![\ underline P, \ overline P \ in \ mathcal P](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8277ea13d20d688f29350bee3d0c459f0dd0aae)
![(P_i) _ {i = 1, \ dots, n} \ in \ mathcal P](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6707eed2d357892869063dfe1c197f5188c8946)
and and are not singular to each other. If and are stochastically independent σ-algebras for all and is a sufficient σ-algebra , then is a distribution-free σ-algebra .
![Pi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3)
![\ mathcal C](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019)
![P \ in \ mathcal P](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb727fbfb3d66abc8c9a7e7b649323f87b85a98d)
![\ mathcal C](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019)
![\ mathcal D](https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966)
Sufficiency of maximum supplements
Be stochastically independent for everyone and be free of distribution. If then , then is sufficient.
![\ mathcal C, \ mathcal D](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f481f9a18d87845978613e8af508a6d753d40b)
![P \ in \ mathcal P](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb727fbfb3d66abc8c9a7e7b649323f87b85a98d)
![\ mathcal D](https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966)
![\ sigma (\ mathcal C, \ mathcal D) = \ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/4768098b6c2363570bdc980c1fc9361bf271d0c7)
![\ mathcal D](https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966)
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