Sufficiency and Variance Reduction via Rao-Blackwell Theorem | UPSC ISS 2024 Paper-2 | Problem-6

Описание: This video explains two important properties related to a sufficient statistic \(T\) for parameter \(\theta\) and an unbiased estimator \(U\):

1. The function \(\phi(T) = \mathbb{E}[U \mid T]\) is independent of \(\theta\). This follows directly from the factorization theorem and the definition of sufficiency, which states that the conditional distribution of the sample given the sufficient statistic does not depend on \(\theta\).

2. By the Rao-Blackwell theorem, \(\phi(T)\) is an unbiased estimator of the parameter with variance less than or equal to that of \(U\), achieving variance reduction. Equality holds if and only if \(U\) is already a measurable function of \(T\).

These foundational concepts illustrate the power of sufficiency and Rao-Blackwellization in improving estimators, key for statistical inference and highly relevant in UPSC preparation.

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