Statistics: Sufficient Statistics Overview

Sufficient Statistics

• A statistic T(X) is sufficient for a parameter θ if the conditional distribution of X given T(X) = t does not depend on θ.
• Sufficient statistics summarize all the information in a sample that is relevant to a parameter.
• They follow the invariance property.
• Necessary and sufficient conditions for a statistic to be sufficient:
  • For a family of distributions f(x; θ), the ratio of any two densities p₂(x)/p₁(x) is a function of T(x) only.
  • Factorization Theorem: P(x; θ) = h(x)g(θ, T(x)) where h(x) does not depend on θ. The statistic T(X) will be sufficient for θ.
• A statistic might not be sufficient if T(x) = T(y) and f(x, θ₁) f(y, θ₂) = f(x, θ₂) f(y, θ₁).
• If a sufficient statistic T exists for θ, any linear combination of T is also sufficient for θ.
• If a statistic based only on a part of the sample instead of the entire sample is sufficient, it is not a minimal sufficient statistic.
• A minimal sufficient statistic is not unique.

Important Properties

• The dimension of a sufficient statistic will always be less than the sample size (n).
• If a distribution belongs to an exponential family, all results about sufficiency, minimal sufficiency, completeness, and best unbiased estimators hold true.
• An ancillary statistic is a statistic whose distribution does not depend on the parameter θ. (e.g. E(statistic) is independent of θ.) Ancillary statistics contain no information about θ.
• If a statistic has no ancillary information, it has reached maximum reduction.
• A complete statistic has the property that if E[h(T)] = 0, then h(T) = 0. -A complete sufficient statistic is a minimal sufficient statistic that is also complete.
  • The converse is not true.
• Based on Basu's theorem, if a complete sufficient statistic exists, there is only one unbiased estimator that is a function of this statistic.

Tables for Specific Distributions

• A table summarizing sufficient, minimal sufficient, ancillary, and complete statistics for several distributions (e.g., binomial, Poisson, geometric, normal) is provided. The table shows which statistics are sufficient, minimal sufficient, ancillary. and complete. This is critical detail for understanding sufficiency properties of various distributions.

This quiz covers the concept of sufficient statistics in the context of statistical inference. Key topics include the definition, invariance property, and the conditions for a statistic to be sufficient. Test your understanding of the Factorization Theorem and minimal sufficient statistics.