A function of observed data that captures all information in the sample relevant to inference about a specified model parameter: conditional on the statistic, the sample provides no further information about the parameter.

Principle

Data reduction without loss for a parameter is achieved when likelihoods factor into a part depending on the statistic and the parameter and a part independent of the parameter; the statistic is sufficient for that parameter.

Demonstration

For a Gaussian model with known variance, the sample mean is a sufficient statistic for the population mean because the likelihood depends on the data only through that mean.

Misapplication

Treating a statistic as sufficient across different model families or for different parameters; assuming a statistic is sufficient because it is low-dimensional without verifying the factorization condition.

Consequence

Correct identification yields lossless compression for parameter inference, enables minimal sufficient statistics and exact reductions of likelihood-based procedures.

Reversal

An ancillary statistic contains no information about the parameter; conditioning on it does not improve estimation of the parameter.

Boundary

Sufficiency is model-dependent: a statistic may be sufficient for one parametric family or parameter and not for another; it presumes a specified probability model and parameterization.

Semantic Tension

Sufficiency versus completeness or minimality: a sufficient statistic need not be minimal or complete, and those refinements require extra conditions.

Synthesis

A sufficient statistic is the model-specific data summary that preserves all inference-relevant information about a given parameter, enabling lossless reduction of the sample for that inferential task.