Statistical models, influence functions, and efficiency

Local changes in a target and the information permitted by a model

A statistical model specifies which distributions and local changes are allowed. A target is a functional on that model. An influence function can represent how the target changes along permitted directions; a separate estimator argument connects that derivative to a sample expansion.

The complete-data mean provides the first example: its centered contribution is both a distributional derivative and the term controlling the sample mean’s uncertainty. Conditional-expectation geometry and likelihood scores then explain how model restrictions affect the possible representations and their variance. A mean with known population stratum weights makes the variance gain concrete before the partially linear regression example.

What you’ll learn

• Represent local target changes: construct valid score paths and derive centered influence functions for means, variances, and ratios.
• Understand model-relative information: distinguish score residualization from gradient projection, and explain which representation gives the regular-estimation variance bound.

Completion means deriving an elementary influence function with its path and moment assumptions, then explaining the projection and variance comparison in a stated model. SP-02–SP-05 and SP-07–SP-09 develop these core skills. SP-06 is the completion target for the quantile extension, which uses a stated probability lemma. The survival page and all additional practice are optional.