Mathematics / advanced
Nonparametric statistical estimation
Estimate flexible functions and distributions while balancing smoothness, bias, variance, adaptivity, and uncertainty.
Your route through mathematics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 advanced
Nonparametric statistical estimation
Estimate flexible functions and distributions while balancing smoothness, bias, variance, adaptivity, and uncertainty.
- 01
Empirical distributions and uniform bands
Estimate a complete cumulative distribution and distinguish a simultaneous finite-sample guarantee from a pointwise interval.
34 min - 02
Kernel density estimation and bandwidth
Normalize a density smoother, derive its leading bias and variance, and separate interior asymptotics from boundary behavior.
37 min - 03
Local polynomial regression
Fit weighted local Taylor models, derive reproduction identities, and understand boundary correction without assuming positive prediction weights.
37 min - 04
Orthogonal series estimation
Translate function estimation into noisy coefficient estimation and separate exact Parseval risk identities from approximate regression analogies.
36 min - 05
Smoothing splines and penalized functions
Control curvature directly through a Sobolev penalty and distinguish the infinite-dimensional variational problem from a restricted polynomial calculation.
37 min - 06
Minimax rates over smoothness classes
Match a constructive bias–variance bound with a testing lower-bound argument, while keeping experiment, dimension and loss explicit.
39 min - 07
Lepski adaptation and model selection
Select a resolution by calibrated comparisons, and distinguish a conditional oracle argument from a complete adaptive risk or confidence theorem.
40 min - 08
Rank tests and distribution-free inference
Derive exact rank and sign null distributions from exchangeability, keeping continuity, symmetry and the tested hypothesis distinct.
37 min - 09
U-statistics and Hoeffding decomposition
Separate first-order information from a degenerate pair remainder and identify when the usual square-root-n normal approximation fails.
40 min - 10
Nonparametric confidence bands and honesty
Control smoothing bias and a whole-function coverage event, then distinguish pointwise validity, uniform honesty and adaptation.
41 min