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Topics in Statistical Theory - Coggle Diagram
Topics in Statistical Theory
Basic Concepts
Parametric vs nonparametric models
Estimating an arbitrary distribution function
Glivenko-Cantelli Theorem
DKW Inequality
Uniform Glivenko-Cantelli Theorem
Order statistics and quantiles
Distribution of \(U_{(j)}\)
Asymptotics
Generalised to general densities
Concentration inequalities
sub-Gaussianity
Propositions
Hoeffding's Ineqeuality
Hoeffding's Lemma
Corollary
Bennett's Inequality
sub-Gamma
Proposition 17
Bernstein's Inequality
Kernel Density Estimation
Introduction
Bin-size, bin positions
Univariate kernel density estimator
Kernel
Bandwidth
Mean squared error of kernel estimators
MSE
MISE
Bounds on the variance and bias
Bounds on Integrated variance and bias
Bandwith selection
Least Squares Cross Validation
Choice of kernel
Epanechnikov kenrel
Multivariate density estimation
Nonparametric Regression
Fixed and random design
Fixed design
homo/heteroscedastic
Random Design
Local polynomial estimators
Splines
Cubic Splines
Natural cubic smoothing splines
Automatic choice of smoothing parameter
Minimax Lower Bounds
Reduction to testing
Lebesgue decomposition theorem, f-divergences
Le Cam's two point lemma
Assouad's lemma
Data processing inequality
Fano's lemma
Application