Shape Constrained Inference: Outline, Bibliographies, and Review

Shape constraints: definitions and starting points.
 Shape constraints: statistical models.
 Density estimation on \RR^+
 Density estimation on \RR
 regression function estimation on \RR
 hazard rate estimation on \RR^+
 mass function estimation on \NN
 mass function estimation on \ZZ
 hazard function estimation on \NN
 density estimation on \RR^d
 problems involving interval censored data
 semiparametric models with shape constraints
 ``white noise'' models and ``canonical'' Gaussian problems
 Shape constraints: approaches to estimation.
 Maximum likelihood
 Penalized maximum likelihood
 Least squares and other minimum contrast estimators
 Bayes estimation
 behavior of estimators under model missspecification
 rearrangement methods
 taut string methods
 approaches via splines
 Shape constraints: theory of estimators.
 Minimax lower bounds
 Global lower bounds
 Local lower bounds
 Maximum likelihood estimation.
 Optimality properties (low dimensions)
 suboptimality properties (high dimensions)
 Optimality of estimators from outside the shapeconstrained
classes
 Theory for Bayes estimation
 Functionals of the estimators
 Smooth functionals
 Mode estimation
 Contour set estimation
 Shape constraints: inference beyond estimation.
 Testing and confidence sets (within the shape constrained class)
 Testing for a given type of shape constraint
 Testing shape against a simpler smaller model
 Shape constraints: computation and algorithms.
 EM
 Active set algorithms
 Interior point methods
 Available R packages
 Avaliable code (other languages).
 Applications of shapeconstrained estimation.
 Applications: ``pure'' shape constraints
 Use with semiparametric models
 Use in other connections (e.g. clustering)
 Examples of applications
 Shape constraints: some open problems.
 Shape constraints and restrictions: some math background
 Some convex analysis
 Facts and principles from optimization theory
 Empirical process theory tools
First draft outline: 27 February 2010.
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