Shape Constrained Inference: Outline, Bibliographies, and Review
First draft outline: 27 February 2010.
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 miss-specification
- 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)
- sub-optimality properties (high dimensions)
- Optimality of estimators from outside the shape-constrained
- 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.
- Active set algorithms
- Interior point methods
- Available R packages
- Avaliable code (other languages).
- Applications of shape-constrained 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
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