Shape Constrained Inference: Outline, Bibliographies, and Review

  1. Shape constraints: definitions and starting points.

    2.  

  2. Shape constraints: statistical models.

    1. Density estimation on \RR^+
    2. Density estimation on \RR
    3. regression function estimation on \RR
    4. hazard rate estimation on \RR^+
    5. mass function estimation on \NN
    6. mass function estimation on \ZZ
    7. hazard function estimation on \NN
    8. density estimation on \RR^d
    9. problems involving interval censored data
    10. semiparametric models with shape constraints
    11. ``white noise'' models and ``canonical'' Gaussian problems
     

  3. Shape constraints: approaches to estimation.

    1. Maximum likelihood
    2. Penalized maximum likelihood
    3. Least squares and other minimum contrast estimators
    4. Bayes estimation
    5. behavior of estimators under model miss-specification
    6. rearrangement methods
    7. taut string methods
    8. approaches via splines
     

  4. Shape constraints: theory of estimators.

    1. Minimax lower bounds
      • Global lower bounds
      • Local lower bounds
    2. Maximum likelihood estimation.
      • Optimality properties (low dimensions)
      • sub-optimality properties (high dimensions)
    3. Optimality of estimators from outside the shape-constrained classes
    4. Theory for Bayes estimation
    5. Functionals of the estimators
      • Smooth functionals
      • Mode estimation
      • Contour set estimation
     

  5. Shape constraints: inference beyond estimation.

    1. Testing and confidence sets (within the shape constrained class)
    2. Testing for a given type of shape constraint
    3. Testing shape against a simpler smaller model
     

  6. Shape constraints: computation and algorithms.

    1. EM
    2. Active set algorithms
    3. Interior point methods
    4. Available R packages
    5. Avaliable code (other languages).
     

  7. Applications of shape-constrained estimation.

    1. Applications: ``pure'' shape constraints
    2. Use with semiparametric models
    3. Use in other connections (e.g. clustering)
    4. Examples of applications
     

  8. Shape constraints: some open problems.

  9. Shape constraints and restrictions: some math background
    1. Some convex analysis
    2. Facts and principles from optimization theory
    3. Empirical process theory tools
First draft outline: 27 February 2010.
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