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lindsay b mixture models theory geometry applications 1995

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Textbook in PDF format The mixture model has long been a challenge to the statistician, whether beginner, practitioner, or theoretician. Recent times have seen great advances in our understanding of some basic mathematical features of this model, and these notes are meant to be a unification of the work I have carried out, jointly with many wonderful collaborators, in this area. Based on lectures given in 1993 at a regional conference of the Conference Board of the Mathematical Sciences, the notes are directed toward a mixed audience of advanced graduate students and research workers in this and related areas. For the sake of newcomers to the mixture model, I will attempt to be complete enough for the text to make sense in itself, but must at some points refer the reader to other more extensive treatments. CHAPTER 1: The Wide Scope 1.1. The finite mixture problem 1.1.1. A simple example 1.1.2. More complicated applications 1.2. The latent (or mixing) distribution 1.2.1. The discrete latent distribution 1.2.2. The continuous latent distribution 1.3. Many more variations and names 1.3.1. Known component densities 1.3.2. Linear inverse problems 1.3.3. Random effects models 1.3.4. Repeated measures models 1.3.5. Latent class and latent trait models 1.3.6. Missing covariates and data 1.3.7. Random coefficient regression models 1.3.8. Empirical and hierarchical Bayes 1.3.9. Nuisance parameter models 1.3.10. Measurement error models 1.3.11. Deconvolution problems 1.3.12. Robustness and contamination models 1.3.13. Overdispersion and heterogeneity 1.3.14. Hidden mixture structures 1.3.15. Clustering: A second kind 1.4. Be aware of limitations 1.4.1. Robustness characteristics 1.4.2. Extracting signal from noise 1.5. The likelihoods 1.5.1. The multinomial likelihood 1.5.2. Partly classified data 1.6. The mixture NPMLE theorem 1.6.1. The fundamental theorem 1. 70 Related nonparametric problems 1. 7 .1. The MLE of an unknown distribution 1.7.2. Accurate and error-prone measurements 1.7.3. Monotone density problems 1.7.4. Censoring problems 1.8. Similar statistical problems CHAPTER 2: Structural Features 2.1. Descriptive features 2.1.1. Some simple moment results 2.1.2. Shape and modality 2.1.3. Overdispersion and sign changes 2.1.4. Log convexity of ratios 2.1.5. Moments and sign changes 2.1.6. Dispersion models 2.2. Diagnostics for exponential families 2.2.1. Empirical ratio plots 2.2.2. Gradient function plots 2.2.3. Comparing gradient and ratio plots 2.3. Geometry of multinomial mixtures 2.3.1. Known component densities 2.3.2. Basic convex geometry 2.3.3. Identifiability of weight parameters 2.3.4. Caratheodory's theorem 2.4. Exponential family geometry 2.4.1. Identifiable functions 2.4.2. Identifiability of weights, m fixed 2.4.3. Full identifiability of m components 2.4.4. Hyperplanes and convex sets 2.4.5. Identifiability of weights and supports 2.4.6. Related problems 2.5. Moment representations0 2.6. Certain nested miXture models 2. 7. Concluding remark CHAPTER 3: Parametric Models 3.1. Discrete versus continuous 3.1.1. Continuous models: The conjugate family 3.2. Discrete latent distribution 3.2.1. Known component distributions 3.2.2. Unknown component parameters 3.3. Properties of the m-component MLE 3.4. EM algorithm 3.4.1. A description of the EM 3.4.2. The EM for finite mixtures 3.4.3. Algorithmic theory 3.5. Multimodality and starting values CHAPTER 4: Testing for Latent Structure 4.1. Dispersion score tests 4.1.1. The dispersion score 4.1.2. Neyman and Scott's C(a) test 4.1.3. Dispersion test optimality 4.1.4. Auxiliary parameters 4.2. LRT for number of components 4.2.1. The testing problem 4.2.2. Historical perspective 4.2.3. Initial observations 4.3. Asymptotic multinomial geometry 4.3.1. The dagger simplex 4.3.2. Maximum likelihood and projections 4.3.3. Type I likelihood ratio testing 4.4. The type II likelihood ratio problem 4.4.1. Parameter constraints 4.4.2. Convex cones 4.4.3. The z-coordinate system 4.4.4. Projections onto convex cones 4.4.5. The dual basis 4.4.6. Sector decomposition and projection 4.4. 7. The type II LRT 4.4.8. Applications 4.5. Asymptotic mixture geometry 4.5.1. Directional score functions 4.5.2. The gradient scores 4.5.3. Other directional scores 4.5.4. Simple binomial examples 4.5.5. The nonparametric LRT 4.5.6. A nonconvex score cone 4.6. The LRT on nonconvex cones 4.6.1. Projections onto nonconvex cones 4.6.2. Measuring distances 4.6.3. Tubes and distributions 4.6.4. Approximations for tubes 4.6.5. The arc length problem 4.6.6. Final comments CHAPTER 5: Nonparametric Maximum Likelihood 5.1 The optimization framework 5.1.1. Reformulating the problem 5.1.2. The feasible region 5.1.3. The objective function 5.2. Basic theorems 5.2.1. Existence and support size 5.2.2. Closed and bounded? 5.2.3. Gradient characterization 5.2.4. Properties of the support set 5.3. Further implications of the theorems 5.3.1. Duality theorem 5.3.2. Gradient bounds on the likelihood 5.3.3. Link to m-component methods 5.3.4. Moment and support point properties 5.4. Applications 5.4.1. A binomial mixture 5.4.2. Empirical CDF 5.4.3. Known component distributions 5.4.4. The multinomial case 5.5. Uniqueness and support size results 5.5.1. The strategy 5.5.2. A geometric approach to Task 1 5.5.3. A gradient function representation CHAPTER 6: Computation: The NPMLE 6.1. The convergence issue 6.2. Using the EM 6.3. Gradient-based algorithms 6.3.1. Design algorithms 6.3.2. Keeping track of the support points 6.3.3. Vertex direction and exchange methods 6.3.4. Intrasimplex direction method 6.3.5. Monotonicity 6.3.6. Using the dual problem 6.4. Ideal slopping rules 6.4.1. The ideal rule 6.4.2. A gradient-based rule 6.4.3. Combining grid and gradient 6.4.4. Bounding the second-order score 6.4.5. A conservative method 6.4.6. Remarks CHAPTER 7: Extending the Method 7 .1. Problems with ratio structure 7 .1.1. Example: Size bias 7.1.2. NPMLE with ratio structure 7.1.3. Example: Size bias 7.1.4. Example: Weibull competing risks 7.1.5. Mixed hazards NPMLE 7.2. NPMLE with constraints on Q 7 .2.1. Profile likelihood 7.2.2. Linear constraints 7.2.3. Examples with linear constraints 7.2.4. The constrained NPMLE 7.2.5. A simple algorithm 7.3. Smooth estimates of Q 7 .3.1. Roughening by smoothing 7.3.2. Deconvolution 7.3.3. Series expansion 7.3.4. A likelihood method CHAPTER 8: The Semiparametric MLE 8.1. An equivalence theorem 8.2. Exponential response models 8.2.1. Example: Rasch model 8.2.2. Type I conditional models 8.2.3. The two-item example 8.2.4. Efficiency theorem 8.2.5. Equivalence theorem for mixture MLE 8.3. Errors-in-variables and case-control studies 8.3.1. The joint sampling model 8.3.2. The retrospective model 8.3.3. Prentice and Pyke's equivalency 8.3.4. The measurement error extension 8.3.5. The extended equivalency result 8.4. A mixture index of fit 8.4.1. The problem 8.4.2. The concept 8.4.3. Application to the multinomial 8.4.4. Maximum likelihood estimation 8.4.5. Inference on the lack-of-fit index Bibliography

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