In All Likelihood Statistical Modelling and Inference Using Likelihood,9780198507659

In All Likelihood Statistical Modelling and Inference Using Likelihood

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ISBN13: 9780198507659
ISBN10: 0198507658
Format: Hardcover
Pub. Date: 2001-08-30
Publisher(s): Oxford University Press
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Summary

This text concentrates on what can be achieved using the likelihood/Fisherian methods of taking into account uncertainty when studying a statistical problem. It takes the concept of the likelihood as the best method for unifying the demands of statistical modeling and theory of inference. Every likelihood concept is illustrated with realistic examples ranging from a simple comparison of two accident rates to complex studies that require generalized linear or semiparametric modeling. The emphasis is on likelihood not as just a device used to produce an estimate, but as an important tool for modeling.

Table of Contents

Introduction
1(21)
Prototype of statistical problems
1(2)
Statistical problems and their models
3(3)
Statistical uncertainty: inevitable controversies
6(2)
The emergence of statistics
8(6)
Fisher and the third way
14(5)
Exercises
19(2)
Elements of likelihood inference
21(32)
Classical definition
21(3)
Examples
24(3)
Combining likelihoods
27(2)
Likelihood ratio
29(1)
Maximum and curvature of likelihood
30(5)
Likelihood-based intervals
35(6)
Standard error and Wald statistic
41(2)
Invariance principle
43(2)
Practical implications of invariance principle
45(3)
Exercises
48(5)
More properties of the likelihood
53(20)
Sufficiency
53(2)
Minimal sufficiency
55(3)
Multiparameter models
58(3)
Profile likelihood
61(3)
Calibration in multiparameter case
64(3)
Exercises
67(6)
Basic models and simple applications
73(44)
Binomial or Bernoulli models
73(3)
Binomial model with under- or overdispersion
76(2)
Comparing two proportions
78(4)
Poisson model
82(2)
Poisson with overdispersion
84(2)
Traffic deaths example
86(1)
Aspirin data example
87(2)
Continuous data
89(6)
Exponential family
95(7)
Box-Cox transformation family
102(2)
Location-scale family
104(3)
Exercises
107(10)
Frequentist properties
117(32)
Bias of point estimates
117(2)
Estimating and reducing bias
119(4)
Variability of point estimates
123(2)
Likelihood and P-value
125(3)
CI and coverage probability
128(3)
Confidence density, CI and the bootstrap
131(3)
Exact inference for Poisson model
134(5)
Exact inference for binomial model
139(1)
Nuisance parameters
140(2)
Criticism of CIs
142(3)
Exercises
145(4)
Modelling relationships: regression models
149(44)
Normal linear models
150(4)
Logistic regression models
154(3)
Poisson regression models
157(3)
Nonnormal continuous regression
160(3)
Exponential family regression models
163(3)
Deviance in GLM
166(8)
Iterative weighted least squares
174(4)
Box-Cox transformation family
178(3)
Location-scale regression models
181(6)
Exercises
187(6)
Evidence and the likelihood principle*
193(20)
Ideal inference machine?
193(1)
Sufficiency and the likelihood principles
194(2)
Conditionality principle and ancillarity
196(1)
Birnbaum's theorem
197(2)
Sequential experiments and stopping rule
199(5)
Multiplicity
204(2)
Questioning the likelihood principle
206(5)
Exercises
211(2)
Score function and Fisher information
213(18)
Sampling variation of score function
213(2)
The mean of S(θ)
215(1)
The variance of S(θ)
216(3)
Properties of expected Fisher information
219(2)
Cramer---Rao lower bound
221(2)
Minimum variance unbiased estimation*
223(3)
Multiparameter CRLB
226(2)
Exercises
228(3)
Large-sample results
231(42)
Background results
231(4)
Distribution of the score statistic
235(3)
Consistency of MLE for scalar θ
238(3)
Distribution of MLE and the Wald statistic
241(2)
Distribution of likelihood ratio statistic
243(1)
Observed versus expected information*
244(3)
Proper variance of the score statistic*
247(1)
Higher-order approximation: magic formula*
247(9)
Multiparameter case: &thetas; &epsis; Rp
256(3)
Examples
259(5)
Nuisance parameters
264(4)
X2 goodness-of-fit tests
268(2)
Exercises
270(3)
Dealing with nuisance parameters
273(24)
Inconsistent likelihood estimates
274(2)
Ideal case: orthogonal parameters
276(2)
Marginal and conditional likelihood
278(3)
Comparing Poisson means
281(2)
Comparing proportions
283(3)
Modified profile likelihood*
286(6)
Estimated likelihood
292(2)
Exercises
294(3)
Complex data structure
297(44)
ARMA models
297(2)
Markov chains
299(3)
Replicated Markov chains
302(3)
Spatial data
305(4)
Censored/survival data
309(5)
Survival regression models
314(2)
Hazard regression and Cox partial likelihood
316(4)
Poisson point processes
320(4)
Replicated Poisson processes
324(7)
Discrete time model for Poisson processes
331(4)
Exercises
335(6)
EM Algorithm
341(24)
Motivation
341(1)
General specification
342(2)
Exponential family model
344(4)
General properties
348(1)
Mixture models
349(3)
Robust estimation
352(2)
Estimating infection pattern
354(2)
Mixed model estimation*
356(3)
Standard errors
359(3)
Exercises
362(3)
Robustness of likelihood specification
365(20)
Analysis of Darwin's data
365(2)
Distance between model and the `truth'
367(3)
Maximum likelihood under a wrong model
370(2)
Large-sample properties
372(3)
Comparing working models with the AIC
375(4)
Deriving the AIC
379(4)
Exercises
383(2)
Estimating equation and quasi-likelihood
385(24)
Examples
387(3)
Computing β in nonlinear cases
390(3)
Asymptotic distribution
393(2)
Generalized estimating equation
395(3)
Robust estimation
398(6)
Asymptotic Properties
404(5)
Empirical likelihood
409(16)
Profile likelihood
409(4)
Double-bootstrap likelihood
413(2)
BCa bootstrap likelihood
415(3)
Exponential family model
418(2)
General cases: M-estimation
420(2)
Parametric versus empirical likelihood
422(2)
Exercises
424(1)
Likelihood of random parameters
425(10)
The need to extend the likelihood
425(2)
Statistical prediction
427(2)
Defining extended likelihood
429(4)
Exercises
433(2)
Random and mixed effects models
435(38)
Simple random effects models
436(3)
Normal linear mixed models
439(3)
Estimating genetic value from family data*
442(2)
Joint estimation of β and b
444(1)
Computing the variance component via β and b
445(3)
Examples
448(4)
Extension to several random effects
452(6)
Generalized linear mixed models
458(2)
Exact likelihood in GLMM
460(2)
Approximate likelihood in GLMM
462(7)
Exercises
469(4)
Nonparametric smoothing
473(30)
Motivation
473(4)
Linear mixed models approach
477(2)
Imposing smoothness using random effects model
479(2)
Penalized likelihood approach
481(1)
Estimate of f given σ2 and σ2b
482(3)
Estimating the smoothing parameter
485(4)
Prediction intervals
489(1)
Partial linear models
489(1)
Smoothing nonequispaced data*
490(2)
Non-Gaussian smoothing
492(5)
Nonparametric density estimation
497(3)
Nonnormal smoothness condition*
500(1)
Exercises
501(2)
Bibliography 503(12)
Index 515

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