Conditional Specification of Statistical Models

by Arnold, Barry C.; Castillo, Enrique; Sarabia, Jose-Maria

• ISBN13: 9780387987613
• ISBN10: 0387987614
• Format: Hardcover
• Copyright: 1999-11-01
• Publisher: Springer Verlag

Summary

The concept of conditional specification of distributions is not new but, except in normal families, it has not been well developed in the literature. Computational difficulties undoubtedly hindered or discouraged developments in this direction. However, such roadblocks are of dimished importance today. Questions of compatibility of conditional and marginal specifications of distributions are of fundamental importance in modeling scenarios. Models with conditionals in exponential families are particularly tractable and provide useful models in a broad variety of settings.

Table of Contents

Preface	vii
Conditional Specification: Concepts and Theorems	1	(18)
Why Conditional Specified Models?	1	(1)
How May One Specify a Bivariate Distribution?	2	(2)
Early Work on Conditionally Specified Models	4	(1)
The Conditional Specification Paradigm	5	(1)
Compatible Conditionals: Finite Discrete Case	5	(3)
Compatibility in More General Settings	8	(3)
Uniqueness	11	(1)
Conditionals in Prescribed Families	12	(2)
An Example	14	(2)
Bibliographic Notes	16	(3)
Exercises	16	(3)
Exact and Near Compatibility	19	(34)
Introduction	19	(1)
Review and Extensions of Compatibility Results	19	(11)
Near Compatibility	30	(1)
Minimal Incompatibility in Terms of Kullback-Leibler Pseudo-Distance	31	(5)
More Than One Expert	36	(1)
Related Discrepancy Measures	37	(4)
Markovian Measures of Discrepancy	41	(2)
ε-Compatibility	43	(7)
Extensions of More General Settings	50	(1)
Bibliographic Notes	50	(3)
Exercises	51	(2)
Distributions with Normal Conditionals	53	(22)
Introduction	53	(1)
Variations on the Classical Bivariate Normal Theme	53	(2)
Normal Conditionals	55	(4)
Properties of the Normal Conditionals Distribution	59	(6)
The Centered Model	65	(7)
Bibliographic Notes	72	(3)
Exercises	72	(3)
Conditionals in Exponential Families	75	(28)
Introduction	75	(1)
Distributions with Conditionals in Given Exponential Families	75	(3)
Dependence in CEF Distributions	78	(2)
Exponential Conditionals	80	(2)
Normal Conditionals	82	(1)
Gamma Conditionals	83	(5)
Weibull Conditionals	88	(1)
Gamma--Normal Conditionals	89	(1)
Power-Function and Other Weighted Distributions as Conditionals	90	(3)
Beta Conditionals	93	(1)
Inverse Gaussian Conditionals	94	(1)
Three Discrete Examples (Binomial, Geometric, and Poisson)	95	(2)
Poisson--Gamma Conditionals	97	(2)
Bibliographic Notes	99	(4)
Exercises	99	(4)
Other Conditionally Specified Families	103	(30)
Introduction	103	(1)
Bivariate Distributions with Pareto Conditionals	104	(3)
Pearson Type VI Conditionals	107	(2)
Bivariate Distributions with Generalized Pareto Conditionals	109	(7)
Bivariate Distributions with Cauchy Conditionals	116	(6)
Bivariate Distributions with Student-t Conditionals	122	(1)
Bivariate Distributions with Uniform Conditionals	123	(1)
Possibly Translated Exponential Conditionals	124	(1)
Bivariate Distributions with Scaled Beta Conditionals	125	(1)
Weibull and Logistic Conditionals	126	(2)
Mixtures	128	(1)
Bibliographic Notes	129	(4)
Exercises	129	(4)
Improper and Nonstandard Models	133	(14)
Introduction	133	(1)
Logistic Regression	134	(1)
Uniform Conditionals	135	(1)
Exponential and Weibull Conditionals	135	(1)
Measurement Error Models	136	(1)
Stochastic Processes and Wohler Fields	136	(8)
Physical Interpretations of the Model Parameters	144	(1)
Bibliographic Notes	145	(2)
Exercises	145	(2)
Characterizations Involving Conditional Moments	147	(28)
Introduction	147	(1)
Mardia's Bivariate Pareto Distribution	148	(1)
Linear Regressions with Conditionals in Exponential Families	149	(1)
Linear Regressions with Conditionals in Location Families	150	(2)
Specified Regressions with Conditionals in Scale Families	152	(1)
Conditionals in Location-Scale Families with Specified Moments	153	(2)
Given One Conditional Distribution and the Other Regression	155	(14)
Conditional Moments Only Given	169	(1)
Bibliographic Notes	170	(5)
Exercises	170	(5)
Multivariate Extensions	175	(22)
Introduction	175	(1)
Extension by Underlining	175	(1)
Compatibility in Three Dimensions	176	(1)
Compatibility in Higher Dimensions	177	(1)
Conditionals in Prescribed Families	177	(1)
Conditionals in Exponential Families	178	(1)
Multivariate Exponential Conditionals Distribution	179	(1)
Multivariate Normal Conditionals Distribution	180	(1)
Multivariate Cauchy Conditionals Distribution	181	(1)
Multivariate Uniform Conditionals Distribution	181	(1)
Multivariate Pareto Conditionals Distribution	182	(1)
Multivariate Beta Conditionals Distribution	183	(2)
Multivariate Binomial Conditionals Distribution	185	(1)
Further Extension by Underlining	185	(1)
Characterization of Multivariate Normality	186	(4)
Multivariate Normality in More Abstract Settings	190	(1)
Characterizing Mardia's Multivariate Pareto Distribution	191	(2)
Bibliographic Notes	193	(4)
Exercises	194	(3)
Estimation in Conditionally Specified Models	197	(32)
Introduction	197	(1)
The Ubiquitous Norming Constant	197	(1)
Maximum Likelihood	198	(6)
Pseudolikelihood Involving Conditional Densities	204	(3)
Marginal Likelihood	207	(1)
An Efficiency Comparison	207	(3)
Method of Moments Estimates	210	(7)
Log-Linear Poisson Regression Estimates	217	(2)
Bayesian Estimates	219	(3)
Multivariate Examples	222	(3)
Bibliographic Notes	225	(4)
Exercises	225	(4)
Marginal and Conditional Specification in General	229	(26)
Introduction	229	(1)
Specifying Bivariate Densities	229	(6)
To Higher Dimensions with Care	235	(1)
Conditional/Marginal Specification in k Dimensions	236	(2)
Checking Uniqueness	238	(2)
Checking Compatibility	240	(6)
Overspecification	246	(1)
Marginals and Conditionals in Specified Families	246	(2)
Xi Given X(i)	248	(1)
X(i) Given Xi	248	(3)
The Case of X Given Y, Y Given Z, and Z Given X	251	(1)
Logistic Regression Models	252	(1)
Bibliographic Notes	253	(2)
Exercises	253	(2)
Conditional Survival Models	255	(20)
Introduction	255	(1)
Conditional Survival in the Bivariate Case	256	(1)
Conditional Survival Functions in Parametric Families	257	(4)
Examples of Distributions Characterized by Conditional Survival	261	(3)
Multivariate Extensions	264	(1)
Conditional Distributions	265	(2)
Conditional Proportional Hazards	267	(2)
Conditional Accelerated Failure	269	(2)
An Alternative Specification Paradigm	271	(3)
Bibliographic Notes	274	(1)
Exercises	274	(1)
Applications to Modeling Bivariate Extremes	275	(18)
Introduction	275	(1)
Univariate and Bivariate Gumbel Distributions	276	(1)
Conditionally Specified Bivariate Gumbel Distributions	277	(5)
Positive or Negative Correlation	282	(1)
Density Contours	282	(5)
Maximal Wind Speeds	287	(3)
More Flexibility Needed	290	(1)
Bibliographic Note	291	(2)
Exercises	291	(2)
Bayesian Analysis Using Conditionally Specified Models	293	(44)
Introduction	293	(1)
Motivation from a Normal Example	294	(1)
Priors with Convenient Posteriors	295	(4)
Conjugate Exponential Family Priors for Exponential Family Likelihoods	299	(2)
Conditionally Specified Priors	301	(3)
Normal Data	304	(8)
Pareto Data	312	(5)
Inverse Gaussian Data	317	(1)
Ratios of Gamma Scale Parameters	318	(4)
Comparison of Normal Means	322	(5)
Regression	327	(3)
The 2 x 2 Contingency Table	330	(1)
Multinomial Data	331	(1)
Change Point Problems	332	(1)
Bivariate Normal	333	(1)
No Free Lunch	334	(1)
Bibliographic Notes	334	(3)
Exercises	334	(3)
Conditional Versus Simultaneous Equation Models	337	(16)
Introduction	337	(1)
Two Superficially Similar Models	338	(1)
General CS and SE Models	339	(1)
Linear Normal Models	340	(4)
Nonlinear Normal Models	344	(2)
Pareto CS and SE Models	346	(3)
Discrete Models	349	(1)
Higher-Dimensional Models	350	(1)
Bibliographic Note	350	(3)
Exercises	351	(2)
Paella	353	(18)
Introduction	353	(1)
Diatomic Conditionals and Stop-Loss Transforms	353	(4)
Failure Rates and Mean Residual Life Functions	357	(1)
Hypothesis Testing	358	(1)
Related Stochastic Processes	359	(2)
Given E(X|Y = y) and FX(x)	361	(1)
Near Compatibility with Given Parametric Families of Distributions	362	(2)
Marginal Maps	364	(1)
A Cautionary Tale	364	(5)
Bibliographic Notes	369	(2)
Exercises	369	(2)
A Simulation	371	(12)
A.1 Introduction	371	(1)
A.2 The Rejection Method	372	(2)
A.3 The Importance Sampling Method	374	(3)
A.4 Application to Models with Conditionals in Exponential Families	377	(1)
A.5 Other Conditionally Specified Models	378	(1)
A.6 A Direct Approach Not Involving Rejection	379	(1)
A.7 Bibliographic Notes	380	(3)
Exercises	380	(3)
B Notation Used in This Book	383	(6)
References	389	(10)
Author Index	399	(4)
Subject Index	403