Inference Framework and Method

This chapter sets the stage for the development of the book. The discussion in this chapter concerns the standard context in which mismeasurement is absent. This chapter lays out a broad framework for parametric inferences where estimation is of central interest. §1.1 outlines the inference framework and the objectives. Important issues concerning modeling and inferences are discussed in §1.2. Representative and useful estimation methodology is reviewed in §1.3. Strategies of handling model misspecification are described in §1.4, and the extension to the regression setting is included in §1.5. Brief bibliographic notes are presented in §1.6.

[1]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[2]  K. Liang.  Estimating functions and approximate conditional likelihood , 1987 .

[3]  Robert Tibshirani,et al.  A Simple Method for the Adjustment of Profile Likelihoods , 1990 .

[4]  Wenxin Jiang,et al.  The Indirect Method: Inference Based on Intermediate Statistics—A Synthesis and Examples , 2004 .

[5]  R. Tibshirani,et al.  An introduction to the bootstrap , 1993 .

[6]  H. Y. Chen.  A unified framework for studying parameter identifiability and estimation in biased sampling designs , 2011 .

[7]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[8]  Randy L. Carter,et al.  An Overview of Normal Theory Structural Measurement Error Models , 2007 .

[9]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[10]  C. Heyde.  Quasi-likelihood and its application : a general approach to optimal parameter estimation , 1998 .

[11]  J. Neyman,et al.  Consistent Estimates Based on Partially Consistent Observations , 1948 .

[12]  R. W. Wedderburn.  Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method , 1974 .

[13]  D. Pollard.  New Ways to Prove Central Limit Theorems , 1985, Econometric Theory.

[14]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[15]  E. Lehmann.  Elements of large-sample theory , 1998 .

[16]  T. Rothenberg.  Identification in Parametric Models , 1971 .

[17]  J. Durbin.  Estimation of Parameters in Time‐Series Regression Models , 1960 .

[18]  Bruce G. Lindsay,et al.  ISSUES AND STRATEGIES IN THE SELECTION OF COMPOSITE LIKELIHOODS , 2011 .

[19]  Z. D. Bai,et al.  On the maximum‐likelihood estimator for the location parameter of a cauchy distribution , 1987 .

[20]  V. P. Godambe.  An Optimum Property of Regular Maximum Likelihood Estimation , 1960 .

[21]  Thomas S. Ferguson,et al.  Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4 , 1978 .

[22]  P. McCullagh.  Quasi-Likelihood Functions , 1983 .

[23]  Richard L. Smith,et al.  Essentials of Statistical Inference: Index , 2005 .

[24]  O. E. Barndorff-Nielsen,et al.  Infereni on full or partial parameters based on the standardized signed log likelihood ratio , 1986 .

[25]  P. Lachenbruch.  Mathematical Statistics, 2nd Edition , 1972 .

[26]  N. Reid,et al.  A Class of Weighted Estimating Equations for Semiparametric Transformation Models with Missing Covariates , 2018 .

[27]  K. Liang,et al.  Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions , 1987 .

[28]  S. Eguchi,et al.  A paradox concerning nuisance parameters and projected estimating functions , 2004 .

[29]  P. Gustafson.  On Model Expansion, Model Contraction, Identifiability and Prior Information: Two Illustrative Scenarios Involving Mismeasured Variables , 2005 .

[30]  C. Roehrig,et al.  Conditions for Identification in Nonparametric and Parametic Models , 1988 .

[31]  C. Gouriéroux,et al.  PSEUDO MAXIMUM LIKELIHOOD METHODS: THEORY , 1984 .

[32]  T. Koopmans,et al.  The Identification of Structural Characteristics , 1950 .

[33]  W. Newey,et al.  Large sample estimation and hypothesis testing , 1986 .

[34]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[35]  R. Royall.  Model robust confidence intervals using maximum likelihood estimators , 1986 .

[36]  J. Kent.  Robust properties of likelihood ratio tests , 1982 .

[37]  Kung-Yee Liang,et al.  Approximate likelihood ratios for general estimating functions , 1995 .

[38]  Scott L. Zeger,et al.  [Inference Based on Estimating Functions in the Presence of Nuisance Parameters]: Rejoinder , 1995 .

[39]  N. Reid,et al.  AN OVERVIEW OF COMPOSITE LIKELIHOOD METHODS , 2011 .

[40]  Tjalling C. Koopmans,et al.  Identification problems in economic model construction , 1949 .

[41]  D. Pollard,et al.  Simulation and the Asymptotics of Optimization Estimators , 1989 .

[42]  David Draper,et al.  Assessment and Propagation of Model Uncertainty , 2011 .

[43]  John D. Kalbfleisch,et al.  Application of Likelihood Methods to Models Involving Large Numbers of Parameters , 1970 .

[44]  David R. Cox,et al.  PRINCIPLES OF STATISTICAL INFERENCE , 2017 .

[45]  Richard L. Smith,et al.  Essentials of statistical inference , 2005 .

[46]  T. Rao,et al.  Identifiability in stochastic models : characterization of probability distributions , 1994 .

[47]  O. Barndorff-Nielsen.  On a formula for the distribution of the maximum likelihood estimator , 1983 .

[48]  C. Matias,et al.  Identifiability of parameters in latent structure models with many observed variables , 2008, 0809.5032.

[49]  Harry Joe,et al.  Composite Likelihood Methods , 2012 .

[50]  Arne Gabrielsen.  Consistency and identifiability , 1978 .

[51]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[52]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[53]  J. Robins,et al.  Estimation of Regression Coefficients When Some Regressors are not Always Observed , 1994 .

[54]  Grace Y. Yi,et al.  A NOTE ON MIS-SPECIFIED ESTIMATING FUNCTIONS , 2010 .

[55]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[56]  David A. Freedman,et al.  Statistical Models: Theory and Practice: References , 2005 .

[57]  C. Heyde,et al.  Multiple roots in general estimating equations , 1998 .