LEADER 05464nam 2200637Ia 450 001 9910678281303321 005 20230421050604.0 010 $a1-283-27403-5 010 $a9786613274038 010 $a1-118-16598-5 010 $a1-118-16597-7 035 $a(CKB)2550000000054407 035 $a(EBL)818938 035 $a(OCoLC)757511752 035 $a(SSID)ssj0000537061 035 $a(PQKBManifestationID)11371248 035 $a(PQKBTitleCode)TC0000537061 035 $a(PQKBWorkID)10551708 035 $a(PQKB)10533542 035 $a(MiAaPQ)EBC818938 035 $a(EXLCZ)992550000000054407 100 $a19960925d1997 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aGeometrical foundations of asymptotic inference$b[electronic resource] /$fRobert E. Kass, Paul W. Vos 210 $aNew York $cWiley$d1997 215 $a1 online resource (378 p.) 225 1 $aWiley series in probability and statistics. Probability and statistics 300 $a"A Wiley Interscience publication." 311 $a0-471-82668-5 320 $aIncludes bibliographical references (p. 340-347) and indexes. 327 $aGeometrical Foundations of Asymptotic Inference; Contents; Preface; 1 Overview and Preliminaries; 1.1 Overview; 1.1.1 Part I; 1.1.2 Part II; 1.1.3 Part III; 1.2 Notation; 1.2.1 Parameter Spaces; 1.2.2 Differentiation; 1.2.3 Tensor Notation; 1.2.4 Connection Notation; PART I ONE-PARAMETER CURVED EXPONENTIAL FAMILIES; 2 First-Order Asymptotics; 2.1 Introduction; 2.2 Exponential Families; 2.2.1 Basic Properties; 2.2.2 Asymptotics; 2.3 Curved Exponential Families: Definition and Examples; 2.3.1 Definition and Basic Properties; 2.3.2 Examples; 2.4 Estimators; 2.4.1 Estimating Equations 327 $a2.4.2 Auxiliary Spaces2.5 Fisher Information; 2.5.1 Information and Sufficiency; 2.5.2 The Information Inner Product; 2.5.3 Observed Information; 2.5.4 The Kullback-Leibler Divergence; 2.6 Consistency, Asymptotic Normality, and Efficiency; 2.6.1 Consistency and Asymptotic Normality; 2.6.2 Efficiency; 2.7 Bibliographical Remarks; 3 Second-Order Asymptotics; 3.1 Introduction; 3.2 Statistical Curvature; 3.2.1 Definition and Calculation; 3.2.2 Examples; 3.3 Information Loss and Local Sufficiency; 3.3.1 Information Loss; 3.3.2 Information Recovery; 3.3.3 Local Sufficiency 327 $a3.4 Other Applications of Statistical Curvature3.4.1 Second-Order Efficiency; 3.4.2 Deficiency; 3.4.3 Large Deviations; 3.4.4 The Fisher Scoring Algorithm; 3.5 Edgeworth Expansions; 3.6 Posterior Expansions; 3.7 Extensions; 3.7.1 Efron's General Formula; 3.7.2 Small-Dispersion Asymptotics; 3.8 Bibliographical Remarks; PART II MULTIPARAMETER CURVED EXPONENTIAL FAMILIES; 4 Extensions of Results from the One-Parameter Case; 4.1 Introduction; 4.2 Multiparameter Curved Exponential Families; 4.3 Curvature; 4.3.1 Curvature and Information Loss; 4.3.2 Asymptotic Risk and Bias 327 $a4.3.3 Interpretation in Nonlinear Regression4.3.4 Statistical Curvature in General Families; 4.4 Information Loss and Sufficiency; 4.5 Multivariate Edgeworth Series; 4.6 Posterior Expansions; 4.7 Bibliographical Remarks; 5 Exponential Family Regression and Diagnostics; 5.1 Introduction; 5.2 Normal Regression; 5.2.1 Normal Regression Model; 5.2.2 Maximum Likelihood Estimate; 5.2.3 Tangent Bundle; 5.3 Exponential Family Regression; 5.3.1 Preliminary Concepts; 5.3.2 A Vector Space Structure; 5.3.3 The Fisher Information Inner Product; 5.3.4 Estimation Algorithms; 5.4 Measures of Influence 327 $a5.4.1 Normal Linear Regression5.4.2 Exponential Family Regression; 5.5 Sensitivity Analysis of the Moment Structure; 5.5.1 Quasi-Likelihood Functions; 5.5.2 The Measures DL and LDLa; 5.5.3 Perturbations of the Moment Structure; 5.6 Bibliographical Remarks; 6 Curvature in Exponential Family Regression; 6.1 Introduction; 6.2 Background on Nonlinear Regression; 6.2.1 Asymptotic Normality; 6.2.2 Curvature Measures of Nonlinearity; 6.3 Curvature in Exponential Family Nonlinear Regression; 6.3.1 Generalizing the Standardized Second-Derivative Array; 6.3.2 Curvature Measures 327 $a6.4 Summaries of the Observed Third-Derivative Array 330 $aDifferential geometry provides an aesthetically appealing and often revealing view of statistical inference. Beginning with an elementary treatment of one-parameter statistical models and ending with an overview of recent developments, this is the first book to provide an introduction to the subject that is largely accessible to readers not already familiar with differential geometry. It also gives a streamlined entry into the field to readers with richer mathematical backgrounds. Much space is devoted to curved exponential families, which are of interest not only because they may be studied g 410 0$aWiley series in probability and statistics.$pProbability and statistics. 606 $aMathematical statistics$xAsymptotic theory 606 $aGeometry, Differential 615 0$aMathematical statistics$xAsymptotic theory. 615 0$aGeometry, Differential. 676 $a519.54 700 $aKass$b Robert E$0423354 701 $aVos$b Paul W.$f1961-$0939630 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910678281303321 996 $aGeometrical foundations of asymptotic inference$92118277 997 $aUNINA