Table Of ContentStatistical Curves and Parameters
Statistical Curves and Parameters:
Choosing an Appropriate Approach
Michael E. Tarter
Boca Raton London New York
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Library of Congress cataloging-in-Publication Data
Tarter, Michael E.
Statistical curves and parameters : choosing an appropriate approach / Michael E.
Tarter.
p. cm.
Includes bibliographical references and index.
ISBN 1-56881-105-5 (alk. paper)
1. Mathematical statistics. I. Title.
QA276.T28 2000
519.5--dc21 99-058803
Contents
Preface xi
1 Introduction 1
1.1 Background................................................................................ 1
1.2 A fictional example.................................................................... 4
1.3 Curves and statistical history.................................................. 5
2 Model and Distribution Terminology 9
2.1 Modeling background .............................................................. 9
2.2 Representative number.............................................................. 10
2.3 Curve types................................................................................ 13
2.4 Distribution and data terminology........................................ 16
2.5 Parameter validity and property existence............................ 18
2.6 Estimator terminology.............................................................. 21
2.7 Degenerate curves .................................................................... 23
3 Variability and Related Curve Properties 25
3.1 Uncertainty and variability..................................................... 25
3.2 The absolute deviation curve property.................................. 26
3.3 The general AD and the ADM curve properties................... 28
3.4 Curve property selection........................................................... 33
3.5 The history of variability appreciation.................................. 36
3.6 Simplistic approaches and the history of probability . . . . 37
4 Moments and Curve Uncertainty 41
4.1 E and Var Geometry................................................................. 41
4.2 Higher order moments and the indicator function................ 45
4.3 Early statistical models........................................................... 47
v
vi
4.4 Early statistical models and higher order moments............. 50
4.5 Curve sub-types and model choice........................................ 52
5 Goodness of fit 55
5.1 Neyman’s and alternative criteria........................................... 55
5.2 Criteria, metrics and estimators............................................... 61
5.3 The KolmogorofF-SmirnofF criteria ........................................ 64
5.4 Bernoulli variation and the Cauchy density......................... 65
5.5 Comparative goodness of fit..................................................... 71
6 Variates, Variables and Regression 73
6.1 Variates and variables.............................................................. 73
6.2 Variates and subjects................................................................. 75
6.3 Expressions, algorithms and life tables.................................. 76
6.4 Distinctions between curve types............................................ 77
6.5 Curve properties and symbols.................................................. 79
6.6 Variates, variables, and Ef(Y\x) regression......................... 80
6.7 μ(χ),Ε/(Υ\χ) and regression alternatives ............................ 83
7 Mixing Parameters and Data-generation models 89
7.1 An introduction to data-generation models ......................... 89
7.2 Error, regression, and probit, models..................................... 91
7.3 Regression and data-generation models.................................... 93
7.4 Probability, proportion, and data-generation models . . . . 94
7.5 The generation of contagious model and mixture model data 99
8 The Association Parameter p 103
8.1 Response, key, and nuisance, variates.........................................103
8.2 The association parameter p .........................................................105
8.3 Conditional, joint and marginal, notation................................110
8.4 The sample and the population correlation coefficient . . . 114
8.5 Correlation geometry.....................................................................116
9 Regression and Association Parameters 123
9.1 The curse of dimensionality.........................................................123
9.2 Multiple variable interdependence...............................................128
9.3 Logit and linear models...............................................................132
9.4 Dual regression functions ............................................................136
vii
10 Parameters, Confounding, and Least Squares 141
10.1 Ideal objects .................................................................................141
10.2 Linear data-generation models and mixture models................146
10.3 Parameter distinctiveness...........................................................148
10.4 Representational uniqueness and model fitting..........................150
10.5 Model-fitting considerations........................................................151
10.6 The variance curve property and bathtub functions............154
10.7 Regression and least squares........................................................155
11 Nonparametric Adjustment 159
11.1 Age-adjustment and logistic regression......................................159
11.2 Crude and specific rates...............................................................163
11.3 Age-adjustment; marginal, joint, and conditional curves . . 164
11.4 Age-adjustment and partial correlation......................................166
11.5 Direct and indirect adjustment..................................................168
11.6 The computation of adjusted rates............................................172
12 Continuous Variate Adjustment 175
12.1 Observed and expected rates.....................................................175
12.2 Trivariate data-generation and additive regression models . 177
12.3 Regression and data generation..................................................179
12.4 Correlation, regression, and nuisance variables..........................180
12.5 Trivariate Normality graphics.....................................................185
13 Procedural Road Maps 189
13.1 The organization of statistical data and statistical methods 189
13.2 Log and log(-log) transformations...............................................193
13.3 Methodological alternatives...........................................................196
13.4 Conditional and joint density models...........................................198
14 Model-based and Generalized Representation 203
14.1 Multiple properties and parameters............................................203
14.2 Specification and generalized representation.............................208
14.3 Identifiability of generalized versus extended
model representation.....................................................................210
14.4 The E(X) curve property’s relationship to location and scale 214
15 Parameters, Transformations, and Quantiles 217
15.1 Location and scale parameter representation of continuous
variates..........................................................................................217
15.2 p-focused transformations and σ-focused transformations . . 220
viii
15.3 Quantiles, quartiles, and box-and-whisker plots...................222
15.4 Normal ranges and box sizes.....................................................226
15.5 Confidence bands and prediction bands ...............................228
15.6 Notches, stems, and leaves........................................................229
15.7 The log transformation and skewness.....................................232
16 Noncentrality Parameters and Degrees of Freedom 237
16.1 The (Ci\A2) case and variate-variable relationships.............237
16.2 Invariance and confounding........................................................243
16.3 ANOVA tables and confounding ............................................247
16.4 Contingency tables and the parameter v ...............................249
16.5 Student-t and Cauchy densities..................................................251
17 Parameter-Based Estimation 255
17.1 Likelihood and BLU estimation..................................................255
17.2 Censoring and incompleteness.....................................................256
17.3 Outliers and errors........................................................................258
17.4 Ordered variates and subscripts..................................................260
17.5 BLU estimators..............................................................................261
17.6 BLU estimation and censoring ..................................................265
17.7 BLU estimators and alternatives...............................................269
18 Inference and Composite Variates 273
18.1 Curves and composite variates ...............................................273
18.2 Specific sampling distributions ..................................................276
18.3 The mean’s variance formula and mixtures............................281
18.4 Inference and a two-valued metric ........................................282
18.5 The one tail z-test........................................................................286
19 Parameters and Test Statistics 293
19.1 The parameter Δ ..................................... 293
19.2 Power and efficiency.....................................................................295
19.3 Power and test considerations.....................................................298
19.4 The sample mean and the sample median.................................302
19.5 Tables and the details of test construction................................304
19.6 Power, efficiency and BLU estimators ..................................306
20 Curve Truncation and the Curve e(x) 311
20.1 Expectation as a limit and the effects of truncation................311
20.2 Truncation symmetry ....................................................................313
20.3 Truncation and bias.....................................................................317
ix
20.4 Truncation and the curve e(x)..................................................327
20.5 When are curve properties relevant and when are model pa
rameters relevant...........................................................................326
I Models and Notation 329
1.1 Notation historical background..................................................329
1.2 Specific models, the Normal........................................................333
1.3 Specific models, lognormals and related curves ........................336
1.4 Model families ..............................................................................340
1.5 Mixtures and Bayesian statistics...............................................342
1.6 Notational conventions about moments and variates . . . . 345
II Variate Independence and Curve Identity 347
II. 1 Independence and identical distribution..................................347
II.2 Regression notation.......................................................................350
III General Statistical and Mathematical Notation 355
References 367
Index 375
