From Algorithms to Z-Scores - matloff - University of California, Davis

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xiv CONTENTS 15.19Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 15.19.1 Example: Prediction of Network RTT . . . . . . . . . . . . . . . . . . . . . . 313 15.19.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 15.19.3 Example: OOP Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 16 Advanced Statistical Estimation and Inference 319 16.1 Slutsky’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 16.1.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 16.1.2 Why It’s Valid to Substitute s for σ . . . . . . . . . . . . . . . . . . . . . . . 320 16.1.3 Example: Confidence Interval for a Ratio Estimator . . . . . . . . . . . . . . 321 16.2 The Delta Method: Confidence Intervals for General Functions of Means or Proportions321 16.2.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 16.2.2 Example: Square Root Transformation . . . . . . . . . . . . . . . . . . . . . . 324 16.2.3 Example: Confidence Interval for σ 2 . . . . . . . . . . . . . . . . . . . . . . . 325 16.2.4 Example: Confidence Interval for a Measurement of Prediction Ability . . . . 328 16.3 The Bootstrap Method for Forming Confidence Intervals . . . . . . . . . . . . . . . . 329 16.3.1 Basic Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 16.3.2 Example: Confidence Intervals for a Population Variance . . . . . . . . . . . 330 16.3.3 Computation in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 16.3.4 General Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 16.3.5 Why It Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 17 Relations Among Variables: Advanced 333 17.1 Nonlinear Parametric Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . 333 17.2 The Classification Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 17.2.1 Classification = Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 17.2.2 Optimality of the Regression Function for 0-1-Valued Y (optional section) . . 335
CONTENTS xv 17.2.3 Logistic Regression: a Common Parametric Model for the Regression Function in Classification Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 336 17.2.3.1 The Logistic Model: Motivations . . . . . . . . . . . . . . . . . . . . 337 17.2.4 Esimation and Inference for Logit Coefficients . . . . . . . . . . . . . . . . . . 339 17.2.5 Example: Forest Cover Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 17.2.6 What If Y Doesn’t Have a Marginal Distribution? . . . . . . . . . . . . . . . 340 17.3 Nonparametric Estimation of Regression and Classification Functions . . . . . . . . . 341 17.3.1 Methods Based on Estimating mY ;X(t) . . . . . . . . . . . . . . . . . . . . . 341 17.3.1.1 Nearest-Neighbor Methods . . . . . . . . . . . . . . . . . . . . . . . 342 17.3.1.2 Kernel-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . 344 17.3.1.3 The Naive Bayes Method . . . . . . . . . . . . . . . . . . . . . . . . 345 17.3.2 Methods Based on Estimating Classification Boundaries . . . . . . . . . . . . 346 17.3.2.1 Support Vector Machines (SVMs) . . . . . . . . . . . . . . . . . . . 346 17.3.2.2 CART . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 17.3.3 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 17.4 Symmetric Relations Among Several Variables . . . . . . . . . . . . . . . . . . . . . 350 17.4.1 Principal Components Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 351 17.4.2 How to Calculate Them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 17.4.3 Example: Forest Cover Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 17.4.4 Log-Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 17.4.4.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 17.4.4.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 17.4.4.3 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 17.4.4.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 356 17.4.4.5 The Goal: Parsimony Again . . . . . . . . . . . . . . . . . . . . . . 357 17.5 Simpson’s (Non-)Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Page 1 and 2: From Algorithms to Z-Scores: Probab
Page 3 and 4: Contents 1 Time Waste Versus Empowe
Page 5 and 6: CONTENTS iii 3.7 A Combinatorial Ex
Page 7 and 8: CONTENTS v 5.5.1.3 Example: Modelin
Page 9 and 10: CONTENTS vii 7.3.1 Properties of Me
Page 11 and 12: CONTENTS ix 10.1 Sampling Distribut
Page 13 and 14: CONTENTS xi 11.9.4 What to Do Inste
Page 15: CONTENTS xiii 15.2 Example Applicat
Page 19 and 20: CONTENTS xvii 19.2 Simulation of Ra
Page 21 and 22: CONTENTS xix 21.4 Loss Models . . .
Page 23 and 24: Preface Why is this book different
Page 25 and 26: Chapter 1 Time Waste Versus Empower
Page 27 and 28: Chapter 2 Basic Probability Models
Page 29 and 30: 2.2. THE CRUCIAL NOTION OF A REPEAT
Page 31 and 32: 2.3. OUR DEFINITIONS 7 2009, cannot
Page 33 and 34: 2.4. “MAILING TUBES” 9 but in m
Page 35 and 36: 2.5. BASIC PROBABILITY COMPUTATIONS
Page 37 and 38: 2.6. BAYES’ RULE 13 Note by the w
Page 39 and 40: 2.8. SOLUTION STRATEGIES 15 2.8 Sol
Page 41 and 42: 2.10. EXAMPLE: A SIMPLE BOARD GAME
Page 43 and 44: 2.11. EXAMPLE: BUS RIDERSHIP 19 Aga
Page 45 and 46: 2.12. SIMULATION 21 1 # roll d dice
Page 47 and 48: 2.12. SIMULATION 23 So, in evaluati
Page 49 and 50: 2.12. SIMULATION 25 3 count
Page 51 and 52: 2.13. COMBINATORICS-BASED PROBABILI
Page 59 and 60: Chapter 3 Discrete Random Variables
Page 61 and 62: 3.4. EXPECTED VALUE 37 3.4.1.1 What
Page 63 and 64: 3.4. EXPECTED VALUE 39 So It turns
Page 65 and 66: 3.4. EXPECTED VALUE 41 • For rand
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3.4. EXPECTED VALUE 43 of two rando
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3.5. VARIANCE 45 ance of U is defin
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3.5. VARIANCE 47 for any constant d
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3.7. A COMBINATORIAL EXAMPLE 49 You
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3.8. A USEFUL FACT 51 Note carefull
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3.10. EXPECTED VALUE, ETC. IN THE A
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3.11. DISTRIBUTIONS 55 3.11.1 Examp
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3.12. PARAMETERIC FAMILIES OF PMFS
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3.13. RECOGNIZING SOME PARAMETRIC D
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3.13. RECOGNIZING SOME PARAMETRIC D
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3.15. A CAUTIONARY TALE 73 T has a
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3.16. WHY NOT JUST DO ALL ANALYSIS
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3.18. RECONCILIATION OF MATH AND IN
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Chapter 4 Introduction to Discrete
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4.3. EXAMPLE: 3-HEADS-IN-A-ROW GAME
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4.4. EXAMPLE: ALOHA 87 The quantity
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4.6. AN INVENTORY MODEL 89 4.6 An I
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Chapter 5 Continuous Probability Mo
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5.3. BUT EQUATION (??) PRESENTS A P
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5.3. BUT EQUATION (??) PRESENTS A P
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5.4. DENSITY FUNCTIONS 97 2(0.1)fX(
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5.4. DENSITY FUNCTIONS 99 5.4.2 Pro
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5.5. FAMOUS PARAMETRIC FAMILIES OF
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5.8. “HYBRID” CONTINUOUS/DISCRE
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5.8. “HYBRID” CONTINUOUS/DISCRE
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Chapter 6 Stop and Review: Probabil
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• famous parametric families of d
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Chapter 7 Covariance and Random Vec
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7.1. MEASURING CO-VARIATION OF RAND
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7.2. SETS OF INDEPENDENT RANDOM VAR
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7.2. SETS OF INDEPENDENT RANDOM VAR
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7.3. MATRIX FORMULATIONS 139 this l
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7.3. MATRIX FORMULATIONS 141 consis
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7.3. MATRIX FORMULATIONS 143 import
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7.3. MATRIX FORMULATIONS 145 since
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7.3. MATRIX FORMULATIONS 147 minimi
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Chapter 8 Multivariate PMFs and Den
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8.1. MULTIVARIATE PROBABILITY MASS
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8.2. MULTIVARIATE DENSITIES 153 So,
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8.2. MULTIVARIATE DENSITIES 155 we
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8.3. MORE ON SETS OF INDEPENDENT RA
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8.4. EXAMPLE: FINDING THE DISTRIBUT
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8.5. PARAMETRIC FAMILIES OF MULTIVA
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Chapter 9 Introduction to Continuou
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9.1. MEMORYLESS PROPERTY OF EXPONEN
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9.3. HOLDING-TIME DISTRIBUTION 187
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Chapter 10 Introduction to Confiden
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10.1. SAMPLING DISTRIBUTIONS 195 Wh
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10.1. SAMPLING DISTRIBUTIONS 197 Ap
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10.3. CONFIDENCE INTERVALS FOR MEAN
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10.4. MEANING OF CONFIDENCE INTERVA
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10.5. GENERAL FORMATION OF CONFIDEN
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10.7. CONFIDENCE INTERVALS FOR PROP
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10.7. CONFIDENCE INTERVALS FOR PROP
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10.8. CONFIDENCE INTERVALS FOR DIFF
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10.10. R COMPUTATION 215 algebra, w
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10.13. OTHER CONFIDENCE LEVELS 217
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10.14. ONE MORE TIME: WHY DO WE USE
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Chapter 11 Introduction to Signific
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11.2. GENERAL TESTING BASED ON NORM
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11.5. ONE-SIDED HA 225 By checking
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11.6. EXACT TESTS 227 It is natural
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11.8. THE POWER OF A TEST 229 11.8
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11.9. WHAT’S WRONG WITH SIGNIFICA
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Chapter 12 General Statistical Esti
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12.1. GENERAL METHODS OF PARAMETRIC
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12.2. BIAS AND VARIANCE 247 people,
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12.2. BIAS AND VARIANCE 249 Moreove
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12.3. MORE ON THE ISSUE OF INDEPEND
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12.4. NONPARAMETRIC DISTRIBUTION ES
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12.5. BAYESIAN METHODS 259 his plan
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12.5. BAYESIAN METHODS 261 it now b
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12.5. BAYESIAN METHODS 263 number
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12.5. BAYESIAN METHODS 265 (b) ˆp,
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Chapter 13 Simultaneous Inference M
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13.2. SCHEFFE’S METHOD 269 You ca
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13.4. OTHER METHODS FOR SIMULTANEOU
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Chapter 14 Introduction to Model Bu
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14.1. “DESPERATE FOR DATA” 275
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14.1. “DESPERATE FOR DATA” 277
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14.2. ASSESSING “GOODNESS OF FIT
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14.4. ROBUSTNESS 281 bin width, or
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14.5. REAL POPULATIONS AND CONCEPTU
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Chapter 15 Relations Among Variable
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15.3. ADJUSTING FOR COVARIATES 287
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15.6. ESTIMATING THAT RELATIONSHIP
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15.6. ESTIMATING THAT RELATIONSHIP
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15.7. EXAMPLE: BASEBALL DATA 293
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15.9. EXAMPLE: BASEBALL DATA (CONT
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15.11. PREDICTION 297 matter, it ma
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15.12. PARAMETRIC ESTIMATION OF LIN
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15.12. PARAMETRIC ESTIMATION OF LIN
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15.14. DUMMY VARIABLES 303 15.14 Du
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15.16. WHAT DOES IT ALL MEAN?—EFF
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15.17. MODEL SELECTION 307 But look
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15.17. MODEL SELECTION 309 Foundati
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15.18. WHAT ABOUT THE ASSUMPTIONS?
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15.19. CASE STUDIES 313 have an und
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15.19. CASE STUDIES 315 Exercises N
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15.19. CASE STUDIES 317 8. Consider
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Chapter 16 Advanced Statistical Est
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16.2. THE DELTA METHOD: CONFIDENCE
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16.3. THE BOOTSTRAP METHOD FOR FORM
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16.3. THE BOOTSTRAP METHOD FOR FORM
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Chapter 17 Relations Among Variable
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17.2. THE CLASSIFICATION PROBLEM 33
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17.3. NONPARAMETRIC ESTIMATION OF R
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17.4. SYMMETRIC RELATIONS AMONG SEV
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17.5. SIMPSON’S (NON-)PARADOX 357
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17.5. SIMPSON’S (NON-)PARADOX 359
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Chapter 18 Describing “Failure”
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18.2. A CAUTIONARY TALE: THE BUS PA
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18.2. A CAUTIONARY TALE: THE BUS PA
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18.3. RESIDUAL-LIFE DISTRIBUTION 36
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Chapter 19 Advanced Multivariate Me
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19.1. CONDITIONAL DISTRIBUTIONS 377
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19.3. MIXTURE MODELS 383 Now that w
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19.4. TRANSFORM METHODS 385 Thus an
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19.4. TRANSFORM METHODS 387 19.4.2
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19.4. TRANSFORM METHODS 389 transfo
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19.5. VECTOR SPACE INTERPRETATIONS
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19.7. CONDITIONAL EXPECTATION AS A
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19.8. PROOF OF THE LAW OF TOTAL EXP
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19.8. PROOF OF THE LAW OF TOTAL EXP
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Chapter 20 Markov Chains One of the
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20.1. DISCRETE-TIME MARKOV CHAINS 4
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20.2. SIMULATION OF MARKOV CHAINS 4
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20.4. CONTINUOUS-TIME MARKOV CHAINS
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20.5. HITTING TIMES ETC. 419 least
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20.5. HITTING TIMES ETC. 421 First
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20.5. HITTING TIMES ETC. 423 valued
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20.5. HITTING TIMES ETC. 425 summon
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Chapter 21 Introduction to Queuing
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21.2. M/M/1 429 busy for approximat
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21.2. M/M/1 431 • Due to the memo
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21.3. MULTI-SERVER MODELS 433 Recal
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21.4. LOSS MODELS 435 1 = i,j,k π
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21.5. NONEXPONENTIAL SERVICE TIMES
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21.6. REVERSED MARKOV CHAINS 439 So
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21.6. REVERSED MARKOV CHAINS 441 21
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21.6. REVERSED MARKOV CHAINS 443 Re
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21.7. NETWORKS OF QUEUES 445 • Gi
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21.7. NETWORKS OF QUEUES 447 Let Li
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Appendix A Review of Matrix Algebra
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A.2. MATRIX TRANSPOSE 451 • Matri
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A.6. EIGENVALUES AND EIGENVECTORS 4
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Appendix B R Quick Start Here we pr
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B.3. FIRST SAMPLE PROGRAMMING SESSI
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B.4. SECOND SAMPLE PROGRAMMING SESS
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B.4. SECOND SAMPLE PROGRAMMING SESS
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B.6. COMPLEX NUMBERS 463 B.6 Comple
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