Search results for "tag:"Author Michael J. Panik""
Wiley Growth Curve Modeling
Book SynopsisSuitable for upper-undergraduate and graduate courses on growth modeling, this title presents an introduction to growth curve modeling and addresses how to monitor the change in variables over time since there is no "one size fits all" approach to growth measurement.Trade Review“Thus, it is an excellent resource for statisticians, public health analysts, biologists, botanists, economists, and demographers who require a modern review of statistical methods for modeling growth curves and analyzing longitudinal data.” (Zentralblatt MATH, 1 April 2015) Table of ContentsPreface xiii 1 Mathematical Preliminaries 1 1.1 Arithmetic Progression 1 1.2 Geometric Progression 2 1.3 The Binomial Formula 4 1.4 The Calculus of Finite Differences 5 1.5 The Number e 9 1.6 The Natural Logarithm 10 1.7 The Exponential Function 11 1.8 Exponential and Logarithmic Functions: Another Look 13 1.9 Change of Base of a Logarithm 14 1.10 The Arithmetic (Natural) Scale versus the Logarithmic Scale 15 1.11 Compound Interest Arithmetic 17 2 Fundamentals of Growth 21 2.1 Time Series Data 21 2.2 Relative and Average Rates of Change 21 2.3 Annual Rates of Change 25 2.3.1 Simple Rates of Change 25 2.3.2 Compounded Rates of Change 26 2.3.3 Comparing Two Time Series: Indexing Data to a Common Starting Point 30 2.4 Discrete versus Continuous Growth 32 2.5 The Growth of a Variable Expressed in Terms of the Growth of its Individual Arguments 36 2.6 Growth Rate Variability 46 2.7 Growth in a Mixture of Variables 47 3 Parametric Growth Curve Modeling 49 3.1 Introduction 49 3.2 The Linear Growth Model 50 3.3 The Logarithmic Reciprocal Model 51 3.4 The Logistic Model 52 3.5 The Gompertz Model 54 3.6 The Weibull Model 55 3.7 The Negative Exponential Model 56 3.8 The von Bertalanffy Model 57 3.9 The Log-Logistic Model 59 3.10 The Brody Growth Model 61 3.11 The Janoschek Growth Model 62 3.12 The Lundqvist–Korf Growth Model 63 3.13 The Hossfeld Growth Model 63 3.14 The Stannard Growth Model 64 3.15 The Schnute Growth Model 64 3.16 The Morgan–Mercer–Flodin (M–M–F) Growth Model 66 3.17 The McDill–Amateis Growth Model 68 3.18 An Assortment of Additional Growth Models 69 3.18.1 The Sloboda Growth Model 71 Appendix 3.A The Logistic Model Derived 71 Appendix 3.B The Gompertz Model Derived 74 Appendix 3.C The Negative Exponential Model Derived 75 Appendix 3.D The von Bertalanffy and Richards Models Derived 77 Appendix 3.E The Schnute Model Derived 81 Appendix 3.F The McDill–Amateis Model Derived 83 Appendix 3.G The Sloboda Model Derived 85 Appendix 3.H A Generalized Michaelis–Menten Growth Equation 86 4 Estimation of Trend 88 4.1 Linear Trend Equation 88 4.2 Ordinary Least Squares (OLS) Estimation 91 4.3 Maximum Likelihood (ML) Estimation 92 4.4 The SAS System 94 4.5 Changing the Unit of Time 109 4.5.1 Annual Totals versus Monthly Averages versus Monthly Totals 109 4.5.2 Annual Totals versus Quarterly Averages versus Quarterly Totals 110 4.6 Autocorrelated Errors 110 4.6.1 Properties of the OLS Estimators when ε Is AR(1) 111 4.6.2 Testing for the Absence of Autocorrelation: The Durbin–Watson Test 113 4.6.3 Detection of and Estimation with Autocorrelated Errors 115 4.7 Polynomial Models in t 126 4.8 Issues Involving Trended Data 136 4.8.1 Stochastic Processes and Time Series 137 4.8.2 Autoregressive Process of Order p 138 4.8.3 Random Walk Processes 141 4.8.4 Integrated Processes 145 4.8.5 Testing for Unit Roots 146 Appendix 4.A OLS Estimated and Related Growth Rates 158 4.A.1 The OLS Growth Rate 158 4.A.2 The Log-Difference (LD) Growth Rate 161 4.A.3 The Average Annual Growth Rate 161 4.A.4 The Geometric Average Growth Rate 162 5 Dynamic Site Equations Obtained from Growth Models 164 5.1 Introduction 164 5.2 Base-Age-Specific (BAS) Models 164 5.3 Algebraic Difference Approach (ADA) Models 166 5.4 Generalized Algebraic Difference Approach (GADA) Models 169 5.5 A Site Equation Generating Function 179 5.5.1 ADA Derivations 180 5.5.2 GADA Derivations 180 5.6 The Grounded GADA (g-GADA) Model 184 Appendix 5.A Glossary of Selected Forestry Terms 186 6 Nonlinear Regression 188 6.1 Intrinsic Linearity/Nonlinearity 188 6.2 Estimation of Intrinsically Nonlinear Regression Models 190 6.2.1 Nonlinear Least Squares (NLS) 191 6.2.2 Maximum Likelihood (ML) 195 Appendix 6.A Gauss–Newton Iteration Scheme: The Single Parameter Case 214 Appendix 6.B Gauss–Newton Iteration Scheme: The r Parameter Case 217 Appendix 6.C The Newton–Raphson and Scoring Methods 220 Appendix 6.D The Levenberg–Marquardt Modification/Compromise 222 Appendix 6.E Selection of Initial Values 223 6.E.1 Initial Values for the Logistic Curve 224 6.E.2 Initial Values for the Gompertz Curve 224 6.E.3 Initial Values for the Weibull Curve 224 6.E.4 Initial Values for the Chapman–Richards Curve 225 7 Yield–Density Curves 226 7.1 Introduction 226 7.2 Structuring Yield–Density Equations 227 7.3 Reciprocal Yield–Density Equations 228 7.3.1 The Shinozaki and Kira Yield–Density Curve 228 7.3.2 The Holliday Yield–Density Curves 229 7.3.3 The Farazdaghi and Harris Yield–Density Curve 230 7.3.4 The Bleasdale and Nelder Yield–Density Curve 231 7.4 Weight of a Plant Part and Plant Density 239 7.5 The Expolinear Growth Equation 242 7.6 The Beta Growth Function 249 7.7 Asymmetric Growth Equations (for Plant Parts) 253 7.7.1 Model I 254 7.7.2 Model II 255 7.7.3 Model III 256 Appendix 7.A Derivation of the Shinozaki and Kira Yield–Density Curve 257 Appendix 7.B Derivation of the Farazdaghi and Harris Yield–Density Curve 258 Appendix 7.C Derivation of the Bleasdale and Nelder Yield–Density Curve 259 Appendix 7.D Derivation of the Expolinear Growth Curve 261 Appendix 7.E Derivation of the Beta Growth Function 263 Appendix 7.F Derivation of Asymmetric Growth Equations 266 Appendix 7.G Chanter Growth Function 269 8 Nonlinear Mixed-Effects Models for Repeated Measurements Data 270 8.1 Some Basic Terminology Concerning Experimental Design 270 8.2 Model Specification 271 8.2.1 Model and Data Elements 271 8.2.2 A Hierarchical (Staged) Model 272 8.3 Some Special Cases of the Hierarchical Global Model 274 8.4 The SAS/STAT NLMIXED Procedure for Fitting Nonlinear Mixed-Effects Model 276 9 Modeling the Size and Growth Rate Distributions of Firms 293 9.1 Introduction 293 9.2 Measuring Firm Size and Growth 294 9.3 Modeling the Size Distribution of Firms 294 9.4 Gibrat’s Law (GL) 297 9.5 Rationalizing the Pareto Firm Size Distribution 299 9.6 Modeling the Growth Rate Distribution of Firms 300 9.7 Basic Empirics of Gibrat’s Law (GL) 305 9.7.1 Firm Size and Expected Growth Rates 305 9.7.2 Firm Size and Growth Rate Variability 308 9.7.3 Econometric Issues 310 9.7.4 Persistence of Growth Rates 312 9.8 Conclusion 313 Appendix 9.A Kernel Density Estimation 314 9.A.1 Motivation 314 9.A.2 Weighting Functions 315 9.A.3 Smooth Weighting Functions: Kernel Estimators 316 Appendix 9.B The Log-Normal and Gibrat Distributions 322 9.B.1 Derivation of Log-Normal Forms 322 9.B.2 Generalized Log-Normal Distribution 325 Appendix 9.C The Theory of Proportionate Effect 326 Appendix 9.D Classical Laplace Distribution 328 9.D.1 The Symmetric Case 328 9.D.2 The Asymmetric Case 330 9.D.3 The Generalized Laplace Distribution 331 9.D.4 The Log-Laplace Distribution 332 Appendix 9.E Power-Law Behavior 332 9.E.1 Pareto’s Power Law 333 9.E.2 Generalized Pareto Distributions 335 9.E.3 Zipf’s Power Law 337 Appendix 9.F The Yule Distribution 338 Appendix 9.G Overcoming Sample Selection Bias 339 9.G.1 Selection and Gibrat’s Law (GL) 339 9.G.2 Characterizing Selection Bias 339 9.G.3 Correcting for Selection Bias: The Heckman (1976 1979) Two-Step Procedure 342 9.G.4 The Heckman Two-Step Procedure Under Modified Selection 345 10 Fundamentals of Population Dynamics 352 10.1 The Concept of a Population 352 10.2 The Concept of Population Growth 353 10.3 Modeling Population Growth 354 10.4 Exponential (Density-Independent) Population Growth 357 10.4.1 The Continuous Case 357 10.4.2 The Discrete Case 359 10.4.3 Malthusian Population Growth Dynamics 361 10.5 Density-Dependent Population Growth 363 10.5.1 Logistic Growth Model 364 10.6 Beverton–Holt Model 371 10.7 Ricker Model 374 10.8 Hassell Model 377 10.9 Generalized Beverton–Holt (B–H) Model 380 10.10 Generalized Ricker Model 382 Appendix 10.A A Glossary of Selected Population Demography/Ecology Terms 389 Appendix 10.B Equilibrium and Stability Analysis 391 10.B.1 Stable and Unstable Equilibria 391 10.B.2 The Need for a Qualitative Analysis of Equilibria 392 10.B.3 Equilibria and Stability for Continuous-Time Models 392 10.B.4 Equilibria and Stability for Discrete-Time Models 394 Appendix 10.C Discretization of the Continuous-Time Logistic Growth Equation 400 Appendix 10.D Derivation of the B–H S–R Relationship 401 Appendix 10.E Derivation of the Ricker S–R Relationship 403 Appendix A 405 Table A.1 Standard Normal Areas (Z Is N(0, 1)) 405 Table A.2 Quantiles of Student’s t Distribution (T Is tv) 407 Table A.3 Quantiles of the Chi-Square Distribution (X Is 𝛘v 2) 408 Table A.4 Quantiles of Snedecor’s F Distribution (F Is Fv1, v2) 410 Table A.5 Durbin–Watson DW Statistic—5% Significance Points dL and dU (n is the sample size and k′ is the number of regressors excluding the intercept) 415 Table A.6 Empirical Cumulative Distribution of τ for ρ = 1 419 References 420 Index 431
£108.86
John Wiley & Sons Inc Introduction to Quantitative Methods in Business
Book SynopsisA well-balanced and accessible introduction to the elementary quantitative methods and Microsoft Office Excel applications used to guide business decision making Featuring quantitative techniques essential for modeling modern business situations, Introduction to Quantitative Methods in Business: With Applications Using Microsoft Office Excel provides guidance to assessing real-world data sets using Excel. The book presents a balanced approach to the mathematical tools and techniques with applications used in the areas of business, finance, economics, marketing, and operations. The authors begin by establishing a solid foundation of basic mathematics and statistics before moving on to more advanced concepts. The first part of the book starts by developing basic quantitative techniques such as arithmetic operations, functions and graphs, and elementary differentiations (rates of change), and integration. After a review of these techniques, theTable of ContentsPreface xiii 1. The Mathematical Toolbox 1 1.1 Introduction 1 1.2 Linear Functions 2 1.3 Solving a Simple Linear Equation for one Unknown Variable 3 1.3.1 Solving Two Simultaneous Linear Equations for Two Unknown Variables 4 1.4 Summation Notation 6 1.5 Sets 12 1.5.1 Subset, Empty Set, Universal Set, and Complement of A Set 13 1.5.2 Intersection and Union 14 1.6 Functions and Graphs 15 1.6.1 Vertical Line Test 16 1.7 Working with Functions 17 1.7.1 Evaluating Functions 17 1.7.2 Graphing Functions 18 1.8 Differentiation and Integration 21 1.8.1 Derivative 22 1.8.2 Derivatives of Logarithmic and Exponential Functions 26 1.8.3 Higher Order Derivatives 26 1.8.4 Integration 28 1.8.5 The Definite Integral 29 1.8.6 Some Rules of Integration 31 1.9 Excel Applications 34 Chapter 1 Review 40 Eercises 41 Appendix 1.A: A Review of Basic Mathematics 45 Eercises 63 2. Applications of Linear and Nonlinear Functions 66 2.1 Introduction 66 2.2 Linear Demand and Supply Functions 66 2.3 Linear Total Cost and Total Revenue Functions 69 2.4 Market Equilibrium 71 2.5 Graphical Presentation of Equilibrium 72 2.6 Applications of Nonlinear Functions 73 2.7 Present Value of an Income Stream 78 2.8 Average Values 79 2.9 Marginal Values 80 2.10 Elasticity 81 2.11 Some Additional Business Applications 84 2.12 Excel Applications 84 Chapter 2 Review 86 Eercises 87 Excel Applications 90 3. Optimization 91 3.1 Introduction 91 3.2 Unconstrained Optimization 91 3.2.1 Models of Profit and Revenue Maximization 91 3.2.2 Solution by Trial and Error (Approximate) Method 92 3.2.3 Solution Using the Calculus Approach 93 3.2.4 Solution by Trial and Error (Approximate) Method 96 3.2.5 Solution Using the Calculus Approach 97 3.3 Models of Cost Minimization: Inventory Cost Functions and Eoq 99 3.3.1 Solution by Trial and Error Method 101 3.3.2 Solution Using the Calculus Approach 103 3.4 Constrained Optimization: Linear Programming 105 3.4.1 Linear Programming: Maximization 106 3.4.1.1 Solution by Graphical Method: First Approach 106 3.4.1.2 Solution by Graphical Method: Second Approach 109 3.4.2 Linear Programming: Minimization 114 3.5 Excel Applications 121 Chapter 3 Review 125 Chapter 3 Eercises 126 Excel Applications 130 4. What Is Business Statistics? 131 4.1 Introduction 131 4.2 Data Description 132 4.2.1 Some Important Concepts in Statistics 132 4.2.2 Scales of Data Measurement 132 4.3 Descriptive Statistics: Tabular and Graphical Techniques 134 4.4 Descriptive Statistics: Numerical Measures of Central Tendency or Location of Data 144 4.4.1 Population Mean 144 4.4.2 Sample Mean 145 4.4.3 Weighted Mean 147 4.4.4 Mean of a Frequency Distribution: Grouped Data 148 4.4.5 Geometric Mean 149 4.4.6 Median 151 4.4.7 Quantiles, Quartiles, 4.5 Descriptive Statistics: Measures of Dispersion—Variability or Spread 155 4.5.1 Range 155 4.5.2 Variance 155 4.5.3 Standard Deviation 158 4.5.4 Coefficient of Variation 160 4.5.5 Some Important Uses of the Standard Deviation 163 4.5.6 Empirical Rule 165 4.6 Measuring Skewness 166 4.7 Excel Applications 169 Chapter 4 Review 186 Eercises 188 Excel Applications 191 5. Probability and Applications 194 5.1 Introduction 194 5.2 Some Useful Definitions 195 5.3 Probability Sources 196 5.3.1 Objective Probability 196 5.3.2 Subjective Probability 196 5.4 Some Useful Definitions Involving Sets of Events in the Sample Space 197 Complement of a Given Set A 199 Mutually Exclusive Events 200 5.5 Probability Laws 200 5.5.1 General Rule of Addition 200 5.5.2 Rule of Complements 202 5.5.3 Conditional Probability 202 5.5.4 General Rule of Multiplication (Product Rule) 203 5.5.5 Independent Events 204 5.5.6 Probability Tree Approach 204 5.6 Contingency Table 208 5.7 Excel Applications 213 Chapter 5 Review 214 Eercises 215 Excel Applications 218 6. Random Variables and Probability Distributions 219 6.1 Introduction 219 6.2 Probability Distribution of a Discrete Random Variable X 220 6.3 Expected Value, Variance, and Standard Deviation of a Discrete Random Variable X 222 6.3.1 Some Basic Rules of Expectation 224 6.3.2 Some Useful Properties of Variance of X 225 6.3.3 Applications of Expected Values 225 6.4 Continuous Random Variables and Their Probability Distributions 230 6.5 A Specific Discrete Probabilty Distribution: the Binomial Case 232 6.5.1 Binomial Probability Distribution 232 6.5.2 Mean and Standard Deviation of the Binomial Random Variable 237 6.5.3 Cumulative Binomial Probability Distribution 238 6.6 Excel Applications 241 Chapter 6 Review 245 Eercises 245 Appendix 6.A 252 About the Companion Website 263 Index 265
£92.70
John Wiley & Sons Inc Introduction to Quantitative Methods in Business
Book SynopsisSet includes Introduction to Quantitative Methods in Business: With Applications Using Microsoft Office Excel ISBN 978-1-119-22097-8 and the accompanying Solutions Manual ISBN 978-1-119-22102-9 A well-balanced and accessible introduction to the elementary quantitative methods and Microsoft Office Excel applications used to guide business decision making Featuring quantitative techniques essential for modeling modern business situations, Introduction to Quantitative Methods in Business: With Applications Using Microsoft Office Excel provides guidance to assessing real-world data sets using Excel. The book presents a balanced approach to the mathematical tools and techniques with applications used in the areas of business, finance, economics, marketing, and operations. The authors begin by establishing a solid foundation of basic mathematics and statistics before moving on to more advanced concepts. The first part of the book starts by develop
£107.96
John Wiley & Sons Inc Solutions Manual to Accompany Introduction to
Book SynopsisSolutions Manual to accompany Introduction to Quantitative Methods in Business: With Applications Using Microsoft(R) Office Excel(R).Table of Contents1. The Mathematical Toolbox: A Summary 1 1.2 Linear Functions 1 1.3.1 Solving Two Simultaneous Linear Equations 1 1.4 Summation Notation 2 1.5 Sets 3 1.6 Functions and Graphs 3 1.7 Working with Functions 4 1.8 Differentiation and Integration 5 Solutions to Odd-Numbered Exercises 8 2. Applications of Linear and Nonlinear Functions: A Summary 32 2.2 Linear Demand and Supply Functions 32 2.3 Linear Total Cost and Total Revenue Functions 33 2.4 Market Equilibrium 33 2.6 Applications of Nonlinear Functions 34 2.7 Present Value of an Income Stream 35 2.8 Average Values 35 2.9 Marginal Values 36 2.10 Elasticity 36 Solutions to Odd-Numbered Exercises 37 3. Optimization: A Summary 47 3.2 Unconstrained Optimization 47 3.2.1 Models of Profit and Revenue Maximization 47 3.2.3 Solution Using the Calculus Approach 47 3.2.5 Solution Using the Calculus Approach 47 3.3 Models of Cost Minimization: Inventory Cost Functions and Economic Order Quantity (EOQ) 48 3.3.2 Solution Using the Calculus Approach 49 3.4 Constrained Optimization: Linear Programming 50 3.4.1 Linear Programming: Maximization 50 3.4.2 Linear Programming: Minimization 51 Solutions to Odd-Numbered Exercises 52 4. What Is Business Statistics? 68 4.3 Descriptive Statistics: Tabular and Graphical Techniques 68 4.4 Descriptive Statistics: Numerical Measures of Central Tendency or Location of Data 70 4.4.1 Population Mean 70 4.4.2 Sample Mean 70 4.4.3 Weighted Mean 70 4.4.4 Mean of a Frequency Distribution: Grouped Data 71 4.4.5 Geometric Mean 71 4.4.6 Median 71 4.4.7 Quantiles, Quartiles, Deciles, and Percentiles 71 4.4.8 Mode 72 4.5 Descriptive Statistics: Measures of Dispersion (Variability or Spread) 73 4.5.2 Variance 73 4.5.3 Standard Deviation 74 4.5.4 Coefficient of Variation 74 4.5.5 Some Important Uses of the Standard Deviation 75 1. Standardization of Values 75 2. Chebysheff’s Theorem 75 4.5.6 Empirical Rule 75 4.6 Measuring Skewness 76 Solutions to Odd-Numbered Exercises 76 5. Probability and Applications 96 5.2 Some Useful Definitions 96 5.3 Probability Sources 96 5.3.1 Objective Probability 96 5.4 Some Useful Definitions Involving Sets of Events in the Sample Space 96 5.5 Probability Laws 97 5.5.2 Rule of Complements 97 5.5.3 Conditional Probability 97 5.5.4 General Multiplication Rule (Product Rule) 97 5.5.5 Independent Events 98 5.5.6 Probability Tree Approach 98 5.6 Contingency Table 98 Solutions to Odd-Numbered Exercises 100 6. Random Variables and Probability Distributions 105 6.2 Probability Distribution of a Discrete Random Variable X 105 6.3 Expected Value, Variance, and Standard Deviation of a Discrete Random Variable 106 6.3.1 Some Basic Rules of Expectation 106 6.3.2 Some Useful Properties of the Variance of X 107 6.4 Continuous Random Variables and Their Probability Distributions 107 6.5 A Specific Discrete Probability Distribution: The Binomial Case 108 6.5.1 Binomial Probability Distribution 108 6.5.2 Mean and Standard Deviation of the Binomial Random Variable 109 6.5.3 Cumulative Binomial Probability Distribution 110 Solutions to Odd-Numbered Exercises 110 Index 119
£24.65
John Wiley & Sons Inc Linear Programming and Resource Allocation
Book SynopsisGuides in the application of linear programming to firm decision making, with the goal of giving decision-makers a better understanding of methods at their disposal Useful as a main resource or as a supplement in an economics or management science course, this comprehensive book addresses the deficiencies of other texts when it comes to covering linear programming theoryespecially where data envelopment analysis (DEA) is concernedand provides the foundation for the development of DEA. Linear Programming and Resource Allocation Modeling begins by introducing primal and dual problems via an optimum product mix problem, and reviews the rudiments of vector and matrix operations. It then goes on to cover: the canonical and standard forms of a linear programming problem; the computational aspects of linear programming; variations of the standard simplex theme; duality theory; single- and multiple- process production functions; sensitivity analysis of the optimal solution; structural changTable of ContentsPreface xi Symbols and Abbreviations xv 1 Introduction 1 2 Mathematical Foundations 13 2.1 Matrix Algebra 13 2.2 Vector Algebra 20 2.3 Simultaneous Linear Equation Systems 22 2.4 Linear Dependence 26 2.5 Convex Sets and n-Dimensional Geometry 29 3 Introduction to Linear Programming 35 3.1 Canonical and Standard Forms 35 3.2 A Graphical Solution to the Linear Programming Problem 37 3.3 Properties of the Feasible Region 38 3.4 Existence and Location of Optimal Solutions 38 3.5 Basic Feasible and Extreme Point Solutions 39 3.6 Solutions and Requirement Spaces 41 4 Computational Aspects of Linear Programming 43 4.1 The Simplex Method 43 4.2 Improving a Basic Feasible Solution 48 4.3 Degenerate Basic Feasible Solutions 66 4.4 Summary of the Simplex Method 69 5 Variations of the Standard Simplex Routine 71 5.1 The M-Penalty Method 71 5.2 Inconsistency and Redundancy 78 5.3 Minimization of the Objective Function 85 5.4 Unrestricted Variables 86 5.5 The Two-Phase Method 87 6 Duality Theory 95 6.1 The Symmetric Dual 95 6.2 Unsymmetric Duals 97 6.3 Duality Theorems 100 6.4 Constructing the Dual Solution 106 6.5 Dual Simplex Method 113 6.6 Computational Aspects of the Dual Simplex Method 114 6.7 Summary of the Dual Simplex Method 121 7 Linear Programming and the Theory of the Firm 123 7.1 The Technology of the Firm 123 7.2 The Single-Process Production Function 125 7.3 The Multiactivity Production Function 129 7.4 The Single-Activity Profit Maximization Model 139 7.5 The Multiactivity Profit Maximization Model 143 7.6 Profit Indifference Curves 146 7.7 Activity Levels Interpreted as Individual Product Levels 148 7.8 The Simplex Method as an Internal Resource Allocation Process 155 7.9 The Dual Simplex Method as an Internalized Resource Allocation Process 157 7.10 A Generalized Multiactivity Profit-Maximization Model 157 7.11 Factor Learning and the Optimum Product-Mix Model 161 7.12 Joint Production Processes 165 7.13 The Single-Process Product Transformation Function 167 7.14 The Multiactivity Joint-Production Model 171 7.15 Joint Production and Cost Minimization 180 7.16 Cost Indifference Curves 184 7.17 Activity Levels Interpreted as Individual Resource Levels 186 8 Sensitivity Analysis 195 8.1 Introduction 195 8.2 Sensitivity Analysis 195 8.2.1 Changing an Objective Function Coefficient 196 8.2.2 Changing a Component of the Requirements Vector 200 8.2.3 Changing a Component of the Coefficient Matrix 202 8.3 Summary of Sensitivity Effects 209 9 Analyzing Structural Changes 217 9.1 Introduction 217 9.2 Addition of a New Variable 217 9.3 Addition of a New Structural Constraint 219 9.4 Deletion of a Variable 223 9.5 Deletion of a Structural Constraint 223 10 Parametric Programming 227 10.1 Introduction 227 10.2 Parametric Analysis 227 10.2.1 Parametrizing the Objective Function 228 10.2.2 Parametrizing the Requirements Vector 236 10.2.3 Parametrizing an Activity Vector 245 10.A Updating the Basis Inverse 256 11 Parametric Programming and the Theory of the Firm 257 11.1 The Supply Function for the Output of an Activity (or for an Individual Product) 257 11.2 The Demand Function for a Variable Input 262 11.3 The Marginal (Net) Revenue Productivity Function for an Input 269 11.4 The Marginal Cost Function for an Activity (or Individual Product) 276 11.5 Minimizing the Cost of Producing a Given Output 284 11.6 Determination of Marginal Productivity, Average Productivity, Marginal Cost, and Average Cost Functions 286 12 Duality Revisited 297 12.1 Introduction 297 12.2 A Reformulation of the Primal and Dual Problems 297 12.3 Lagrangian Saddle Points 311 12.4 Duality and Complementary Slackness Theorems 315 13 Simplex-Based Methods of Optimization 321 13.1 Introduction 321 13.2 Quadratic Programming 321 13.3 Dual Quadratic Programs 325 13.4 Complementary Pivot Method 329 13.5 Quadratic Programming and Activity Analysis 335 13.6 Linear Fractional Functional Programming 338 13.7 Duality in Linear Fractional Functional Programming 347 13.8 Resource Allocation with a Fractional Objective 353 13.9 Game Theory and Linear Programming 356 13.9.1 Introduction 356 13.9.2 Matrix Games 357 13.9.3 Transformation of a Matrix Game to a Linear Program 361 13.A Quadratic Forms 363 13.A.1 General Structure 363 13.A.2 Symmetric Quadratic Forms 366 13.A.3 Classification of Quadratic Forms 367 13.A.4 Necessary Conditions for the Definiteness and Semi-Definiteness of Quadratic Forms 368 13.A.5 Necessary and Sufficient Conditions for the Definiteness and Semi-Definiteness of Quadratic Forms 369 14 Data Envelopment Analysis (DEA) 373 14.1 Introduction 373 14.2 Set Theoretic Representation of a Production Technology 374 14.3 Output and Input Distance Functions 377 14.4 Technical and Allocative Efficiency 379 14.4.1 Measuring Technical Efficiency 379 14.4.2 Allocative, Cost, and Revenue Efficiency 382 14.5 Data Envelopment Analysis (DEA) Modeling 385 14.6 The Production Correspondence 386 14.7 Input-Oriented DEA Model under CRS 387 14.8 Input and Output Slack Variables 390 14.9 Modeling VRS 398 14.9.1 The Basic BCC (1984) DEA Model 398 14.9.2 Solving the BCC (1984) Model 400 14.9.3 BCC (1984) Returns to Scale 401 14.10 Output-Oriented DEA Models 402 References and Suggested Reading 405 Index 411
£89.96
Taylor & Francis Ltd Mathematical Analysis and Optimization for Economists
a huge range and FREE tracked UK delivery on ALL orders.
£999.99
Taylor & Francis Ltd Mathematical Analysis and Optimization for Economists
a huge range and FREE tracked UK delivery on ALL orders.
£999.99
John Wiley & Sons Inc Stochastic Differential Equations
Book SynopsisA beginner's guide to stochastic growth modeling The chief advantage of stochastic growth models over deterministic models is that they combine both deterministic and stochastic elements of dynamic behaviors, such as weather, natural disasters, market fluctuations, and epidemics. This makes stochastic modeling a powerful tool in the hands of practitioners in fields for which population growth is a critical determinant of outcomes. However, the background requirements for studying SDEs can be daunting for those who lack the rigorous course of study received by math majors. Designed to be accessible to readers who have had only a few courses in calculus and statistics, this book offers a comprehensive review of the mathematical essentials needed to understand and apply stochastic growth models. In addition, the book describes deterministic and stochastic applications of population growth models including logistic, generalized logistic, Gompertz, negative exponentiTrade Review"An indispensable resource for students and practitioners with limited exposure tomathematics and statistics, Stochastic Differential Equations: An Introduction withApplications in Population Dynamics Modeling is an excellent fit for advanced under-graduates and beginning graduate students, as well as practitioners who need a gentleintroduction to SDEs" Mathematical Reviews, October 2017Table of ContentsDedication x Preface xi Symbols and Abbreviations xiii 1 Mathematical Foundations 1: Point-Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration 1 1.1 Set Notation and Operations 1 1.1.1 Sets and Set Inclusion 1 1.1.2 Set Algebra 2 1.2 Single-Valued Functions 4 1.3 Real and Extended Real Numbers 6 1.4 Metric Spaces 7 1.5 Limits of Sequences 8 1.6 Point-Set Theory 10 1.7 Continuous Functions 12 1.8 Operations on Sequences of Sets 13 1.9 Classes of Subsets of Ω 15 1.9.1 Topological Space 15 1.9.2 σ-Algebra of Sets and the Borel σ-Algebra 15 1.10 Set and Measure Functions 17 1.10.1 Set Functions 17 1.10.2 Measure Functions 18 1.10.3 Outer Measure Functions 19 1.10.4 Complete Measure Functions 21 1.10.5 Lebesgue Measure 21 1.10.6 Measurable Functions 23 1.10.7 Lebesgue Measurable Functions 26 1.11 Normed Linear Spaces 27 1.11.1 Space of Bounded Real-Valued Functions 27 1.11.2 Space of Bounded Continuous Real-Valued Functions 28 1.11.3 Some Classical Banach Spaces 29 1.12 Integration 31 1.12.1 Integral of a Non-negative Simple Function 32 1.12.2 Integral of a Non-negative Measurable Function Using Simple Functions 33 1.12.3 Integral of a Measurable Function 33 1.12.4 Integral of a Measurable Function on a Measurable Set 34 1.12.5 Convergence of Sequences of Functions 35 2 Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables 37 2.1 Probability Spaces 37 2.2 Probability Distributions 42 2.3 The Expectation of a Random Variable 49 2.3.1 Theoretical Underpinnings 49 2.3.2 Computational Considerations 50 2.4 Moments of a Random Variable 52 2.5 Multiple Random Variables 54 2.5.1 The Discrete Case 54 2.5.2 The Continuous Case 59 2.5.3 Expectations and Moments 63 2.5.4 The Multivariate Discrete and Continuous Cases 69 2.6 Convergence of Sequences of Random Variables 72 2.6.1 Almost Sure Convergence 73 2.6.2 Convergence in Lp,p>0 73 2.6.3 Convergence in Probability 75 2.6.4 Convergence in Distribution 75 2.6.5 Convergence of Expectations 76 2.6.6 Convergence of Sequences of Events 78 2.6.7 Applications of Convergence of Random Variables 79 2.7 A Couple of Important Inequalities 80 Appendix 2.A The Conditional Expectation E(X|Y) 81 3 Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion 85 3.1 Stochastic Processes 85 3.1.1 Finite-Dimensional Distributions of a Stochastic Process 86 3.1.2 Selected Characteristics of Stochastic Processes 88 3.1.3 Filtrations of A 89 3.2 Martingales 91 3.2.1 Discrete-Time Martingales 91 3.2.1.1 Discrete-Time Martingale Convergence 93 3.2.2 Continuous-Time Martingales 96 3.2.2.1 Continuous-Time Martingale Convergence 97 3.2.3 Martingale Inequalities 97 3.3 Path Regularity of Stochastic Processes 98 3.4 Symmetric Random Walk 99 3.5 Brownian Motion 100 3.5.1 Standard Brownian Motion 100 3.5.2 BM as a Markov Process 104 3.5.3 Constructing BM 106 3.5.3.1 BM Constructed from N(0, 1) Random Variables 106 3.5.3.2 BM as the Limit of Symmetric Random Walks 108 3.5.4 White Noise Process 109 Appendix 3.A Kolmogorov Existence Theorem: Another Look 109 Appendix 3.B Nondifferentiability of BM 110 4 Mathematical Foundations 4: Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation 113 4.1 Introduction 113 4.2 Stochastic Integration: The Itô Integral 114 4.3 One-Dimensional Itô Formula 120 4.4 Martingale Representation Theorem 126 4.5 Multidimensional Itô Formula 127 Appendix 4.A Itô’s Formula 129 Appendix 4.B Multidimensional Itô Formula 130 5 Stochastic Differential Equations 133 5.1 Introduction 133 5.2 Existence and Uniqueness of Solutions 134 5.3 Linear SDEs 136 5.3.1 Strong Solutions to Linear SDEs 137 5.3.2 Properties of Solutions 147 5.3.3 Solutions to SDEs as Markov Processes 152 5.4 SDEs and Stability 154 Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 159 5.A.1 Linear Homogeneous Variety 159 5.A.2 Linear Variety 161 Appendix 5.B Integrating Factors and Variation of Parameters 162 5.B.1 Integrating Factors 163 5.B.2 Variation of Parameters 164 6 Stochastic Population Growth Models 167 6.1 Introduction 167 6.2 A Deterministic Population Growth Model 168 6.3 A Stochastic Population Growth Model 169 6.4 Deterministic and Stochastic Logistic Growth Models 170 6.5 Deterministic and Stochastic Generalized Logistic Growth Models 174 6.6 Deterministic and Stochastic Gompertz Growth Models 177 6.7 Deterministic and Stochastic Negative Exponential Growth Models 179 6.8 Deterministic and Stochastic Linear Growth Models 181 6.9 Stochastic Square-Root Growth Model with Mean Reversion 182 Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect 184 Appendix 6.B Reducible SDEs 189 7 Approximation and Estimation of Solutions to Stochastic Differential Equations 193 7.1 Introduction 193 7.2 Iterative Schemes for Approximating SDEs 194 7.2.1 The EM Approximation 194 7.2.2 Strong and Weak Convergence of the EM Scheme 196 7.2.3 The Milstein (Second-Order) Approximation 196 7.3 The Lamperti Transformation 199 7.4 Variations on the EM and Milstein Schemes 203 7.5 Local Linearization Techniques 205 7.5.1 The Ozaki Method 205 7.5.2 The Shoji–Ozaki Method 207 7.5.3 The Rate of Convergence of the Local Linearization Method 211 Appendix 7.A Stochastic Taylor Expansions 212 Appendix 7.B The EM and Milstein Discretizations 217 7.B.1 The EM Scheme 217 7.B.2 The Milstein Scheme 218 Appendix 7.C The Lamperti Transformation 219 8 Estimation of Parameters of Stochastic Differential Equations 221 8.1 Introduction 221 8.2 The Transition Probability Density Function Is Known 222 8.3 The Transition Probability Density Function Is Unknown 227 8.3.1 Parameter Estimation via Approximation Methods 228 8.3.1.1 The EM Routine 228 8.3.1.2 The Ozaki Routine 230 8.3.1.3 The SO Routine 233 Appendix 8.A The ML Technique 235 Appendix 8.B The Log-Normal Probability Distribution 238 Appendix 8.C The Markov Property, Transitional Densities, and the Likelihood Function of the Sample 239 Appendix 8.D Change of Variable 241 Appendix A: A Review of Some Fundamental Calculus Concepts 245 Appendix B: The Lebesgue Integral 259 Appendix C: Lebesgue–Stieltjes Integral 261 Appendix D: A Brief Review of Ordinary Differential Equations 263 References 275 Index 279
£999.99
John Wiley & Sons Inc Statistical Inference
Book SynopsisThis concise, easily accessible introduction to descriptive and inferential techniques presents the essentials of basic statistics for readers seeking to acquire a working knowledge of statistical concepts, measures, and procedures.Trade Review“The book is addressed to courses on probability, mathematical statistics, and statistical inference at the upper-undergraduate and graduate levels. It also serves as a valuable reference for researchers and practitioners who would like to develop further insights into essential statistical tools.” (Zentralblatt Math, 1 August 2013) “If an undergraduate student seeks a guide that will introduce the basic ideas of statistics, or a lecturer wants interesting life examples and a source of valid intuitions to improve his teaching skills, then this book is a great place to start. . . This book, with its explanations of basic intuitions, its many examples, the easy language, and a minimal requirement for mathematical training, is a good self-contained starting point to prepare one for the jump into those heavier works.” (Computing Reviews, 30 September 2013)Table of ContentsPreface xv 1 The Nature of Statistics 1 1.1 Statistics Defined 1 1.2 The Population and the Sample 2 1.3 Selecting a Sample from a Population 3 1.4 Measurement Scales 4 1.5 Let us Add 6 Exercises 7 2 Analyzing Quantitative Data 9 2.1 Imposing Order 9 2.2 Tabular and Graphical Techniques: Ungrouped Data 9 2.3 Tabular and Graphical Techniques: Grouped Data 11 Exercises 16 Appendix 2.A Histograms with Classes of Different Lengths 18 3 Descriptive Characteristics of Quantitative Data 22 3.1 The Search for Summary Characteristics 22 3.2 The Arithmetic Mean 23 3.3 The Median 26 3.4 The Mode 27 3.5 The Range 27 3.6 The Standard Deviation 28 3.7 Relative Variation 33 3.8 Skewness 34 3.9 Quantiles 36 3.10 Kurtosis 38 3.11 Detection of Outliers 39 3.12 So What Do We Do with All This Stuff? 41 Exercises 47 Appendix 3.A Descriptive Characteristics of Grouped Data 51 3.A.1 The Arithmetic Mean 52 3.A.2 The Median 53 3.A.3 The Mode 55 3.A.4 The Standard Deviation 57 3.A.5 Quantiles (Quartiles, Deciles, and Percentiles) 58 4 Essentials of Probability 61 4.1 Set Notation 61 4.2 Events within the Sample Space 63 4.3 Basic Probability Calculations 64 4.4 Joint, Marginal, and Conditional Probability 68 4.5 Sources of Probabilities 73 Exercises 75 5 Discrete Probability Distributions and Their Properties 81 5.1 The Discrete Probability Distribution 81 5.2 The Mean, Variance, and Standard Deviation of a Discrete Random Variable 85 5.3 The Binomial Probability Distribution 89 5.3.1 Counting Issues 89 5.3.2 The Bernoulli Probability Distribution 91 5.3.3 The Binomial Probability Distribution 91 Exercises 96 6 The Normal Distribution 101 6.1 The Continuous Probability Distribution 101 6.2 The Normal Distribution 102 6.3 Probability as an Area Under the Normal Curve 104 6.4 Percentiles of the Standard Normal Distribution and Percentiles of the Random Variable X 114 Exercises 116 Appendix 6.A The Normal Approximation to Binomial Probabilities 120 7 Simple Random Sampling and the Sampling Distribution of the Mean 122 7.1 Simple Random Sampling 122 7.2 The Sampling Distribution of the Mean 123 7.3 Comments on the Sampling Distribution of the Mean 127 7.4 A Central Limit Theorem 130 Exercises 132 Appendix 7.A Using a Table of Random Numbers 133 Appendix 7.B Assessing Normality via the Normal Probability Plot 136 Appendix 7.C Randomness, Risk, and Uncertainty 139 7.C.1 Introduction to Randomness 139 7.C.2 Types of Randomness 142 7.C.2.1 Type I Randomness 142 7.C.2.2 Type II Randomness 143 7.C.2.3 Type III Randomness 143 7.C.3 Pseudo-Random Numbers 144 7.C.4 Chaotic Behavior 145 7.C.5 Risk and Uncertainty 146 8 Confidence Interval Estimation of m 152 8.1 The Error Bound on X as an Estimator of m 152 8.2 A Confidence Interval for the Population Mean m (s Known) 154 8.3 A Sample Size Requirements Formula 159 8.4 A Confidence Interval for the Population Mean m (s Unknown) 160 Exercises 165 Appendix 8.A A Confidence Interval for the Population Median MED 167 9 The Sampling Distribution of a Proportion and its Confidence Interval Estimation 170 9.1 The Sampling Distribution of a Proportion 170 9.2 The Error Bound on ^p as an Estimator for p 173 9.3 A Confidence Interval for the Population Proportion (of Successes) p 174 9.4 A Sample Size Requirements Formula 176 Exercises 177 Appendix 9.A Ratio Estimation 179 10 Testing Statistical Hypotheses 184 10.1 What is a Statistical Hypothesis? 184 10.2 Errors in Testing 185 10.3 The Contextual Framework of Hypothesis Testing 186 10.3.1 Types of Errors in a Legal Context 188 10.3.2 Types of Errors in a Medical Context 188 10.3.3 Types of Errors in a Processing or Control Context 189 10.3.4 Types of Errors in a Sports Context 189 10.4 Selecting a Test Statistic 190 10.5 The Classical Approach to Hypothesis Testing 190 10.6 Types of Hypothesis Tests 191 10.7 Hypothesis Tests for m (s Known) 194 10.8 Hypothesis Tests for m (s Unknown and n Small) 195 10.9 Reporting the Results of Statistical Hypothesis Tests 198 10.10 Hypothesis Tests for the Population Proportion (of Successes) p 201 Exercises 204 Appendix 10.A Assessing the Randomness of a Sample 208 Appendix 10.B Wilcoxon Signed Rank Test (of a Median) 210 Appendix 10.C Lilliefors Goodness-of-Fit Test for Normality 213 11 Comparing Two Population Means and Two Population Proportions 217 11.1 Confidence Intervals for the Difference of Means when Sampling from Two Independent Normal Populations 217 11.1.1 Sampling from Two Independent Normal Populations with Equal and Known Variances 217 11.1.2 Sampling from Two Independent Normal Populations with Unequal but Known Variances 218 11.1.3 Sampling from Two Independent Normal Populations with Equal but Unknown Variances 218 11.1.4 Sampling from Two Independent Normal Populations with Unequal and Unknown Variances 219 11.2 Confidence Intervals for the Difference of Means when Sampling from Two Dependent Populations: Paired Comparisons 224 11.3 Confidence Intervals for the Difference of Proportions when Sampling from Two Independent Binomial Populations 227 11.4 Statistical Hypothesis Tests for the Difference of Means when Sampling from Two Independent Normal Populations 228 11.4.1 Population Variances Equal and Known 229 11.4.2 Population Variances Unequal but Known 229 11.4.3 Population Variances Equal and Unknown 229 11.4.4 Population Variances Unequal and Unknown (an Approximate Test) 230 11.5 Hypothesis Tests for the Difference of Means when Sampling from Two Dependent Populations: Paired Comparisons 234 11.6 Hypothesis Tests for the Difference of Proportions when Sampling from Two Independent Binomial Populations 236 Exercises 239 Appendix 11.A Runs Test for Two Independent Samples 243 Appendix 11.B Mann–Whitney (Rank Sum) Test for Two Independent Populations 245 Appendix 11.C Wilcoxon Signed Rank Test when Sampling from Two Dependent Populations: Paired Comparisons 249 12 Bivariate Regression and Correlation 253 12.1 Introducing an Additional Dimension to our Statistical Analysis 253 12.2 Linear Relationships 254 12.2.1 Exact Linear Relationships 254 12.3 Estimating the Slope and Intercept of the Population Regression Line 257 12.4 Decomposition of the Sample Variation in Y 262 12.5 Mean, Variance, and Sampling Distribution of the Least Squares Estimators ^b0 and ^b1 264 12.6 Confidence Intervals for b0 and b1 266 12.7 Testing Hypotheses about b0 and b1 267 12.8 Predicting the Average Value of Y given X 269 12.9 The Prediction of a Particular Value of Y given X 270 12.10 Correlation Analysis 272 12.10.1 Case A: X and Y Random Variables 272 12.10.1.1 Estimating the Population Correlation Coefficient r 274 12.10.1.2 Inferences about the Population Correlation Coefficient r 275 12.10.2 Case B: X Values Fixed, Y a Random Variable 277 Exercises 278 Appendix 12.A Assessing Normality (Appendix 7.B Continued) 280 Appendix 12.B On Making Causal Inferences 281 12.B.1 Introduction 281 12.B.2 Rudiments of Experimental Design 282 12.B.3 Truth Sets, Propositions, and Logical Implications 283 12.B.4 Necessary and Sufficient Conditions 285 12.B.5 Causality Proper 286 12.B.6 Logical Implications and Causality 287 12.B.7 Correlation and Causality 288 12.B.8 Causality from Counterfactuals 289 12.B.9 Testing Causality 292 12.B.10 Suggestions for Further Reading 294 13 An Assortment of Additional Statistical Tests 295 13.1 Distributional Hypotheses 295 13.2 The Multinomial Chi-Square Statistic 295 13.3 The Chi-Square Distribution 298 13.4 Testing Goodness of Fit 299 13.5 Testing Independence 304 13.6 Testing k Proportions 309 13.7 A Measure of Strength of Association in a Contingency Table 311 13.8 A Confidence Interval for s2 under Random Sampling from a Normal Population 312 13.9 The F Distribution 314 13.10 Applications of the F Statistic to Regression Analysis 316 13.10.1 Testing the Significance of the Regression Relationship Between X and Y 316 13.10.2 A Joint Test of the Regression Intercept and Slope 317 Exercises 318 Appendix A 323 Table A.1 Standard Normal Areas [Z is N(0,1)] 323 Table A.2 Quantiles of the t Distribution (T is tv) 325 Table A.3 Quantiles of the Chi-Square Distribution (X is w2v) 327 Table A.4 Quantiles of the F Distribution (F is Fv1;v2 ) 329 Table A.5 Binomial Probabilities P(X;n,p) 334 Table A.6 Cumulative Binomial Probabilities 338 Table A.7 Quantiles of Lilliefors’ Test for Normality 342 Solutions to Exercises 343 References 369 Index 373
£999.99
