ABSTRACT
CONTENTS 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 8.2 Basic Concepts of Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 8.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
8.3.1 Sample Space and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 8.3.2 Axioms, Interpretations, and Properties of Probability . . . . . . . . . . . . . . . . . . . 418 8.3.3 Borel σ-Field, Random Variables, and Convergence . . . . . . . . . . . . . . . . . . . . 418 8.3.4 Some Important Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
8.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 8.4.2 Desirable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 8.4.3 Methods of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 8.4.4 Method of Moment Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 8.4.5 Bayes Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
8.5 Linear and Nonlinear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 8.5.1 Linear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
8.5.1.1 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 8.5.2 Nonlinear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
8.6 Introduction to Multivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 8.7 Joint and Marginal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 8.8 Multinomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 8.9 Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 8.10 Multivariate Student t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 8.11 Wishart Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 8.12 Multivariate Extreme Value Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 8.13 MLE Estimates of Parameters (Related to MND Only) . . . . . . . . . . . . . . . . . . . . . . . 458 8.14 Copula Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 8.15 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 8.16 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
8.16.1 Mathematical Formulation of Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . 464 8.16.2 Estimation in Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 8.16.3 Principal Component Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 8.16.4 Maximum Likelihood Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 8.16.5 General Working Principle for FA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
8.17 Multiple Analysis of Variance and Multiple Analysis of Covariance . . . . . . . . . . . . . . 471 8.17.1 Introduction to Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 8.17.2 Multiple Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
8.18 Conjoint Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 413
8.19 Canonical Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 8.19.1 Formulation of Canonical Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . 477 8.19.2 Standardized Form of CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 8.19.3 Correlation between Canonical Variates and Their Component Variables . . . . . . 479 8.19.4 Testing the Test Statistics in CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 8.19.5 Geometric and Graphical Interpretation of CCA . . . . . . . . . . . . . . . . . . . . . . . 485 8.19.6 Conclusions about CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
8.20 Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 8.20.1 Clustering Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
8.21 Multiple Discriminant and Classification Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 493 8.22 Multidimensional Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8.23 Structural Equation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 8.24 Future Areas of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
ABSTRACT The chapter of Statistical Methods starts with the basic concepts of data analysis and then leads into the concepts of probability, important properties of probability, limit theorems, and inequalities. The chapter also covers the basic tenets of estimation, desirable properties of estimates, before going on to the topic of maximum likelihood estimation, general methods of moments, Baye’s estimation principle. Under linear and nonlinear regression different concepts of regressions are discussed. After which we discuss few important multivariate distributions and devote some time on copula theory also. In the later part of the chapter, emphasis is laid on both the theoretical content as well as the practical applications of a variety of multivariate techniques like Principle Component Analysis (PCA), Factor Analysis, Analysis of Variance (ANOVA), Multivariate Analysis of Variance (MANOVA), Conjoint Analysis, Canonical Correlation, Cluster Analysis, Multiple Discriminant Analysis, Multidimensional Scaling, Structural Equation Modeling, etc. Finally, the chapter ends with a good repertoire of information related to softwares, data sets, journals, etc., related to the topics covered in this chapter.
