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Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.

This page reproduces the prescribed outline for the theory paper and for Section B of its practical, so that the teaching pages can be checked against it line by line. It is the syllabus, not a summary of it. Nothing about marks, duration or examination pattern appears on this site.

Course Objectives

  1. To understand the extension from univariate to multivariate probability distributions and its importance, necessity and real time applications.
  2. To derive the basic properties, applications and problems on multivariate probability distributions (Multinomial and Multivariate normal) and Multivariate Sampling distributions (Wishart, Hotelling T² and Wilks Lambda).
  3. Basic concepts of various multivariate Statistical techniques for data analysis (classification, identification, testing, clustering etc.)

Course Outcomes

  1. Able to solve and derive the common and special properties of multinomial and multivariate normal probability distributions and multivariate sampling distributions.
  2. Able to carry out the analysis for any multivariate data set using the multivariate tools Linear discriminant, Principal component, multi-dimensional scaling, Factor Analysis and Cluster analysis.
  3. Able to identify the real time applications of multivariate statistical tools.

Unit I

AS PRESCRIBED

Multinomial distribution. Multivariate normal distribution, marginal, conditional distributions. Independence of multivariate vectors. Random sampling from a multivariate normal distribution. Maximum likelihood estimators of parameters. Distribution of sample mean vector. Independence of sample mean vector and variance-covariance matrix.

→ Unit 1 notes

Unit II

AS PRESCRIBED

Wishart matrix, Wishart distribution and its properties. Distribution of sample generalized variance. Null distribution of simple correlation coefficients. Null distribution of partial and multiple correlation coefficients. Distribution of sample regression coefficients. Application in testing and interval estimation.

→ Unit 2 notes

Unit III

AS PRESCRIBED

Null distribution of Hotelling's T² statistic. Application in tests on mean vector for one and more multivariate normal populations and also on equality of the components of a mean vector in a multivariate normal population. Mahalanobi's D² statistic. Wilk's Λ-criterion and statement of its properties with simple applications. Linear Discriminant Analysis: Classification and discrimination procedures for discrimination between two multivariate normal populations – sample discriminant function, tests associated with discriminant functions, probabilities of misclassification and their estimation, classification into two multivariate normal populations with equal covariance matrices.

→ Unit 3 notes

Unit IV

AS PRESCRIBED

Principal component analysis and its properties and applications. Canonical variables and canonical correlations: definition, use, estimation and computation. Cluster analysis: Definitions, Agglomerative hierarchical clustering methods, Single complete and average linkages, K-means, KNN clustering. Multi-dimensional scaling methods (metric & non metric). Introduction to Factor analysis, orthogonal factor model. Path analysis, Correspondence analysis, conjoint analysis.

→ Unit 4 notes

Practical Paper STS-205, Section B — List of Practicals

The practical paper is Estimation Theory and Multivariate Analysis (Conventional). Section B belongs to this paper; Section A belongs to Estimation Theory (STS-201) and is written with that paper.

  1. MLE of Mean vector and variance covariance Matrix based on the sample drawn from p-Normal population.
  2. Writing the density function based on Mean vector and covariance matrix and identification of parameters from the p-variate normal density.
  3. Hotelling's T² for test the mean vector based on single sample.
  4. Mahalanobi's D² for test the mean vector based on single sample.
  5. Hotelling's T² for testing equality of the mean vectors based on two samples.
  6. Mahalanobi's D² for testing equality of the mean vectors based on two samples.
  7. Computation of Principal Components.
  8. Classification between two normal populations by discriminant analysis using Maximum likelihood ratio approach and Bayesian misclassification.
  9. Cluster analysis using Single, Complete and Average linkages.
  10. Computation of Canonical variables and correlation.
  11. Computation of Orthogonal Factor Model.
  12. Computation of Path coefficients and drawing Path diagram.
  13. Computation of Multidimensional Scaling.

→ Practical notes, Section B

References

  1. Johnson, R. A. and Wichern, D. W. (2012): Applied Multivariate Statistical Analysis, PHI, 6th edition.
  2. Anderson, T. W. (1984): An Introduction to Multivariate Statistical Analysis, 2nd edition, Wiley.
  3. Kshirsagar, A. M. (1972): Multivariate Analysis, Marcel Dekker.
  4. Morrison, D. F. (1976): Multivariate Statistical Methods, 2nd edition, McGraw Hill.

Author and publisher spellings above follow the published books: the printed list gives “Kshirasagar” for Kshirsagar and “Marcel Decker” for Marcel Dekker, and the first title is given in full so that the book can be found.

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