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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 A 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 procedures for the estimation of parameters involved in the probability distribution functions using various methods.
  2. To understand the procedures for examining the properties of the estimators like consistency, unbiasedness, efficiency, sufficiency, completeness, minimum variance bound, CAN, BAN, etc.
  3. Estimation of density function based on the sample in nonparametric approach.
  4. Estimation of parameters based on the resampling techniques.

Course Outcomes

  1. Able to estimate the parameters involved in the probability distribution functions using various methods.
  2. Able to examine the properties of the estimators like consistency, unbiasedness, efficiency, sufficiency, completeness, minimum variance bound, CAN, BAN, etc.
  3. Able to estimate the density function based on the sample.
  4. Able to evaluate the confidence limits for the parameters for any distribution.

Stated Pre-requisite

ASSUMED BEFORE THIS PAPER

Concepts of point estimation and interval estimation; the method of maximum likelihood and the method of moments, with related simple problems; criteria for good estimation — consistency, unbiasedness, efficiency and sufficiency — and their related problems; and simple problems on interval estimation using the pivot method.

Covered by Inferential Statistics, Unit 1 — Theory of Estimation, which sets out all four criteria and builds confidence intervals by the pivot method.

Unit I

AS PRESCRIBED

Estimation: Point and Interval estimation, Simple problems related to criterion for good estimator, Minimum Variance Unbiased Estimator, UMVU estimation, Fisher's information, Cramer-Rao inequality, and Bhattacharya bounds. Rao–Blackwell theorem.

→ Unit 1 notes

Unit II

AS PRESCRIBED

Completeness, Lehmann–Scheffé's theorem and their applications and its related problems. MLE and its properties (statements only). Consistency and asymptotic normality of the consistent solutions of likelihood equations. Definition of CAN and BAN estimators and their properties, related examples. Estimation of bias and standard deviation of point estimators of Jackknife and Bootstrap methods with examples.

→ Unit 2 notes

Unit III

AS PRESCRIBED

Concept of U-statistics, Kernel and examples. Statement of Asymptotic distributions of U-statistics. Interval estimation: confidence level CI using pivots and shortest length CI. Confidence intervals for the parameters for Normal, Exponential, Binomial and Poisson Distributions. Confidence Intervals for quantiles. Concept of tolerance limits and examples.

→ Unit 3 notes

Unit IV

AS PRESCRIBED

Concepts of loss, risk and decision functions, admissible and optimal decision functions, estimation and testing viewed as decision problems, apriori, aposteriori distributions, conjugate families, Baye's and minimax decision functions with applications to estimation with quadratic loss. Concepts of nonparametric estimation: Density estimates, survey of existing methods. Rosenblatt's naïve density estimator, its bias and variance. Consistency of Kernel density estimators and its MSE.

→ Unit 4 notes

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

The practical paper is Estimation Theory and Multivariate Analysis (Conventional). Section A belongs to this paper; Section B belongs to Multivariate Analysis (STS-202) and is written with that paper.

  1. Computation of Jackknife estimates.
  2. Computation of Boot-strap estimates.
  3. MLE by Scoring method for Cauchy population.
  4. Confidence limits for parameters of normal population.
  5. Large sample confidence limits in case of Binomial, Poisson, Exponential distributions.

→ Practical notes, Section A

References

  1. Goon, A. M., Gupta, M. K. and Dasgupta, B. (1993): An Outline of Statistical Theory, Vol. 2, The World Press.
  2. Rohatgi, V. K. (1993): An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern.
  3. Rao, C. R. (1973): Linear Statistical Inference and Its Applications, John Wiley.
  4. Gray, H. L. and Schucany, W. R. (1972): The Generalized Jackknife Statistic, Marcel Dekker.
  5. Efron, B. and Tibshirani, R. J. (1994): An Introduction to the Bootstrap, Chapman and Hall.
  6. Lehmann, E. L. (1983): Theory of Point Estimation, John Wiley.

Author and title spellings above follow the published books: the printed list gives “Lehman” for Lehmann, “Marcel Decker” for Marcel Dekker and “Chapmen” for Chapman, and the short titles are expanded to the titles the books actually carry, so that each one can be found.

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