Useful for UGC NET · ASRB NET · ISS
This is the complete study package for Estimation Theory (STS-201) together with Section A of STS-205, Estimation Theory and Multivariate Analysis (Conventional). The Foundation course asked which estimator to use; this course asks a harder question — is there a better one, and how would you know? It answers it three times over: by a lower bound no unbiased estimator can beat, by a construction that turns any unbiased estimator into the best one, and by a decision-theoretic framework in which “best” is stated as a risk to be minimised.
Why a criterion alone does not pick an estimator; Fisher's information and its two equal forms; the Cramér–Rao inequality derived from the Cauchy–Schwarz inequality rather than quoted; when the bound is attainable and Bhattacharya's sharper bounds when it is not; and the Rao–Blackwell theorem, with the variance reduction measured on a worked case.
UNIT 2Completeness as the uniqueness half of the argument; the Lehmann–Scheffé theorem with the uniform case worked in full; maximum likelihood and its large-sample properties; CAN and BAN estimators, with the sample median's efficiency computed; and bias and standard error by the jackknife and the bootstrap.
UNIT 3Kernels and U-statistics, with the sample variance exhibited as one; the asymptotic distribution stated and used; confidence intervals by pivots for the normal, exponential, binomial and Poisson; the shortest-length interval and how much it actually saves; distribution-free intervals for quantiles; and tolerance limits.
UNIT 4Loss, risk and decision functions; admissibility and why it is a weak requirement; prior and posterior distributions and conjugate families; Bayes and minimax estimators under quadratic loss, including the constant-risk minimax for a proportion; then non-parametric density estimation — Rosenblatt's naive estimator, its bias and variance, and kernel estimators worked by hand.
PRACTICALAll five prescribed Section A practicals, each worked by hand with the full arithmetic shown: jackknife estimates, bootstrap estimates, maximum likelihood by the method of scoring for a Cauchy population, confidence limits for the parameters of a normal population, and large-sample confidence limits for the binomial, Poisson and exponential.
REFERENCEThe prescribed unit-wise outline for STS-201 and the Section A practical list for STS-205, as printed, with the objectives, outcomes, the stated pre-requisite and the reading list.
| What is built here | Where it is used |
|---|---|
| Fisher information and the Cramér–Rao bound | Every large-sample standard error — the tests in Inferential Statistics, Unit 3 divide by one; and the scoring method in Practical 3 |
| Sufficiency, completeness and Lehmann–Scheffé | The exponential family of Distribution Theory, Unit 2 is what makes the theorem applicable; the same route gives the MLE of the mean vector and covariance matrix in Multivariate Analysis (STS-202) |
| Maximum likelihood and its large-sample properties | Fitting the lifetime distributions in STS-107; the likelihood ratio behind every test of a composite hypothesis |
| Jackknife and bootstrap | A standard error wherever no formula exists — including statistics whose exact distribution is unknown, such as a correlation or a ratio |
| Interval estimation by pivots, and shortest-length intervals | The confidence limits quoted in Inferential Statistics, Unit 1, which are built here from the sampling distributions of Distribution Theory, Unit 3 |
| Loss, risk, Bayes and minimax | The decision-theoretic reading of testing; conjugate families, which reappear wherever a prior is chosen for convenience |
| Kernel density estimation | Smoothing a histogram into a density curve — Data Science using Python (STS-208), and any exploratory plot of a continuous variable |