Useful for UGC NET · ASRB NET · ISS
This is the complete study package for Sampling Theory (STS-204) together with Section B of STS-206, Designs and Analysis of Experiments and Sampling Theory (Conventional). The Foundation course selected units with equal probability and estimated a mean from what came back. This course asks the two questions that follow from that: what if some units deserve a larger chance of selection, and what if something is already known about every unit before the sample is drawn. The first leads to unequal probability sampling and the estimators that keep it unbiased; the second to ratio and regression estimation. The last two units then relax the assumption that a unit is cheap to reach and honest when asked.
Begins where the Foundation PPS section stops. The variance of the Hansen–Hurwitz estimator derived and its \(1/n\) behaviour explained; Lahiri's method worked draw by draw and proved to give exactly \(p_i = x_i/X\); inclusion probabilities \(\pi_i\) and \(\pi_{ij}\), and why \(\pi_i = np_i\) is a design requirement and not an identity; the Horvitz–Thompson estimator, its unbiasedness, and the Yates–Grundy form of its variance with the condition under which it cannot go negative.
UNIT 2The bias of the ratio estimator computed exactly over every possible sample of a small population, then compared with the \(\operatorname{Cov}(\hat R, \bar x)\) expression and the usual bound; the difference estimator with a pre-assigned slope, and what is lost by estimating the slope instead; and the separate and combined forms in stratified sampling, with the rule that decides between them.
UNIT 3The design effect \(1 + (M-1)\rho\) derived, and the exact finite-population identity it approximates — on a four-cluster population the familiar form understates the loss by \(18\%\). Optimum cluster size under a cost function, tabulated; and clusters of unequal size, where three estimators compete and the choice is not obvious.
UNIT 4Two-stage sampling with equal first-stage units, its two-part variance and the optimum sub-sample size, with both corner solutions worked; non-sampling error decomposed into bias and variance, and why increasing \(n\) does not touch the bias; Warner's randomized response model and the unrelated question model, with the price of the protection measured; and direct, synthetic and composite estimators for a small area.
PRACTICALAll seven prescribed experiments: four worked in the units, plus the selection mechanics of a PPS draw, the ratio and regression estimators in simple random sampling compared with the sample mean on one data set, and separate against combined regression estimators in stratified sampling.
REFERENCEThe prescribed unit-wise outline for STS-204 and the Section B practical list for STS-206, as printed, with the objectives, outcomes, the stated pre-requisite and the reading list.
| What is built here | Where it is used |
|---|---|
| Unbiasedness of an estimator under a design, not a model | The contrast with Estimation Theory, Unit 1, where unbiasedness is taken over a model; the two notions of expectation are different and the distinction is worth stating in an answer |
| Ratio and regression estimators | The same least squares slope as Linear Algebra and Linear Models, Unit 4, used here to improve an estimate of a mean rather than to predict |
| Intra-class correlation and the design effect | Every multi-stage survey ever reported; the effective sample size that a complex design is worth |
| Optimum allocation under a cost constraint | The same Cauchy–Schwarz argument as Neyman allocation in Sampling Techniques, Unit 3, applied to cluster and sub-sample sizes |
| Non-sampling error as bias plus variance | The mean square error decomposition of Estimation Theory, Unit 1, applied to a quantity no larger sample can remove |
| Randomized response | Any survey on a sensitive attribute; the trade of variance for truthfulness |