Useful for UGC NET · ASRB NET · ISS
Every other paper on these pages carries a syllabus number, because a prescribed syllabus document was supplied for it. For this topic no such document is in hand — the syllabus these pages follow sets out Semesters I and II, and this topic is not among them.
Rather than invent a code and a unit split, the course is written to the topic. The mathematics of testing does not vary between universities; only the packaging does. What follows is the postgraduate core as every syllabus that examines it states it, and the table below says where it is examined.
| Examination | Unit | What it asks for |
|---|---|---|
| Indian Statistical Service (UPSC) |
Statistics-II (Objective), section (ii) | “Hypothesis testing: Simple and composite hypotheses. Two kinds of error. Critical region. Different types of critical regions and similar regions. Power function. Most powerful and uniformly most powerful tests. Neyman-Pearson fundamental lemma. Unbiased test. Randomized test. Likelihood ratio test. Wald's SPRT, OC and ASN functions. Elements of decision theory.” |
| UGC NET Statistics | Unit V — Testing of Hypotheses | the same list, plus the non-parametric tests, as an examination-shaped summary with model questions |
| CSIR NET | Statistics, inference | Neyman–Pearson, UMP, likelihood ratio, sequential tests |
| University MSc | varies — commonly Paper V, or “Statistical Inference II” | the same content; the course number is the only thing that differs |
The ISS wording above is quoted from Examination Notice No. 07/2026-IES/ISS, the Union Public Service Commission notification for the Indian Economic Service and Indian Statistical Service Examination, Appendix-I, Section-II. It is reproduced as printed.
None of that is written again. Each unit below opens by naming what it is continuing from.
The test function in place of the critical region, and why a discrete distribution forces it: on \(\text{Bin}(10, 0.3)\) the achievable sizes jump from \(0.150268\) straight to \(0.047349\), so no set has size \(0.05\) and the lemma's optimality claim is out of reach. The randomization constant solved exactly — \(\gamma = 13255063/514596726\) — and the existence and necessity halves of the Neyman–Pearson lemma proved, the two halves the Foundation treatment does not give.
UNIT 2When one test can be best against a whole family at once: monotone likelihood ratio, the Karlin–Rubin theorem proved in three steps including the level check on the composite null that is usually skipped, and a worked demonstration with plotted power curves that no UMP test exists against a two-sided alternative — the one-sided test has power \(0.000831\) at \(\mu = -0.5\), far below its own level, and is not merely sub-optimal but biased. Then unbiased tests, similar regions and Neyman structure.
UNIT 3The reduction carried out rather than asserted: four steps to \(\lambda = (1 + t^{2}/(n-1))^{-n/2}\), so the likelihood ratio test is the \(t\) test at every sample size. Then Wilks' approximation measured against the truth — at \(n = 10\) it reports \(p = 0.021652\) where the exact answer is \(0.033862\) — and the likelihood ratio, Wald and Rao score statistics on one binomial sample where the three disagree with each other and all three are wrong.
UNIT 4Wald's SPRT with its boundaries derived from the overshoot argument, Wald's equation and fundamental identity, and the operating characteristic and average sample number computed for a proportion — \(22.782\) observations under \(H_0\) and \(17.245\) under \(H_1\) against a fixed-sample \(39\), but \(31.630\) in the worst case between them. Then testing as a decision problem, where the Neyman–Pearson constant turns out to be a ratio of prior-weighted losses.
| Thread | Where it appears |
|---|---|
| The likelihood ratio orders the sample space | Unit 1 makes it the most powerful test; Unit 2 asks when the ordering does not depend on the alternative; Unit 3 generalises it to composite hypotheses; Unit 4 accumulates it over time |
| Optimality is traded for existence | UMP, then UMPU, then invariant, then merely constructible — each step down in Unit 2's closing table buys existence with a weaker claim |
| Asymptotic results are checked, not trusted | Unit 3 measures Wilks' error at \(n = 10\); Unit 4 measures Wald's overshoot and finds the ASN peak is not where it is usually said to be |
| Exact tests beat approximations where they exist | the binomial example of Unit 3: three asymptotic \(p\) values spanning \(0.051\) to \(0.074\), against an exact \(0.115318\) |