Statistical Inference
GATE ST tests estimation and testing as decision problems: which estimator is admissible, which test is most powerful, and what a confidence interval actually covers. Numerical questions often reduce to sufficiency, likelihood, or a rejection region.
Core rules
- An estimator is unbiased if E(θ̂)=θ, consistent if θ̂→p θ, and efficient if it attains the Cramér–Rao lower bound under regularity.
- Sufficiency is checked by the factorization theorem: f(x;θ)=g(T(x),θ)h(x). A complete sufficient statistic plus an unbiased estimator yields the UMVUE by Lehmann–Scheffé.
- MLEs are invariant under transformations: if θ̂ is the MLE of θ, then g(θ̂) is the MLE of g(θ). MLEs are not automatically unbiased.
- Neyman–Pearson gives the most powerful test for simple versus simple hypotheses via the likelihood ratio. UMP tests exist for one-sided hypotheses in monotone likelihood ratio families.
- A confidence interval is dual to a test: values not rejected form the interval. Coverage is a long-run property, not a probability statement about a fixed parameter.
Worked example
Let X1,…,Xn be i.i.d. Bernoulli(p). The likelihood is p^{Σxi}(1−p)^{n−Σxi}, so the MLE is p̂=X̄. It is unbiased because E(X̄)=p, and its variance is p(1−p)/n. By factorization, T=ΣXi is sufficient; in fact it is complete. Thus X̄ is the UMVUE of p. For testing H0:p=p0 against H1:p=p1 with p1>p0, the Neyman–Pearson test rejects for large ΣXi.
Traps
Do not claim an MLE is unbiased without checking. The Cramér–Rao bound needs regularity and may not apply when the support depends on the parameter. A UMP test may fail to exist for two-sided alternatives. Do not interpret a 95% confidence interval as “95% probability the parameter is inside” after the data are observed.