6. Data-Driven Methods

Statistical Foundations — Quiz

Test your understanding of statistical foundations with 5 practice questions.

Practice Questions

Question 1

In the context of parametric statistical models, if $I(\theta)$ denotes the Fisher information contained in $n$ i.i.d. observations, what is the Cramér–Rao lower bound for the variance of any unbiased estimator $\hat{\theta}$?

Question 2

Given i.i.d. observations $X_1,\dots,X_n\sim\mathrm{Exp}(\lambda)$, what is the maximum likelihood estimator $\hat{\lambda}$ for $\lambda$?

Question 3

Using the delta method, what is the approximate asymptotic variance of $\ln\hat{\lambda}$ given that $\hat{\lambda}$ is the MLE for $\lambda$ based on $n$ i.i.d. observations from $\mathrm{Exp}(\lambda)$?

Question 4

Given a sample of size $n=20$ from a normal distribution with unknown variance and observed sample variance $S^2=4$, what is a 95\\% confidence interval for the true variance $\sigma^2$?

Question 5

In a likelihood‐ratio test for $H_0:\mu=\mu_0$ versus $H_a:\mu\neq\mu_0$ with known $\sigma^2$, under $H_0$ the statistic $-2\ln\Lambda$ asymptotically follows which distribution?