Quiz: Lecture 4 — Fréchet Variance and ANOVA

This quiz tests your understanding of the core concepts from Lecture 4. There are 10 multiple-choice questions, each worth 1 mark. After submitting, your score is shown, and any questions you got wrong are highlighted — but the correct answer is not revealed. You can then redo only the questions you missed.

Q1. Which statement correctly distinguishes the two scalar quantities \(V\) and \(\sigma^2\)?

A. Both are definitions of the Fréchet variance B. \(V\) is the variance of \(\hat\mu_n\), while \(\sigma^2\) is the variance of \(\mu\) C. \(V=\mathbb E d^2(X,\mu)\), while \(\sigma^2=\operatorname{var}\{d^2(X,\mu)\}\) is the scalar CLT variance D. \(V\) measures location differences, while \(\sigma^2\) measures curvature

Q2. Under assumptions (A1)–(A3), the scalar Fréchet-variance CLT is:

A. \(n(\hat V_n-V)\Rightarrow\chi^2_1\) B. \(\sqrt n(\hat V_n-V)\Rightarrow N(0,\sigma^2)\) C. \(\sqrt n\,d(\hat\mu_n,\mu)\Rightarrow N(0,V)\) D. \(\hat V_n/V\Rightarrow\chi^2_{n-1}\)

Q3. What is the main role of assumption (A1)?

A. It requires all groups to have equal Fréchet variances B. It supplies the \(\chi^2_{k-1}\) critical value C. It requires \(\Omega\) to be a Euclidean vector space D. It gives existence, uniqueness, and separation of the relevant Fréchet mean, supporting consistency of \(\hat\mu_n\)

Q4. What does \(F_n=\hat V_p-\sum_j\lambda_{j,n}\hat V_j\) measure?

A. The loss from using one pooled center instead of separate group centers B. The maximum pairwise distance between observations C. Only differences among group Fréchet variances D. The covariance between \(\hat\mu_p\) and the group means

Q5. Under \(H_0^\mu:\mu_1=\cdots=\mu_k\), which statement is correct?

A. \(nF_n\Rightarrow\chi^2_{k-1}\) B. \(F_n\) is exactly zero in every finite sample C. \(\sqrt n F_n=o_P(1)\), so its normalized square is an omnibus component rather than a separately calibrated test D. \(F_n\) tests equality of the full distributions \(P_1,\ldots,P_k\)

Q6. Under \(H_0^V:V_1=\cdots=V_k\), the studentized variance statistic has which limit?

A. \(N(0,1)\) B. \(\chi^2_{k-1}\) C. \(\chi^2_k\) D. A degenerate point mass at zero

Q7. What is the omnibus null hypothesis associated with \(T_n\)?

A. Only \(H_0^\mu\) B. Only \(H_0^V\) C. Equality of the complete group distributions D. \(H_0^\mu\cap H_0^V\): equal Fréchet means and equal Fréchet variances

Q8. Which rates are sufficient for uniform consistency against shrinking alternatives?

A. \(\sqrt n\,a_n\to\infty\) for \(F\ge a_n\), and \(n b_n\to\infty\) for \(U\ge b_n\) B. \(n a_n\to0\) and \(\sqrt n\,b_n\to0\) C. \(a_n=b_n=n^{-2}\) D. No shrinking alternatives can be detected consistently

Q9. Under the log-Euclidean metric, the sample Fréchet mean of SPD matrices \(\Sigma_1,\ldots,\Sigma_m\) is:

A. \(\frac1m\sum_i\Sigma_i\) B. \(\left(\frac1m\sum_i\Sigma_i^{-1}\right)^{-1}\) C. \(\exp\{\frac1m\sum_i\log\Sigma_i\}\) D. The matrix formed by averaging eigenvalues only

Q10. Why does the SPD application use non-overlapping return blocks, and what happens at separation \(s=1\)?

A. Non-overlap makes the matrices diagonal, and \(s=1\) maximizes separation B. Non-overlap produces independent covariance objects, and \(s=1\) gives the same population covariance in all groups C. Non-overlap is needed only for numerical positive definiteness, and \(s=1\) changes correlations but not variances D. Non-overlap guarantees exact finite-sample \(\chi^2\) calibration, and \(s=1\) tests equality of full distributions without assumptions