Quiz: Lecture 14 — Distribution-on-Distribution Regression

This quiz tests your understanding of regression when both predictor and response are probability distributions — transformation-based approaches, intrinsic Wasserstein-based methods, transport-map regression, and their trade-offs. 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. The two broad strategies for regression with distributional data are:

A. Linear regression and logistic regression B. Transformation-based (map to a Hilbert space, fit standard models, map back) and intrinsic/Wasserstein-based (work directly with Wasserstein distances and transport maps) C. Bayesian and frequentist regression D. Parametric and nonparametric regression

Q2. The log-quantile-density transform (LQDT) of Petersen & Müller (2016) works by:

A. Taking the Fourier transform of the density and truncating high frequencies B. Mapping a suitable density \(f\) on \([0,1]\) to \(\psi(f)(u)=\log q(u)\), where \(q(u)=f\{Q(u)\}^{-1}\) is the quantile density, and using a normalized inverse to recover a density C. Applying a kernel density estimator with a fixed bandwidth D. Computing the cumulative distribution function and applying a logit transformation

Q3. The three complementary intrinsic/Wasserstein-based regression formulations are:

A. Linear regression, ridge regression, and lasso regression B. ANOVA, ANCOVA, and MANOVA C. Fréchet regression (Petersen & Müller 2019), tangent-space Wasserstein regression (Chen, Lin, Müller 2023), and optimal-transport-map regression (Ghodrati & Panaretos 2022) D. K-means, hierarchical clustering, and DBSCAN

Q4. In transport-map regression (Ghodrati & Panaretos 2022), the regression function is modeled as:

A. \(f(x) = T_0\#x\), where \(T_0\) is an optimal transport map and \(\#\) denotes the push-forward — the response distribution is the push-forward of the predictor distribution B. \(f(x) = \beta_0 + \beta_1 x\) — a simple linear model C. \(f(x) = \Phi(\beta x)\) — a probit model D. \(f(x) = \exp(\beta x)\) — an exponential model

Q5. The Fréchet least-squares criterion for transport-map regression is strictly convex because:

A. The objective is linear in \(T\) B. It uses the \(L^1\) norm instead of the \(L^2\) norm C. The regularization parameter is chosen by cross-validation D. The criterion \(M_n(T) = \frac{1}{2n}\sum_i \int |T - T_{X_i \to Y_i}|^2 dX_i\) is a sum of strictly convex squared-error terms, and the monotonicity constraint defines a convex feasible set

Q6. In 1D transport-map regression, the monotone map \(T_0\) is estimated via:

A. Kernel ridge regression B. Weighted isotonic regression using the pool-adjacent-violators algorithm (PAVA), yielding a \(n^{-1/3}\) convergence rate C. Maximum likelihood estimation D. Ordinary least squares

Q7. The convergence rate \(n^{-1/3}\) for transport-map regression (PAVA) arises because:

A. The sample size is cubed B. The estimator uses only one-third of the data C. Three iterations are required for convergence D. Isotonic regression estimates a monotone function under a shape constraint — the \(n^{-1/3}\) rate is characteristic of nonparametric estimation under monotonicity

Q8. In transport-map regression, residual diagnostics use the residual map defined as:

A. \(\hat{R}_i = Y_i - \hat{Y}_i\), the difference of the density functions B. \(\hat{R}_i=T_{\hat{Y}_i\to Y_i}\), the optimal transport map from the fitted to the observed response — under correct specification, it should fluctuate around the identity map C. \(\hat{R}_i = W_2(Y_i, \hat{Y}_i)\), a scalar Wasserstein distance D. \(\hat{R}_i = \|Y_i - \hat{Y}_i\|_2\), the Euclidean norm of the difference

Q9. Why is the transport-map regression approach (Ghodrati & Panaretos 2022) restricted to one-dimensional distributions?

A. One-dimensional data is always sufficient for statistical inference B. The PAVA algorithm only works for scalar inputs C. Monotone rearrangements and the ordering that defines isotonic regression are inherently one-dimensional — in higher dimensions, optimal transport maps are gradients of convex functions and lack a simple ordering constraint D. Wasserstein distance is undefined in higher dimensions

Q10. For an income predictor distribution \(X\), what is the correct interpretation of \(\hat T_n\{Q_X(u)\}\)?

A. It is the Gini coefficient of the follow-up distribution B. It identifies the follow-up income of the same individual who was at percentile \(u\) C. It is the fitted follow-up income at percentile \(u\); it describes marginal quantile change, not individual mobility D. It is the cumulative income share below percentile \(u\)