Alexander Steinicke:
Worst-Case Optimal Investment in Incomplete Markets

We study and solve the worst-case optimal portfolio problem of an investor with logarithmic preferences facing the possibility of a market crash. Our setting takes place in a Lévy-market and we assume stochastic market coefficients. To tackle this problem, we enhance the martingale approach developed by F. Seifried in 2010. A utility crash-exposure transformation into a backward stochastic differential equation (BSDE) setting allows us to characterize the optimal indifference strategies. Further, we deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. To numerically compute the strategies, we solve the corresponding (non-Lipschitz) BSDEs through their associated PDEs and need to analyze continuity and boundedness properties of CIR forward processes. We demonstrate our approach for Heston’s stochastic volatility model, Bates’ stochastic volatility model including jumps, and Kim-Omberg’s model for a stochastic excess return.

Xinwei Shen:
Distributional Learning: From Methodology to Applications

Estimating the full (conditional) distribution is crucial to many applications. However, existing methods such as quantile regression typically struggle with high-dimensional response variables. To this end, distributional learning models the target distribution via a generative model, which enables inference via sampling. In this talk, we introduce a distributional learning method called engression. We then demonstrate the applications of engression to several statistical problems including extrapolation in nonparametric regression, causal effect estimation, and dimension reduction, as well as scientific problems such as climate downscaling.

Davy Paindaveine:
Rank Tests for PCA Under Weak Identifiability

In a triangular array framework where n observations are randomly sampled from a p-dimensional elliptical distribution with shape matrix V_n, we consider the problem of testing the null hypothesis H_0: theta=theta_0, where theta is the (fixed) leading unit eigenvector of V_n and theta_0 is a given unit p-vector. The dependence of the shape matrix on the sample size allows us to consider challenging asymptotic scenarios in which the parameter of interest theta is unidentified in the limit, because the ratio between both leading eigenvalues of V_n converges to one. We study the corresponding limiting experiments under such weak identifiability, and we show that these may be LAN or non-LAN. While earlier work in the framework was strictly limited to Gaussian distributions, where the study of local log-likelihood ratios could simply rely on explicit expressions, our asymptotic investigation allows for essentially arbitrary elliptical distributions. Even in non-LAN experiments, our results enable us to investigate the asymptotic null and non-null properties of multivariate rank tests. These nonparametric tests are shown to exhibit an excellent behavior under weak identifiability: not only do they maintain the target nominal size irrespective of the amount of weak identifiability, but they also keep their uniform efficiency properties under such non-standard scenarios.

Sirio Legramanti:
Leveraging Covariates in Bayesian Nonparametric Clustering: An Application to Transportation Networks

In clustering, observed individual data are often accompanied by covariates that can assist the clustering process itself. This is the case, for example, of transportation networks, where each node has spatial coordinates, and it is often desirable that clusters of nodes are spatially cohesive. In fact, the obtained clusters may be used to inform public policy decisions, and it may be preferrable that such policies are uniform over neighboring areas. Naturally, depending on the application, different notions of closeness can be used to define such neighborhoods, thus potentially requiring proper transformations of the spatial covariates. Motivated by real-world data about the monthly subscriptions to the public transportation system of the Bergamo province (Italy), we show how to incorporate properly-transformed spatial covariates into a state-of-the-art stochastic block model, while allowing to weight the contribution of covariates.

(joint work with Valentina Ghidini and Raffaele Argiento)

Lukas Steinberger:
Statistical Efficiency in Local Differential Privacy

We develop a theory of asymptotically efficient estimation in regular parametric models when data confidentiality is ensured by local differential privacy (LDP). The idea of LDP is that individual data owners should be able to release an anonymized or sanitized version Z_i of their possibly sensitive information X_i by drawing Z_i from a pre-specified conditional distribution Q that satisfies the formal α-differential privacy constraint. The problem is now to identify a randomization mechanism Q, generating Z_i, and an estimator θ^, that uses the sanitized data to estimate the population parameter, with minimal variance among all data-generation and estimation schemes satisfying the privacy constraint. Starting from a regular parametric model for the iid unobserved sensitive data X₁, . . . ,X_n, we establish local asymptotic mixed normality (along subsequences) of the model describing the sanitized observations Z₁, . . . ,Z_n. This result readily implies convolution and local asymptotic minimax theorems. In case p = 1, the optimal asymptotic variance is found to be the inverse of the supremal Fisher-Information, where the supremum runs over all α-differentially private (marginal) privacy mechanisms. We present a numerical algorithm for finding a (nearly) optimal privacy mechanism and an estimator based on the corresponding sanitized data that achieves this asymptotically optimal variance under mild assumptions. In special cases, such as the Gaussian location model, our theory also enables us to identify exact closed form expressions of efficient privacy mechanism and estimators.