The Inference Report

We prove $F_{\mathrm{weighted}}(2,2,3)<11/6$ for minimum-Frobenius-norm least-squares regression with two input and two output coordinates, improving the upper bound $15/8$. More precisely, a universal $\varepsilon_0>0$ satisfies $\mathcal T_3\le(5/6-\varepsilon_0)E$ for every finite residual frame. The proof combines an exact scalar certificate, rigidity of equality in planar selection, and a compactness argument that controls disappearing weights. Together with the known lower bound, this gives $13/8\le F_{\mathrm{weighted}}(2,2,3)<11/6$. No numerical value of $\varepsilon_0$ is claimed.

We propose a new estimator of low-frequency skewness (monthly) that exploits high-frequency data (5 min) through a direct functional mapping consisting of layers of Convolutional Neural Networks followed by Multi Layer Perceptrons. Monte Carlo experiments involving stochastic volatility models, with and without jumps, show that the estimator reduces finite-sample bias relative to existing realized-skewness estimators and remains accurate under model misspecification. Empirically, our estimator exhibits temporal stability and delivers cross-sectional performance in actual skewness-sorted portfolios. We characterize the finite-sample bias of conventional plug-in skewness estimators and establish risk consistency properties of the proposed NN estimator under standard regularity conditions.

Conformal prediction gives finite-sample coverage when calibration and test points are exchangeable, but exchangeability can fail when deployment changes the data distribution. We study conformal prediction in performative settings, where the deployed predictor affects the data distribution, and seek prediction sets valid at performative equilibrium: coverage must hold under $D(f^\star)$ for a performatively stable predictor $f^\star$. Under total-variation $L$-sensitivity of the distribution map, we prove a phase transition at $L=1$. If $L \ge 1$, no finite-sample procedure can uniformly guarantee both equilibrium-valid coverage and nontrivial prediction sets. If $L<1$, the minimax excess miscoverage over informative procedures is $\widetildeΩ(1/((1-L)\sqrt{n}))$, implying a sample-complexity inflation of order $1/(1-L)^2$ for fixed excess miscoverage. Our main technical tool is a transfer theorem: any procedure returning a predictor within $\varepsilon$ of equilibrium yields equilibrium-valid coverage up to an additive $L\varepsilon$ term after split-conformal calibration. We instantiate the theorem with two retraining algorithms: a model-free method for discretizable classes and a parametric tail-averaged method that matches the $n$- and $L$-dependence of this rate up to logarithmic factors. Experiments, including semi-synthetic credit scoring, demonstrate the predicted threshold behavior and scaling laws.

We consider nonparametric maximum likelihood estimation of the intensity function of a Poisson process, allowing the intensity to depend on observed covariates. Such models describe how event rates vary with external conditions, including conditions under which the intensity becomes arbitrarily small or vanishes. However, risk analysis based on the Kullback-Leibler (KL) divergence typically requires the true intensity to be bounded away from zero, a restrictive assumption in applications. We develop a unified approach that avoids this assumption and derive an oracle inequality for the covariate-averaged Hellinger distance between the true intensity and the nonparametric maximum likelihood estimator (NPMLE). As an application of the oracle inequality, we establish convergence rates for NPMLEs based on deep neural networks (DNNs). Our results show that, as in nonparametric regression, the ability of DNNs to capture underlying structure leads to improved convergence rates. Specifically, DNN-based NPMLEs mitigate the curse of dimensionality when the true intensity has a compositional structure and automatically adapt to a low-dimensional Riemannian manifold structure of the event space. For intensity functions with a compositional structure, DNN-based NPMLEs achieve nearly minimax-optimal convergence rates without explicitly incorporating this structure into the network architecture. Simulations demonstrate that DNN-based estimators outperform kernel estimators in most settings, while manifold kernels remain competitive in some designs. Finally, we analyze stimulus-evoked neural spiking using data from the Allen Brain Observatory Visual Coding Neuropixels project. DNN-based NPMLEs provide better out-of-sample performance than a conditional kernel estimator, as evaluated by held-out log-likelihood.

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ of an outcome on contemporaneous covariates can be regarded as a field on covariate space. We study how this field evolves when the covariates of individual units move as a diffusion and the outcome attached to each unit changes according to its own dynamics. Under mild regularity we show that $m_t$ satisfies a second-order evolution equation whose diffusion term is $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, the trace of the Hessian of the regression function weighted by the instantaneous covariance $Σ$ of the covariate increments, whose transport term is the drift of the time-reversed covariate process, and whose source term is the conditional mean rate of change of the outcome. The equation is a regression-theoretic reading of a classical transport law, and it yields a Feynman-Kac representation of $m_t$ along reversed covariate paths. We connect the Hessian appearing in the equation to the Hessian estimated by quadratic regression (under Gaussian design the best quadratic approximation has Hessian $E[\nabla^2 m(X)]$ exactly) and to the leading bias of kernel smoothers. For binary outcomes governed by bistable replicator dynamics with imitation among units with similar covariates, the closed equation is of Nagumo type, its planar travelling fronts are exactly logistic in $(x,t)$, and a single logistic regression of the outcome on covariates and time identifies the ratio of diffusion to reaction and the direction of invasion. We discuss identification under panel versus repeated cross-sectional designs and a two-stage estimation strategy, and we illustrate the results with simulations.

We study minimax rates for conditional generative learning under integrated Wasserstein loss, with Hölder continuous conditional laws and an unrestricted covariate distribution. We evaluate unrestricted generators through their conditional output distributions. For expected Wasserstein loss, the minimax rate agrees with the rate under uniform covariates and follows from balancing localization bias against unconditional distribution estimation error. Higher moments of the loss can instead be dominated by unobserved covariate regions, which force an additional risk contribution of order $n^{-1}$, where $n$ is the sample size. An adaptive localization argument gives matching upper bounds. The results cover unrestricted response laws and possibly vanishing Besov densities for $W_1$, and extend to higher Wasserstein orders under uniform response density bounds and integrability conditions. A multiscale lower bound closes the logarithmic gap in unconditional $W_1$ minimax rates in dimension two for both unrestricted distributions and Besov density classes. We also establish minimax rates for conditional Brenier maps under density, integrability, and uniform curvature conditions.