arXiv subjects

Robert Lunde

Publications and source records attributed to Robert Lunde.

11 recordsLinked to original sources

arXiv2026

Conformal Prediction for Dyadic Regression Under Complex Missingness

We develop a framework for conformal prediction in dyadic regression problems under complex missingness mechanisms. At the theoretical level, we develop general technical tools for establishing finite-sample validity of conformal prediction under distributional invariance conditions weaker than exchangeability. A key result handles the case where the sample itself is a random subset of the index set, a setting not covered by existing theory, via a novel bijection argument that constructs an explicit measure-preserving correspondence between events. In addition, we propose conformal prediction procedures for jointly exchangeable arrays, including full conformal, split conformal, a row-column approach exploiting similarities within rows and columns, and a selective conformal procedure achieving mask-conditional validity. For missing elements, we establish asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism. We further establish conditional validity results for both continuous and discrete responses; to the best of our knowledge, this is the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption. The proposed methods are illustrated on synthetic and real network data.

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arXiv2025

Assumption-lean Inference for Network-linked Data

We consider statistical inference for network-linked regression problems, where covariates may include network summary statistics computed for each node. In settings involving network data, it is often natural to posit that latent variables govern connection probabilities in the graph. Since the presence of these latent features makes classical regression assumptions even less tenable, we propose an assumption-lean framework for linear regression with jointly exchangeable regression arrays. We establish an analog of the Aldous-Hoover representation for such arrays, which may be of independent interest. Moreover, we consider two different projection parameters as potential targets and establish conditions under which asymptotic normality and bootstrap consistency hold when commonly used network statistics, including local subgraph frequencies and spectral embeddings, are used as covariates. In the case of linear regression with local count statistics, we show that a bias-corrected estimator allows one to target a more natural inferential target under weaker sparsity conditions compared to the OLS estimator. Our inferential tools are illustrated using both simulated data and real data related to the academic climate of elementary schools.

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arXiv2025

Statistical Inference for Subgraph Frequencies of Exchangeable Hyperedge Models

In statistical network analysis, models for binary adjacency matrices satisfying vertex exchangeability are commonly used. However, such models may fail to capture key features of the data-generating process when interactions, rather than nodes, are fundamental units. We study statistical inference for subgraph counts under an exchangeable hyperedge model. We introduce several classes of subgraph statistics for hypergraphs and develop inferential tools for subgraph frequencies that account for edge multiplicity. We show that a subclass of these subgraph statistics is robust to the deletion of low-degree nodes, enabling inference in settings where low-degree nodes are more likely to be missing. We also examine a more traditional notion of subgraph frequency that ignores multiplicity, showing that while inference based on limiting distributions is feasible in some cases, a non-degenerate limiting distribution may not exist in others. Empirically, we assess our methods through simulations and newly collected real-world hypergraph data on academic and movie collaborations, where our inferential tools outperform traditional approaches based on binary adjacency matrices.

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arXiv2023

On the Validity of Conformal Prediction for Network Data Under Non-Uniform Sampling

We study the properties of conformal prediction for network data under various sampling mechanisms that commonly arise in practice but often result in a non-representative sample of nodes. We interpret these sampling mechanisms as selection rules applied to a superpopulation and study the validity of conformal prediction conditional on an appropriate selection event. We show that the sampled subarray is exchangeable conditional on the selection event if the selection rule satisfies a permutation invariance property and a joint exchangeability condition holds for the superpopulation. Our result implies the finite-sample validity of conformal prediction for certain selection events related to ego networks and snowball sampling. We also show that when data are sampled via a random walk on a graph, a variant of weighted conformal prediction yields asymptotically valid prediction sets for an independently selected node from the population.

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arXiv2023

Conformal Prediction for Network-Assisted Regression

An important problem in network analysis is predicting a node attribute using both network covariates, such as graph embedding coordinates or local subgraph counts, and conventional node covariates, such as demographic characteristics. While standard regression methods that make use of both types of covariates may be used for prediction, statistical inference is complicated by the fact that the nodal summary statistics are often dependent in complex ways. We show that under a mild joint exchangeability assumption, a network analog of conformal prediction achieves finite sample validity for a wide range of network covariates. We also show that a form of asymptotic conditional validity is achievable. The methods are illustrated on both simulated networks and a citation network dataset.

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arXiv2022

Bootstrapping the error of Oja's algorithm

We consider the problem of quantifying uncertainty for the estimation error of the leading eigenvector from Oja's algorithm for streaming principal component analysis, where the data are generated IID from some unknown distribution. By combining classical tools from the U-statistics literature with recent results on high-dimensional central limit theorems for quadratic forms of random vectors and concentration of matrix products, we establish a weighted $χ^2$ approximation result for the $\sin^2$ error between the population eigenvector and the output of Oja's algorithm. Since estimating the covariance matrix associated with the approximating distribution requires knowledge of unknown model parameters, we propose a multiplier bootstrap algorithm that may be updated in an online manner. We establish conditions under which the bootstrap distribution is close to the corresponding sampling distribution with high probability, thereby establishing the bootstrap as a consistent inferential method in an appropriate asymptotic regime.

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arXiv2022

Trading off Accuracy for Speedup: Multiplier Bootstraps for Subgraph Counts

We propose a new class of multiplier bootstraps for count functionals, ranging from a fast, approximate linear bootstrap tailored to sparse, massive graphs to a quadratic bootstrap procedure that offers refined accuracy for smaller, denser graphs. For the fast, approximate linear bootstrap, we show that $\sqrt{n}$-consistent inference of the count functional is attainable in certain computational regimes that depend on the sparsity level of the graph. Furthermore, even in more challenging regimes, we prove that our bootstrap procedure offers valid coverage and vanishing confidence intervals. For the quadratic bootstrap, we establish an Edgeworth expansion and show that this procedure offers higher-order accuracy under appropriate sparsity conditions. We complement our theoretical results with a simulation study and real data analysis and verify that our procedure offers state-of-the-art performance for several functionals.

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arXiv2020

On the Theoretical Properties of the Network Jackknife

We study the properties of a leave-node-out jackknife procedure for network data. Under the sparse graphon model, we prove an Efron-Stein-type inequality, showing that the network jackknife leads to conservative estimates of the variance (in expectation) for any network functional that is invariant to node permutation. For a general class of count functionals, we also establish consistency of the network jackknife. We complement our theoretical analysis with a range of simulated and real-data examples and show that the network jackknife offers competitive performance in cases where other resampling methods are known to be valid. In fact, for several network statistics, we see that the jackknife provides more accurate inferences compared to related methods such as subsampling.

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arXiv2019

Subsampling Sparse Graphons Under Minimal Assumptions

We establish a general theory for subsampling network data generated by the sparse graphon model. In contrast to previous work for networks, we demonstrate validity under minimal assumptions; the main requirement is weak convergence of the functional of interest. We study the properties of two procedures: vertex subsampling and $p$-subsampling. For the first, we prove validity under the mild condition that the number of subsampled vertices is $o(n)$. For the second, we establish validity under analogous conditions on the expected subsample size. For both procedures, we also establish conditions under which uniform validity holds. Furthermore, under appropriate sparsity conditions, we derive limiting distributions for the nonzero eigenvalues of the adjacency matrix of a low rank sparse graphon. Our weak convergence result immediately yields the validity of subsampling for the nonzero eigenvalues under suitable assumptions.

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arXiv2019

Sample Splitting and Weak Assumption Inference For Time Series

We consider the problem of inference after model selection under weak assumptions in the time series setting. Even when the data are not independent, we show that sample splitting remains asymptotically valid as long as the process satisfies appropriate weak dependence conditions and the functional of interest is suitably well-behaved. In addition, if the inference targets are appropriately defined, we demonstrate that valid statistical inference is possible without assuming stationarity. As a working example, we consider post-selection inference for regression coefficients under a random design assumption, in which the pair $(Y_i, X_i) \in \mathbb{R}^{p_n}$ is assumed to be an observation from a weakly dependent triangular array. We establish (asymptotic) sample splitting validity for regression coefficients under both $β$-mixing and $τ$-dependence assumptions. To facilitate statistical inference in the non-stationary, weakly dependent regime, we extend a central limit theorem of Doukhan and Wintenberger (2007). To extend their result, we derive some properties of the variance of a normalized sum of a weakly dependent process. In particular, we show that, under very general conditions, the variance is often well-approximated by independent blocks. Using this result, we derive the validity of the block multiplier bootstrap under $θ$-dependence and demonstrate the validity of an inference procedure that combines sample splitting with the bootstrap under weak assumptions.

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arXiv2017

Bootstrapping Generalization Error Bounds for Time Series

We consider the problem of finding confidence intervals for the risk of forecasting the future of a stationary, ergodic stochastic process, using a model estimated from the past of the process. We show that a bootstrap procedure provides valid confidence intervals for the risk, when the data source is sufficiently mixing, and the loss function and the estimator are suitably smooth. Autoregressive (AR(d)) models estimated by least squares obey the necessary regularity conditions, even when mis-specified, and simulations show that the finite- sample coverage of our bounds quickly converges to the theoretical, asymptotic level. As an intermediate step, we derive sufficient conditions for asymptotic independence between empirical distribution functions formed by splitting a realization of a stochastic process, of independent interest.

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