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Jeffrey Näf

Publications and source records attributed to Jeffrey Näf.

At least 19 recordsLinked to original sources
arXiv2026

Parametric MMD Estimation with Missing Values: Robustness to Missingness and Data Model Misspecification

In the missing data literature, the Maximum Likelihood Estimator (MLE) is celebrated for its ignorability property under missing at random (MAR) data. However, its sensitivity to misspecification of the (complete) data model, even under MAR, remains a significant limitation. This issue is further exacerbated by the fact that the MAR assumption may not always be realistic, introducing an additional source of potential misspecification through the missingness mechanism. To address this, we propose a novel M-estimation procedure based on the Maximum Mean Discrepancy (MMD), which is provably robust to both model misspecification and deviations from the assumed missingness mechanism. Our approach offers strong theoretical guarantees and improved reliability in complex settings. We establish the consistency and asymptotic normality of the estimator under missing completely at random (MCAR), provide an efficient stochastic gradient descent algorithm, and derive error bounds that explicitly separate the contributions of model misspecification and missingness bias. Furthermore, we analyze missing not at random (MNAR) scenarios where our estimator maintains controlled error, including a Huber setting where both the missingness mechanism and the data model are contaminated. Our contributions refine the understanding of the limitations of the MLE and provide a robust and principled alternative for handling missing data.

stat.ME↗
arXiv2026

Asymptotics of Nonparametric Estimation under General Non-monotone MAR Missingness: A Nonparametric Maximum Likelihood Approach

Missing data constitute a pervasive challenge in empirical research. Consequently, there is an ever-growing number of methods designed to address this challenge, with multiple imputation and inverse probability weighting the dominant strategies. Despite this, theoretical guarantees remain limited, particularly in the challenging case of non-monotone missing at random (MAR). When guarantees exist, they are often confined to simplified settings such as missing completely at random, monotone or block-wise missingness, or rest on restrictive assumptions about the missingness mechanism. In this paper, we utilize the theory of sieve maximum likelihood to establish a general rate of convergence under MAR that requires no modeling of the missingness mechanism and no restriction on the configuration of missing patterns, beyond MAR itself and a natural positivity condition. Applying this result to density estimation, we show that the complete-data density can be estimated at the minimax rate over a Hölder class, up to a logarithmic factor, for any prescribed smoothness level. The missingness does not affect the rate and enters only through a constant. The estimator is approximated in practice by a simple expectation-maximization (EM) algorithm operating on the incomplete data directly. In simulations, it performs comparably to the kernel density estimator supplied with the complete data across a wide range of missingness levels.

stat.ME↗
arXiv2026

Asymptotics of Nonparametric Estimation under general non-monotone MAR missingness: A Bayesian Approach

Missing values are ubiquitous in statistical practice, with potentially detrimental consequences for any statistical analysis. As such, a wealth of methods and theoretical results have been developed in the last decades. However, many questions remain open, in particular in the case of general non-monotone missing at random (MAR), where nonparametric results are still lacking. In this paper, we extend nonparametric Bayesian theory to this MAR setting. We introduce a general theorem of posterior contraction under MAR and an additional positivity condition and apply this result to density estimation as well as regression problems. In particular, we show that, despite the missing values, the complete-data density can be estimated with the minimax posterior contraction rate up to logarithmic factors. To the best of our knowledge, this is the first nonparametric result showing that the complete-data distribution can be consistently estimated under Rubin's MAR definition. As a consequence, we obtain an algorithm that takes incomplete data and returns a sample from a consistent estimate of the complete-data distribution.

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arXiv2026

Generative Modeling under Non-Monotone MAR Missingness via Approximate Wasserstein Gradient Flows

The prevalence of missing values in data science poses a substantial risk to any further analyses. Despite a wealth of research, principled nonparametric methods to deal with general non-monotone missingness are still scarce. Instead, ad-hoc imputation methods are often used, for which it remains unclear whether the correct distribution can be recovered. In this paper, we propose FLOWGEM, a principled iterative method for generating a complete dataset from a dataset with values Missing at Random (MAR). Motivated by convergence results of the ignoring maximum likelihood estimator, our approach minimizes the expected Kullback-Leibler (KL) divergence between the observed data distribution and the distribution of the generated sample over different missingness patterns. To minimize the KL divergence, we employ a discretized particle evolution of the corresponding Wasserstein Gradient Flow, where the velocity field is approximated using a local linear estimator of the density ratio. This construction yields a data generation scheme that iteratively transports an initial particle ensemble toward the target distribution. Simulation studies and real-data benchmarks demonstrate that FLOWGEM achieves state-of-the-art performance across a range of settings, including the challenging case of non-monotone MAR mechanisms. Together, these results position FLOWGEM as a principled and practical alternative to existing imputation methods, and a decisive step towards closing the gap between theoretical rigor and empirical performance.

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arXiv2026

CLVAE: A Variational Autoencoder for Long-Term Customer Revenue Forecasting

Predicting customers' long-term revenue from sparse and irregular transaction data is central to marketing resource allocation in non-contractual settings, yet existing approaches face a trade-off. Traditional probabilistic customer base models deliver robust long-horizon forecasts by imposing strong structural assumptions, while flexible machine-learning models often require substantial training data and careful tuning. We propose a variational-autoencoder-based model that preserves the process-based likelihood of established attrition-transaction-spend models conditional on customer heterogeneity, but replaces the restrictive parametric mixing distribution with a flexible latent representation learned by encoder-decoder networks. The resulting approach (i) provides a single model for customer attrition, transactions and spending, (ii) remains reliable when contextual covariates are unavailable, and (iii) flexibly incorporates rich covariates and nonlinear effects when they are available. This design balances structural stability with the flexibility needed to capture complex purchase dynamics. Across multiple real-world datasets and prediction horizons, the proposed model improves upon the latest benchmarks. Businesses benefit directly, as a better assessment of customers' future revenues improves the efficiency of campaign targeting. For research, this work provides guidance on how to embed domain-specific models into the variational autoencoder framework, enabling flexible representation learning while retaining an econometrically meaningful process structure.

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arXiv2026

Causal-DRF: Conditional Kernel Treatment Effect Estimation using Distributional Random Forest

The conditional average treatment effect (CATE) is a commonly targeted statistical parameter for measuring the effect of a treatment conditional on covariates. However, the CATE will fail to capture effects of treatments beyond differences in conditional expectations. Inspired by causal forests for CATE estimation, we develop a forest-based method to estimate the conditional kernel treatment effect (CKTE), based on the recently introduced Distributional Random Forest (DRF) algorithm. Adapting the splitting criterion of DRF, we show how one forest fit can be used to obtain a consistent and asymptotically normal estimator of the CKTE, as well as an approximation of its sampling distribution. This allows to study the difference in distribution between control and treatment group and thus yields a more comprehensive understanding of the treatment effect.

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arXiv2026

Estimating Individual Customer Lifetime Values with R: The CLVTools Package

Customer lifetime value (CLV) describes a customer's long-term economic value for a business. This metric is widely used in marketing, for example, to select customers for a marketing campaign. However, modeling CLV is challenging. When relying on customers' purchase histories, the input data is sparse. Additionally, given its long-term focus, prediction horizons are often longer than estimation periods. Probabilistic models are able to overcome these challenges and, thus, are a popular option among researchers and practitioners. The latter also appreciate their applicability for both small and big data as well as their robust predictive performance without any fine-tuning requirements. Their popularity is due to three characteristics: data parsimony, scalability, and predictive accuracy. The R package CLVTools provides an efficient and user-friendly implementation framework to apply key probabilistic models such as the Pareto/NBD and Gamma-Gamma model. Further, it provides access to the latest model extensions to include time-invariant and time-varying covariates, parameter regularization, and equality constraints. This article gives an overview of the fundamental ideas of these statistical models and illustrates their application to derive CLV predictions for existing and new customers.

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arXiv2026

What Is a Good Imputation Under MAR Missingness?

Missing values pose a persistent challenge in modern data science. Consequently, there is an ever-growing number of publications introducing new imputation methods in various fields. The present paper attempts to take a step back and provide a more systematic analysis. Starting from an in-depth discussion of the Missing at Random (MAR) condition for nonparametric imputation, we first investigate whether the widely used fully conditional specification (FCS) approach indeed identifies the correct conditional distributions. Based on this analysis, we propose three essential properties an ideal imputation method should meet, thus enabling a more principled evaluation of existing methods and more targeted development of new methods. In particular, we introduce a new imputation method, denoted mice-DRF, that meets two out of the three criteria. We also discuss ways to compare imputation methods, based on distributional distances. Finally, numerical experiments illustrate the points made in this discussion.

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arXiv2025

Do we Need Dozens of Methods for Real World Missing Value Imputation?

Missing values pose a persistent challenge in modern data science. Consequently, there is an ever-growing number of publications introducing new imputation methods in various fields. While many studies compare imputation approaches, they often focus on a limited subset of algorithms and evaluate performance primarily through pointwise metrics such as RMSE, which are not suitable to measure the preservation of the true data distribution. In this work, we provide a systematic benchmarking method based on the idea of treating imputation as a distributional prediction task. We consider a large number of algorithms and, for the first time, evaluate them not only on synthetic missing mechanisms, but also on real-world missingness scenarios, using the concept of Imputation Scores. Finally, while the focus of previous benchmark has often been on numerical data, we also consider mixed data sets in our study. The analysis overwhelmingly confirms the superiority of iterative imputation algorithms, especially the methods implemented in the mice R package.

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arXiv2025

How to rank imputation methods?

Imputation is an attractive tool for dealing with the widespread issue of missing values. Consequently, studying and developing imputation methods has been an active field of research over the last decade. Faced with an imputation task and a large number of methods, how does one find the most suitable imputation? Although model selection in different contexts, such as prediction, has been well studied, this question appears not to have received much attention. In this paper, we follow the concept of Imputation Scores (I-Scores) and develop a new, reliable, and easy-to-implement score to rank missing value imputations for a given data set without access to the complete data. In practice, this is usually done by artificially masking observations to compare imputed to observed values using measures such as the Root Mean Squared Error (RMSE). We discuss how this approach of additionally masking observations can be misleading if not done carefully and that it is generally not valid under MAR. We then identify a new missingness assumption and develop a score that combines a sensible masking of observations with proper scoring rules. As such the ranking is geared towards the imputation that best replicates the distribution of the data, allowing to find imputations that are suitable for a range of downstream tasks. We show the propriety of the score and discuss an estimation algorithm involving energy scores. Finally, we show the efficacy of the new score in simulated data examples, as well as a downstream task.

stat.ME↗
arXiv2024

MMD-based Variable Importance for Distributional Random Forest

Distributional Random Forest (DRF) is a flexible forest-based method to estimate the full conditional distribution of a multivariate output of interest given input variables. In this article, we introduce a variable importance algorithm for DRFs, based on the well-established drop and relearn principle and MMD distance. While traditional importance measures only detect variables with an influence on the output mean, our algorithm detects variables impacting the output distribution more generally. We show that the introduced importance measure is consistent, exhibits high empirical performance on both real and simulated data, and outperforms competitors. In particular, our algorithm is highly efficient to select variables through recursive feature elimination, and can therefore provide small sets of variables to build accurate estimates of conditional output distributions.

stat.ML↗
arXiv2023

Confidence and Uncertainty Assessment for Distributional Random Forests

The Distributional Random Forest (DRF) is a recently introduced Random Forest algorithm to estimate multivariate conditional distributions. Due to its general estimation procedure, it can be employed to estimate a wide range of targets such as conditional average treatment effects, conditional quantiles, and conditional correlations. However, only results about the consistency and convergence rate of the DRF prediction are available so far. We characterize the asymptotic distribution of DRF and develop a bootstrap approximation of it. This allows us to derive inferential tools for quantifying standard errors and the construction of confidence regions that have asymptotic coverage guarantees. In simulation studies, we empirically validate the developed theory for inference of low-dimensional targets and for testing distributional differences between two populations.

math.ST↗
arXiv2023

R-NL: Covariance Matrix Estimation for Elliptical Distributions based on Nonlinear Shrinkage

We combine Tyler's robust estimator of the dispersion matrix with nonlinear shrinkage. This approach delivers a simple and fast estimator of the dispersion matrix in elliptical models that is robust against both heavy tails and high dimensions. We prove convergence of the iterative part of our algorithm and demonstrate the favorable performance of the estimator in a wide range of simulation scenarios. Finally, an empirical application demonstrates its state-of-the-art performance on real data.

stat.ME↗
arXiv2022

Imputation Scores

Given the prevalence of missing data in modern statistical research, a broad range of methods is available for any given imputation task. How does one choose the `best' imputation method in a given application? The standard approach is to select some observations, set their status to missing, and compare prediction accuracy of the methods under consideration of these observations. Besides having to somewhat artificially mask observations, a shortcoming of this approach is that imputations based on the conditional mean will rank highest if predictive accuracy is measured with quadratic loss. In contrast, we want to rank highest an imputation that can sample from the true conditional distributions. In this paper, we develop a framework called "Imputation Scores" (I-Scores) for assessing missing value imputations. We provide a specific I-Score based on density ratios and projections, that is applicable to discrete and continuous data. It does not require to mask additional observations for evaluations and is also applicable if there are no complete observations. The population version is shown to be proper in the sense that the highest rank is assigned to an imputation method that samples from the correct conditional distribution. The propriety is shown under the missing completely at random (MCAR) assumption but is also shown to be valid under missing at random (MAR) with slightly more restrictive assumptions. We show empirically on a range of data sets and imputation methods that our score consistently ranks true data high(est) and is able to avoid pitfalls usually associated with performance measures such as RMSE. Finally, we provide the R-package Iscores available on CRAN with an implementation of our method.

stat.AP↗
arXiv2022

PKLM: A flexible MCAR test using Classification

We develop a fully non-parametric, easy-to-use, and powerful test for the missing completely at random (MCAR) assumption on the missingness mechanism of a dataset. The test compares distributions of different missing patterns on random projections in the variable space of the data. The distributional differences are measured with the Kullback-Leibler Divergence, using probability Random Forests. We thus refer to it as "Projected Kullback-Leibler MCAR" (PKLM) test. The use of random projections makes it applicable even if very few or no fully observed observations are available or if the number of dimensions is large. An efficient permutation approach guarantees the level for any finite sample size, resolving a major shortcoming of most other available tests. Moreover, the test can be used on both discrete and continuous data. We show empirically on a range of simulated data distributions and real datasets that our test has consistently high power and is able to avoid inflated type-I errors. Finally, we provide an R-package PKLMtest with an implementation of our test.

stat.ME↗
arXiv2022

High Probability Lower Bounds for the Total Variation Distance

The statistics and machine learning communities have recently seen a growing interest in classification-based approaches to two-sample testing. The outcome of a classification-based two-sample test remains a rejection decision, which is not always informative since the null hypothesis is seldom strictly true. Therefore, when a test rejects, it would be beneficial to provide an additional quantity serving as a refined measure of distributional difference. In this work, we introduce a framework for the construction of high-probability lower bounds on the total variation distance. These bounds are based on a one-dimensional projection, such as a classification or regression method, and can be interpreted as the minimal fraction of samples pointing towards a distributional difference. We further derive asymptotic power and detection rates of two proposed estimators and discuss potential uses through an application to a reanalysis climate dataset.

math.ST↗
arXiv2022

Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression

Random Forest (Breiman, 2001) is a successful and widely used regression and classification algorithm. Part of its appeal and reason for its versatility is its (implicit) construction of a kernel-type weighting function on training data, which can also be used for targets other than the original mean estimation. We propose a novel forest construction for multivariate responses based on their joint conditional distribution, independent of the estimation target and the data model. It uses a new splitting criterion based on the MMD distributional metric, which is suitable for detecting heterogeneity in multivariate distributions. The induced weights define an estimate of the full conditional distribution, which in turn can be used for arbitrary and potentially complicated targets of interest. The method is very versatile and convenient to use, as we illustrate on a wide range of examples. The code is available as Python and R packages drf.

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