arXiv subjects

Aaditya Ramdas

Publications and source records attributed to Aaditya Ramdas.

At least 55 records · Page 3Linked to original sources

arXiv2026

Gaffke's confidence interval for the mean of bounded data is inadmissible but asymptotically efficient

Given observations $\mathbf x=(x_1,\dots,x_n)$, Gaffke (2005) defined \[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \] and conjectured that it is a $p$-value whenever the inputs are independent e-values. Recently, Vlassis and Thomas (2026) proved this conjecture. Inverting the tests for observations in $[0,1]$ gives the confidence interval studied by Learned-Miller and Thomas (2020), which reduces to Clopper--Pearson for Bernoulli data. We give a finite- and large-sample account of Gaffke's test and interval. First, for every $\mathbf x\in[0,\infty)^n$ and every elementary symmetric polynomial $e_k$, $ K_n(\mathbf x)e_k(\mathbf x)\le {n\choose k}, $ so the Gaffke $p$-value never larger than the SymPol $p$-value of Ming et al. (2026). However, Gaffke's p-value is inadmissible. For $n=2$, we construct a valid rule that is strictly smaller on mixed configurations and is the unique admissible rule that dominates $K_2$. A neutral-face extension proves inadmissibility of $K_n$ for every $n\ge2$. If one independent uniform random variable is allowed, there is an even simpler full-dimensional improvement: on the upper orthant, where $K_n(\mathbf x)=1/\prod_i x_i$, replace it by $U/\prod_i x_i$. The equal-tail Gaffke confidence interval $I_n$ is nevertheless first-order asymptotically efficient: for iid observations on $[0,1]$ with unknown variance $σ^2>0$, \[ \sqrt n\,\operatorname{Width}(I_n)\longrightarrow 2σz_{1-α/2}\qquad\text{almost surely}. \] Our simulations also find that, among a variety of bounded-mean intervals considered, the Gaffke interval is the shortest, including comparisons with a recent empirical Berry--Esseen procedure having the same first-order Gaussian target.

math.ST↗

arXiv2026

Strong duality for the GROW criterion

This paper presents general strong duality results when testing hypotheses by betting against them. A bet is an e-variable for a composite null hypothesis $\Pcal$: a nonnegative random variable $X$ whose expected value is at most one under every $P \in \mathcal P$. Following Kelly, Breiman, Cover, Shafer, and Grunwald et al. (2024), we study a natural minimax \emph{log-optimality} criterion: given a composite alternative $\Qcal$, we characterize the ``GROW value'' $\sup_{X} \inf_{Q} \E_{Q}[\log X]$. This paper generalizes the results of Larsson et al. (2025) from (arbitrary $\mathcal P$ and) simple $\mathcal Q$ to arbitrary $\mathcal Q$. We prove that there always exists a minimizing information-projection pair between the weak-$*$ closures of the convex hulls of arbitrary $\mathcal P$ and $\mathcal Q$, and show that the GROW value for \emph{bounded} e-variables always equals their relative entropy. We also prove a similarly general strong duality for the REGROW criterion with bounded e-variables and arbitrary bounded offsets. Under various assumptions our results extend to unbounded e-variables, and examples show that without any assumptions such extensions fail. Our results are analogous to those in Larsson et al. (2026), swapping tests for bounded e-variables, minimax risk for the GROW criterion, and total variation for relative entropy.

math.ST↗

arXiv2026

Testing by Betting while Borrowing and Bargaining

Testing by betting has been a cornerstone of the game-theoretic statistics literature. One bets against the null hypothesis, and the accumulated wealth $W_t$ quantifies the evidence against the null hypothesis after $t$ rounds, and the null can be rejected at level $α$ whenever $W_t \geq 1/α$. A key assumption permeating the literature is that one cannot bet more money than they currently have (the wealth must stay nonnegative). In this work, we examine the consequences of allowing the bettor to borrow money in each round (for example after going bankrupt). Specifically, we ask how the threshold of $1/α$ must be accordingly adjusted to retain the desired level $α$. Our findings are twofold. First, if the new rejection rule is $W_t \geq g(α,L_t)$ where $L_t$ is the total liability at time $t$, then we show that $g(α,0)>1/α$ if $g(α,L_t)<\infty$ for any $L_t > 0$; in words, we must pay for the possibility of borrowing, even if in fact we do not borrow. Second, and in contrast to the first, if one employs a path dependent threshold $h(α,W_0,L_1,\dots,W_{t-1},L_t)$, that is a function of past leverage ratios, then there is in fact no extra price to pay for the possibility of borrowing.

math.ST↗

arXiv2026

E-values and sequential power-one tests for monotonicity and unimodality

We develop e-values and e-processes testing the null hypothesis that a distribution over nonnegative integers is monotone, and that a distribution over integers is unimodal given a certain mode. Our e-processes lead to tests of power one under any non-null distribution with a sequence of i.i.d. observations, and consistent set-valued mode estimators that eventually equal the true set of modes. Additionally, we characterize the set of all e-values, and therefore the set of all valid tests, with one monotone and unimodal observation, as well as the most powerful e-value for a fixed alternative. We then show that many of our results can be generalized to continuous random variables, relating them to the existing results in the shape-constrained inference literature.

math.ST↗

arXiv2026

Improving online FDR procedures via online analogs of e-closure and compound e-values

In many scientific applications, hypotheses are generated and tested continuously in a stream. We develop a framework for improving online multiple testing procedures with false discovery rate (FDR) control under arbitrary dependence. Our approach is two-fold: we construct methods via the online e-closure principle, as well as a novel formulation of online compound e-values that is defined through donations. This yields strict power improvements over state-of-the-art e-value and p-value procedures while retaining FDR control. We further derive algorithms that compute the decision at time $t$ in $O(\log t)$ time, and we demonstrate improved empirical performance on synthetic and real data.

stat.ME↗

arXiv2026

Distribution-free changepoint localization after sequential change detection

This paper introduces a distribution-free framework for constructing post-detection confidence sets for changepoints after stopping a sequential change detection procedure. It is well known that conformal test martingales can be used to sequentially detect changes in distribution, but by themselves provide no inference for the time at which a proclaimed change occurred. Past work on post-detection inference requires pre- and post-change classes of distributions to be known, but this paper accomplishes localization of the changepoint without any distributional assumptions. We establish finite-sample coverage guarantees (conditional on correct detection). We provide non-asymptotic bounds on the conditional expected size of the confidence sets. Under suitable asymptotic regimes, we prove that the conditional expected size of the confidence set remains uniformly bounded and demonstrate strong empirical performance on simulated and real data. To the best of our knowledge, this is the first general distribution-free framework for sequential changepoint localization with valid post-detection coverage.

stat.ML↗

arXiv2026

Optimal Rates for Differentially Private Hypothesis Testing with E-values

E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable e-power when testing $X\sim \mathbb{P}^n$ against $X\sim\mathbb{Q}^n$ with e-values that satisfy $\varepsilon$-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it exactly. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, we give matching upper and lower bounds on the stopping times of any private e-process. Numerical experiments confirm the practicality of our algorithms, which require less data than the recently proposed DP-SPRT across a range of sequential testing problems and privacy levels.

cs.CR↗

arXiv2026

M-estimation with e-statistics

We present a theory of point estimation with e-statistics (e-values and e-processes) by introducing the "ME-estimator": the parameter that minimizes the corresponding e-statistic, or the evidence against it. Our approach is based on the intuitive idea of e-statistics as a measure of evidence and betting pay-off, and naturally generalizes the classical method of maximum likelihood estimation. First, we establish the consistency as well as the almost sure convergence rate for ME-estimators relating to the high-probability bounds on the size of the confidence set derived from thresholding the e-statistics, an approach that sets ME-estimators apart from traditional M-estimators. Second, we conduct classical M-estimator-style analysis on the consistency and asymptotic normality of ME-estimators in the bounded mean estimation setting, discussing the notion of efficiency (or lack thereof) from various choices of betting strategy. Our work brings e-statistics, a fundamental tool for inference and uncertainty quantification, to the space of estimation.

stat.ME↗

arXiv2026

Online monotone density estimation and log-optimal calibration

We study the problem of online monotone density estimation, where density estimators must be constructed in a predictable manner from sequentially observed data. We propose two online estimators: an online analogue of the classical Grenander estimator, and an expert aggregation estimator inspired by exponential weighting methods from the online learning literature. In the well-specified stochastic setting, where the underlying density is monotone, we show that the expected cumulative log-likelihood gap between the online estimators and the true density admits an $O(n^{1/3})$ bound. We further establish a $\sqrt{n\log{n}}$ pathwise regret bound for the expert aggregation estimator relative to the best offline monotone estimator chosen in hindsight, under minimal regularity assumptions on the observed sequence. As an application of independent interest, we show that the problem of constructing log-optimal p-to-e calibrators for sequential hypothesis testing can be formulated as an online monotone density estimation problem. We adapt the proposed estimators to build empirically adaptive p-to-e calibrators and establish their optimality. Numerical experiments illustrate the theoretical results.

stat.ML↗

arXiv2026

Distribution-free root cause analysis

We study distribution-free root cause analysis in multi-stream data, where an evolving underlying system is observed through multiple data streams that may each undergo distributional changes at unknown timepoints. In such settings, the stream exhibiting the earliest change provides a natural starting point for investigating the underlying cause, which we refer to as the root-cause index. Leveraging conformal $p$-values, we propose a novel framework, Conformal Root Cause Analysis (CROC), which constructs finite-sample valid confidence sets for the root-cause index under minimal assumptions: the data streams are independent, and within each stream the pre- and post-change observations are sampled exchangeably from arbitrary and unknown distributions. We further establish a universality property, showing that any distribution-free method for root cause localization can be represented within the CROC framework. In addition, under mild regularity conditions and principled score design, our method yields asymptotically sharp confidence sets that efficiently isolate the root cause. We further extend CROC to efficiently handle cross-stream dependence when present. Extensive simulations demonstrate accurate localization of the root stream, supporting our theoretical guarantees.

stat.ME↗

arXiv2026

Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity

The study of self-normalized processes plays a crucial role in a wide range of applications, from sequential decision-making to econometrics. While the behavior of self-normalized concentration has been widely investigated for scalar-valued processes, vector-valued processes remain comparatively underexplored, especially outside of the sub-Gaussian framework. In this contribution, we provide concentration bounds for self-normalized processes with light tails beyond sub-Gaussianity (such as Bennett or Bernstein bounds). We illustrate the relevance of our results in the context of online linear regression, with applications in (kernelized) linear bandits.

stat.ML↗

arXiv2026

Optimal sequential tests yield log-optimal e-processes

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$α$ sequential test can be obtained by thresholding an e-process at $1/α$. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.

math.ST↗

arXiv2026

Almost sure null bankruptcy of testing-by-betting strategies

The bounded mean betting procedure serves as a crucial interface between the domains of (1) sequential, anytime-valid statistical inference, and (2) online learning and portfolio selection algorithms. While recent work in both domains has established the exponential wealth growth of numerous betting strategies under any alternative distribution, the tightness of the inverted confidence sets, and the pathwise minimax regret bounds, little has been studied regarding the asymptotics of these strategies under the null hypothesis. Under the null, a strategy induces a wealth martingale converging to some random variable that can be zero (bankrupt) or non-zero (non-bankrupt, e.g. when it eventually stops betting). In this paper, we show the conceptually intuitive but technically nontrivial fact that these strategies (universal portfolio, Krichevsky-Trofimov, GRAPA, hedging, etc.) all go bankrupt with probability one, under any non-degenerate null distribution. Part of our analysis is based on the subtle almost sure divergence of various sums of $\sum_n O_p(n^{-1})$ type, a result of independent interest. We also demonstrate the necessity of null bankruptcy by showing that non-bankrupt strategies are all improvable in some sense. Our results significantly deepen our understanding of these betting strategies as they qualify their behavior on "almost all paths", whereas previous results are usually on "all paths" (e.g. regret bounds) or "most paths" (e.g. concentration inequalities and confidence sets).

math.PR↗

arXiv2026

The optimal betting wealth growth rate

This paper characterizes the best possible rate of growth of wealth in a Kelly betting game when repeatedly betting against a general i.i.d. null hypothesis $\mathscr{P}$, but the data are drawn i.i.d from an arbitrary alternative $Q$. We prove that it equals $\lim_{n \to \infty}n^{-1}\inf_{P \in (\mathscr P)^n)^{\circ\circ}} \mathrm{KL}(Q^n,P)$, where ${\mathscr P}^n = \{P^n: P \in \mathscr{P}\}$ and $(\mathscr {P}^n)^{\circ\circ}$ is its bipolar, i.e., this rate is achievable and one cannot do better. This quantity is in general smaller than a more popular quantity in the literature, $\mathrm{KL}_{\inf}(Q,\mathscr{P}) := \inf_{P \in \mathscr P}\mathrm{KL}(Q,P)$. If $\mathrm{KL}_{\mathrm{inf}}(\cdot,\mathscr P)$ is weakly lowersemicontinuous (w.l.s.c.) at $Q$, we show that the two quantities are equal; in particular, this happens when $\mathscr P$ is weakly compact. For simple alternatives, we provide the first matching necessary and sufficient condition for when power-one sequential tests exist (without assumptions on $\mathscr P, Q$). We also derive the optimal worst-case growth rate against composite $\mathscr Q$. We emphasize that test supermartingales on reduced filtrations suffice for all i.i.d. testing problems, and more general e-processes are not required. We thus completely generalize the recent results of Larsson et al.~\cite{larsson2025numeraire} to the sequential setting.

math.ST↗

arXiv2026

Besag-Clifford e-values for unnormalized testing

Unnormalized probability distributions are frequently used in machine learning for modeling complex data generating processes. Though Markov chain Monte Carlo (MCMC) algorithms can approximately sample from unnormalized distributions, intractability of their normalizing constants renders likelihood ratio testing infeasible. We propose to use the parallel method of Besag and Clifford to generate samples that are exchangeable with the data under the null, to then generate valid e-values for any number of iterations or algorithmic steps. We show that as the number of samples grows, these Besag-Clifford e-values constructed using the unnormalized likelihood ratio are actually log-optimal up to a multiplicative term that diminishes with the mixing time of the Markov chain. Additionally, averaging over the output of multiple chains retains validity while increasing the e-power. We extend Besag-Clifford e-values to the general problem of unnormalized test statistics, which allows application to composite hypotheses, uncertainty quantification, generative model evaluation, and sequential testing. Through simulations and an application to galaxy velocity modeling, we empirically verify our theory, explore the impact of autocorrelation and mixing, and evaluate the performance of Besag-Clifford e-values.

stat.ME↗

arXiv2026

Optimal e-variables under constraints

E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well understood, one may need/wish to impose additional structural constraints, including differential privacy, quantization, boundedness, or moment restrictions. We show that under these constraints, log-optimal constrained e-variables can often be constructed by a simple \emph{optimize-then-constrain} principle: first compute the unconstrained log-optimal e-variable, then impose the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different LFD pair; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.

stat.ME↗

arXiv2026

Eventually LIL Regret: Almost Sure $\ln\ln T$ Regret for a sub-Gaussian Mixture on Unbounded Data

We prove that a classic sub-Gaussian mixture proposed by Robbins in a stochastic setting actually satisfies a path-wise (deterministic) regret bound. For every path in a natural ``Ville event'' $\mathcal E_α$, this regret till time $T$ is bounded by $\ln^2(1/α)/V_T + \ln (1/α) + \ln \ln V_T$ up to universal constants, where $V_T$ is a nonnegative, nondecreasing, cumulative variance process. (The bound reduces to $\ln(1/α) + \ln \ln V_T$ if $V_T \geq \ln(1/α)$.) If the data were stochastic, then one can show that $\mathcal E_α$ has probability at least $1-α$ under a wide class of distributions (eg: sub-Gaussian, symmetric, variance-bounded, etc.). In fact, we show that on the Ville event $\mathcal E_0$ of probability one, the regret on every path in $\mathcal E_0$ is eventually bounded by $\ln \ln V_T$ (up to constants). We explain how this work helps bridge the world of adversarial online learning (which usually deals with regret bounds for bounded data), with game-theoretic statistics (which can handle unbounded data, albeit using stochastic assumptions). In short, conditional regret bounds serve as a bridge between stochastic and adversarial betting.

cs.LG↗

arXiv2026

Tiny but uniform improvements of adaptive BH procedures via compound e-values

After the seminal Benjamini-Hochberg (BH) procedure for controlling the false discovery rate (FDR) was proposed, dozens of papers have attempted to improve its power by adapting to the unknown proportion of nulls. We observe that most null proportion estimates are simply compound e-values in disguise, and thus most adaptive FDR procedures can be interpreted as instances of the e-weighted BH (ep-BH) procedure of Ignatiadis, Wang, and Ramdas [2024], i.e., the BH procedure weighted by compound e-values. This lens helps us show that most existing procedures are inadmissible, and we provide uniform improvements to them. While the improvements are small in practice, they still come for free (without additional assumptions), and help unify the literature. We also use our "leave-one-out ep-BH method" to design a new method with finite-sample FDR control for the simultaneous t-test setting.

stat.ME↗