Papers

This paper develops a decision-theoretic interpretation of local expected utility on Abstract Wiener space. The framework is intended for economic acts that are naturally stochastic paths, such as income paths, consumption paths, investment payoffs, insurance losses, or laboratory stimuli unfolding over a continuum of contingencies. In this setting the object of choice is an act or payoff path, while Hermite functions serve as state-feature coordinates rather than objects of preference. Abstract Wiener space supplies a Gaussian reference measure on the Banach space of paths and a Hilbert subspace on which local utility is represented by orthogonal coefficients. The associated Wiener-integral weights are generally signed stochastic functionals; they become admissible decision weights only after a specified nonnegative normalization. This distinction clarifies the relation between the model, state-dependent utility, subjective expected utility, rank-dependent utility, and cumulative prospect theory. The main representation result gives finite primitive conditions under which projected path-valued acts admit a normalized local-utility representation. The corresponding elicitation result identifies composite utility-weight coefficients from projected acts and recovers normalized decision weights when the local utility scale is independently elicited. The paper also states testable restrictions: normalized weights must be nonnegative, add to one, preserve monotonicity, and satisfy additional rank or gain-loss restrictions when the subjective expected utility, rank-dependent utility, or cumulative prospect theory submodels are imposed. Numerical illustrations show how the representation operates in finite approximations.

We design a method for measuring the risk preferences of agents in the deep past. The method combines a structural model of crop choice as a portfolio allocation with machine-learning prediction of expected crop returns, using historic agronomic and climate data. We estimate county-level risk preferences for the United States and farmer-level preferences in Kansas from 1889 to 1929. More risk averse farmers leveraged less, were less likely to purchase novel WWI Liberty Bonds, and were more likely to participate in local risk-sharing institutions. We show that higher risk aversion predicts slower tractor adoption and farm mechanization during the 1920s.

We study how many observations are needed to determine the causal direction between two linearly related variables. Classical LiNGAM theory shows that independent non-Gaussian disturbances identify the direction, but does not quantify the difficulty when the causal effect is weak or the disturbances are nearly Gaussian. Let $\beta$ bound the absolute structural coefficient from below, let $\nu$ measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in $[\underline\sigma,\overline\sigma]$. We prove the sharp local minimax law \[ N_2^\star(\beta,\nu,\delta) \asymp \frac{\log(1/\delta)} {d_\beta^2+\beta^2\nu^2}, \qquad d_\beta= \left[\beta^2- \left(1-\frac{\underline\sigma^2}{\overline\sigma^2}\right)\right]_+. \] Previous theory established population identifiability or assumed a fixed separation between the two directions. By contrast, we establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty, and characterize when identification comes from non-Gaussian dependence or from covariance alone. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.

In Shapley-Scarf housing markets, Ma (1994) shows that top trading cycles (TTC) is the unique mechanism satisfying individual rationality (IR), Pareto efficiency (PE), and strategy-proofness. We ask what other mechanisms become possible when strategy-proofness is replaced by a weaker condition called non-obvious manipulability (NOM), introduced by Troyan and Morrill (2020). We first show that this weaker condition does not help on its own: every IR and PE mechanism is already NOM. We therefore introduce a new condition: NOM with groups, under which each agent knows the preferences of the other members of her group, but not those of agents outside the group. This condition reduces to strategy-proofness when all agents belong to one group, and to standard NOM when every group is a singleton. Also, we introduce a participation condition called group rationality (GR), which requires that no group do worse than it would by trading only among its own members. We then define a class of mechanisms called TTC with super-groups, whose members satisfy GR, PE, and NOM with groups. The class includes mechanisms that differ from standard TTC, including mechanisms that are not strategy-proof. Furthermore, we show that every mechanism that satisfies GR, PE, and NOM with groups has the same best- and worst-case outcomes as every TTC with super-groups mechanism.

Route and activity choice are distinct transportation problems that both require models of feasible decisions unfolding over networks and time. This critical integrative review connects transportation choice modeling with inverse reinforcement learning (IRL) and imitation learning (IL), while distinguishing evidence from transportation applications, transferable methods from other fields, and emerging proposals. We develop a four-layer sequential mobility choice framework comprising the environment, behavioral objective, stochastic choice mechanism, and observation process. Under stated assumptions, recursive logit, logit dynamic discrete choice, and maximum-entropy IRL use the same soft Bellman recursion linking future opportunities to current choice probabilities. Expected state-action visitation also satisfies conservation equations analogous to network flows. These mathematical connections do not make utility, reward, policy, occupancy, constraints, and observation error behaviorally interchangeable. Transportation evidence is strongest for network-scale planning, context-dependent reward learning, inference from incomplete trajectories, and activity-schedule generation, but remains limited for actual interventions and transfer across networks. We therefore propose a behaviorally disciplined hybrid architecture that keeps feasible actions, interpretable trade-offs, observation processes, and system feedback explicit while using machine learning for scalable computation, contextual representation, heterogeneity, and data integration.

We propose a family of control variate estimators for variance reduction in design-based survey sampling and causal inference, with and without interference. In these settings, inverse probability weighting (IPW) estimators are widely used, but may have large variance when sampling, treatment, or exposure probabilities are small. Building on the observation that several common estimators, including the Hajek estimator, the normalized estimator, the augmented inverse probability weighting (AIPW) estimator, and the targeted maximum likelihood estimator (TMLE), all correct the Horvitz-Thompson estimator by canceling part of its randomness, we provide a unified interpretation of these estimators as special cases of a general control variate estimator. We then construct optimal control variates that can reduce the finite sample variance compared to these common estimators. We parameterize the proposed control variates by their bases and characterize the optimal bases through a stochastic optimization formulation. In survey sampling and causal inference without interference, the optimal bases are characterized by leading eigenvectors of matrices that depend on both the design-based sampling structure and the model-based outcome uncertainty. In causal inference under network interference, the optimal bases solve a nonconvex quadratic optimization problem; we provide a $\frac{1}{2}$-approximate solution and an alternating local search heuristic. We apply the control variate estimators to the Swiss Environmental Panel survey data and the Chinese social network data, and conduct extensive simulations to show that the proposed control variate estimators can achieve substantial variance reduction.

A honeytoken, a decoy credential or record whose use reveals an intruder, produces defence intelligence only if an adversary sometimes treats it as genuine. The adversary can instead pay to fingerprint it. I model this problem as information design followed by receiver information acquisition. Without fingerprinting, a defender whose installed trap prevalence exceeds the adversary's participation boundary places the attacked pool exactly at that boundary: calibrated bait. That posterior also maximises the gross value of any fixed informative diagnostic with finitely many signal outcomes. For a perfect covert test, the resulting appearance design has a closed-form, prior-dependent characterisation: expensive testing preserves calibrated bait, intermediate-cost testing produces under-calibrated bait, and cheap testing yields pooling, protected segmentation or abstention. Free and costless adjustment of trap prevalence makes non-degenerate segmentation unnecessary, but a sufficiently large adjustment cost around a sufficiently high inherited composition can sustain it. Finally, along an exogenously declining fingerprinting-cost path, lower cost reduces trap activation without reducing genuine compromise within the under-calibrated regime. Evaluation against fresh attackers can therefore overstate the durability of reusable bait.

Neurofinance shows that financial valuation depends on evolving neural states, while temporal experience is itself state dependent. Yet intertemporal models typically treat time as exogenous and ask how delay affects valuation. This paper examines the converse question: can valuation-related neural dynamics generate subjective financial time? We develop a continuous-time model on a joint financial–neuro-evaluative state space in which subjective financial time is accumulated valuation along dynamically admissible histories. We characterise realised temporal-rate dispersion and show that, under matched financial dynamics, distinct neuro-evaluative trajectories can generate different subjective financial times. Financial equivalence therefore need not imply temporal equivalence. Valuation curvature further determines local path dependence. A proof-of-concept behavioural–fMRI analysis uses \(1{,}183\) observations, \(1{,}126\) valuation transitions, and \(798\) financially matched pairs. Using a medial-prefrontal valuation-state coordinate based on vmPFC and dmPFC responses, the strongest participant exhibits a positive association between neural-state separation and subsequent valuation divergence (\(\rho=0.253,\ p_{\mathrm{perm}}=0.001\)), robust to residual financial distance (\(\rho=0.251\)) and tighter matching (\(\rho=0.207\)). Participant effects are heterogeneous, although combined evidence gives \(p=0.0216\). The data support the neural state-separation premise rather than directly identifying subjective financial time. The framework establishes a theoretical and empirically grounded basis for endogenous subjective time in neurofinance. \keywords{Neurofinance $\cdot$ Subjective financial time $\cdot$ Neural valuation $\cdot$ Relative subjective value $\cdot$ Intertemporal choice $\cdot$ Temporal dispersion $\cdot$ Path dependence

Economics journals ration access to peer review with time and money. A July 2026 census of 500 RePEc-ranked economics journals accepting unsolicited submissions finds that 113 (22.6 per cent) charge a fee, with incidence falling from 68 per cent in the top 50 to 2 per cent in the bottom 50. A sequential-submission model shows that waiting pays part of the capacity-clearing price, leaving the fee as a residual. Because rejected manuscripts move down the ranking, a fee reduces system-wide referee demand only when authors stop, shorten their submission chains or switch to journals requiring fewer reports.

This study investigates the methodological and theoretical properties of session handover in applications that use large language models. A task may continue in a new session when the context reaches the model's input limit, when the application restarts, or when another agent is asked to finish the task. The application must then decide which information from the earlier session to pass on. We formulate handover as the transfer of a task-relative in-context learning (ICL) state and distinguish exact recovery of earlier material from preservation of the target distribution. Under an exogeneity condition, predictive equivalence characterizes the coarsest deterministic sufficient handover and gives a fixed-length bit requirement. The analysis isolates the effects of the memory constraint, the writer, and the continuation procedure, and quantifies the cost of writing before the realized downstream query is known. We propose a three-part record that stores decisions and constraints exactly, uses task-justified statistics for repeated evidence, and retains original observations whose effect is not preserved by those statistics. Gaussian linear regression gives an exact finite-dimensional handover and finite-bit perturbation bounds, while nonparametric regression gives upper and lower bounds that relate memory to squared prediction error. These results provide a theory and method for deciding what a handover must retain and how its memory requirement depends on the continuation task.

We propose a measure of lottery complexity that combines probabilistic dispersion with dissimilarity between outcomes. Taking degenerate lotteries, which yield a single outcome with certainty, as the simplest alternatives, we measure complexity by proximity to this class. Specifically, for each degenerate lottery, we compute the expected distance between its certain outcome and the lottery's outcomes, and take the minimum across all degenerate lotteries. We provide an axiomatic characterization establishing uniqueness up to positive rescaling, study how complexity changes under outcome compression, and characterize maximally complex lotteries. Finally, we introduce Complexity-Adjusted Expected Utility preferences and characterize adherence to stochastic dominance and strong risk aversion.

AI is increasingly being used to help with AI R&D. Under certain conditions this feedback loop might be able to produce an intelligence explosion, with rapidly escalating AI capabilities. I explore the mathematics of the most explosive possibilities, with an eye to understanding what drives the dynamics. I show that singular growth (towards a vertical asymptote) is harder to achieve than would be expected from recent economics-inspired modelling, and that there is an important but neglected class of growth rates that are faster than exponential but don't lead to a vertical asymptote. I draw out the generation time (the time to go around the feedback loop) as a neglected parameter that plays a pivotal role in determining the behaviour of any intelligence explosion — one cannot have singular growth unless the generation time rapidly approaches zero.

This paper studies how information censoring enables a myopic cutoff rule in Bayesian sequential search. Under full information, Bayesian learning generally destroys the monotonicity of continuation values, preventing simple cutoff rules. We show that one-sided censoring restores monotonicity by limiting posterior fluctuations, thereby making a myopic cutoff rule optimal. By decomposing the intertemporal change in the marginal value of search into a fallback-value effect and a learning effect, we derive necessary and sufficient conditions for monotonicity under lower censoring and characterize the optimal cutoff rule. In contrast, under full revelation, monotonicity requires highly restrictive conditions. We further show that expected monotonicity (i.e., the supermartingale property) is characterized by the same conditions under both lower censoring and full revelation, owing to Bayes plausibility and the affine structure of the problem. Thus, censoring restores monotonicity not by altering expected learning, but by reducing posterior volatility. Finally, we apply our framework to job search, consumer price search, and product experimentation.

This paper employs a Threshold Bayesian Vector Autoregression (TBVAR) to estimate the regime-dependent macroeconomic effects of capital regulation in Hungary. Using the Factor-based Index of Systemic Stress (FISS) as the threshold variable, the model identifies normal and stress regimes consistent with the occasionally binding constraints literature. The TBVAR offers a practical multivariate alternative to Growth-at-Risk for data-constrained economies. Generalised impulse responses reveal a pronounced asymmetry: releasing regulatory capital during stress raises GDP growth at the peak, with effects persisting for roughly twenty months, while the cost of accumulating capital in the normal regime is economically negligible. These findings are robust to alternative Cholesky orderings, sample periods, and credit variable definitions, providing direct empirical support for the countercyclical operation of the capital buffer.

We formulate the voter's strategic information acquisition to control the future self's action as a Bayesian persuasion problem. Our main result shows that acquiring information that is irrelevant to the voter's objective can be a worst-case optimal solution: it can reduce the possibility that the future self is swayed by additional information whose content is ambiguous to the current voter.

Academic freedom has come under growing strain worldwide. To study whether and how academics respond to political pressure, we exploit a natural experiment: the publication in early 2025 of a "blacklist" of words flagged by the U.S. government. We find that the release of this list led to a sharp reduction in the use of these flagged words among economists at universities that rely heavily on federal funding, relative to scholars from institutions that are less dependent on federal funding or based in the UK. The drop is driven by content related to gender, race, and environment. We show that changes are not simply semantic but reflect actual paper content and that neither the individual funding status nor time-invariant author characteristics are driving the effects. We also document interesting heterogeneous effects by department quality and author gender and ethnicity. Our findings are consistent with the idea that scholars respond strongly to political pressure.

Ownership matters for the media's watchdog role. We map the ownership networks behind thousands of online news outlets in the U.S., Canada, and Europe. The networks reveal who is ultimately responsible for the news: for over half of the outlets, a single entity. The rest sit behind multi-layered structures, making responsibility hard to trace. Market concentration, measured comparably across countries, is largely low to moderate. Looking at content, we find that co-owned outlets report more similarly, even within fixed outlet pairs, as ownership changes – not least in the U.S., where reader demand is often thought dominant.

We develop an axiomatic model of robust Bayesian persuasion where the sender cannot fully control the information available to the receiver. After selecting an information structure, the sender expects that more informative structures might be implemented. We model this by allowing the sender to assess each information structure under worst-case information leakage, represented by a set of more informative structures. The model encompasses a wide range of examples of information leakage, which we also explore.

We develop a model of interbank networks with random liquidity shocks. Networks of dilutable debt—e.g., long-term, unsecured—facilitate efficient liquidity transfers: Shocked banks pledge interbank claims as collateral for new senior debt, diluting existing debt. Unlike with non-dilutable debt, indebtedness and connectedness are sources of stability, not fragility. Dilution is thus a “backdoor bail-in” that reallocates losses absent a resolution authority, trigger security, or ex post renegotiation. We uncover a class of networks, “exponential networks,” that implement optimal contingent transfers via plain debt. Yet exponential networks are not pairwise stable, whereas some core–periphery networks are, rationalizing observed interbank structures and their under-insurance against crises.

We develop a method for estimating and testing a single block of a macroeconomic model with heterogeneous agents, without placing assumptions on the structure of the rest of the economy. In a large class of models, individual agents' decisions depend on the macroeconomy only through their expectations of the evolution of a finite-dimensional vector of "sufficient statistics" (e.g., asset returns or aggregate earnings). Our estimator selects the structural parameters that provide the best model-consistent fit between empirical impulse responses with respect to identified macro shocks of (a) cross-sectional moments of agent choices (e.g., moments of consumption) and (b) the vector of sufficient statistics. In a simulation illustration, we estimate a two-asset heterogeneous household model block without restricting production, firm investment, financial intermediation, monetary policy, trade, etc.

While technological innovations typically increase wage inequality by favoring skilled workers, we show that instant payment systems instead reduce it. We study the labor market effects of instant payment systems in the context of Brazil's Pix rollout. Using matched employer-employee data, we implement a triple-difference design that exploits pre-Pix mobile penetration across municipalities, the differential benefits of Pix for small versus large establishments, and the timing of Pix. We find that wages in small establishments rise significantly relative to large establishments after Pix. These gains are concentrated in cash-intensive sectors such as retail and services, with no effects in wholesale or manufacturing. Crucially, wage inequality declines, driven by wage gains in the lower half of the distribution, with no effect at the top. Our evidence points to increased small-firm labor demand, consistent with lower payment frictions. These effects are amplified where low-skill labor is scarce. A calibrated monopsony model implies that uniform Pix adoption would reduce both the within- and between-municipality components of wage dispersion, amplifying the aggregate inequality reduction.

Limited dependent variable models are central to empirical economics, but likelihood-based inference is infeasible when likelihoods involve high-dimensional integration over latent variables. This paper proposes Stochastically Estimated Gradient Ascent (SEGA), a scalable estimation approach for limited dependent variable models. Using Fisher's identity, SEGA replaces the intractable likelihood score with an unbiased augmented data score evaluated at a single conditional draw of the latent variables, and embeds this score in a stochastic gradient ascent algorithm. With sufficiently many iterations, we show that SEGA is asymptotically equivalent to the infeasible maximum likelihood estimator. A variance estimator based on Fisher's and Louis' identities is proposed that allows inference to proceed in the usual manner. Applications to brand choice and household demand demonstrate the usefulness of SEGA for conducting inference in large-scale discrete choice and censored demand models.

Article 14 of the EU AI Act requires that a high-risk system be overseen by natural persons who understand its limits, remain alert to automation bias, and can disregard or override its output. That capability is invisible in the output and decays precisely when the system is good. A provider can certify it only by an outcome-contingent liability commitment, and how large a commitment courts will enforce is itself open. After Cavendish the enforceable multiple of actual loss is measured against the legitimate interest the clause protects; where that interest is preserved oversight capability, it exists only if the market separates, which requires a sufficiently permissive doctrine. We model the enforceable ceiling as a fixed point of an expectations map on a complete lattice. Existence follows from Knaster-Tarski. Because no case separates at compensation, compensation is always an equilibrium; once the doctrinal uplift clears a threshold set by the case population, a permissive equilibrium and a watershed appear, and the map inherits the provider's solvency cap, so an insurance withdrawal deep enough and long enough can destroy the permissive equilibrium, which returning cover does not restore. Perturbing the adjustment yields a closed-form long-run tipping point, exact recovery times, and the noise levels at which irreversibility fails. The welfare cost of the resulting trap is capped at the drafting cost of primary-obligation substitutes. The order-theoretic core is machine-checked in Lean 4 with an axiom-free report. Both AI-side parameters, shared-base-model intensity and provability, are matched to instruments available in 2026.

We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $\rho_n^*$ of an estimation problem with $2$ unknown parameters. We leverage this reduction to establish a second-order risk expansion $\rho_n^* = n^{-1} - Cn^{-4/3} + o_n(n^{-4/3})$ for an explicit constant $C$ related to the Airy function. The minimax risk is attained by Bernoulli randomization with a nonlinear shrinkage estimator. Our results show that standard procedures such as complete randomization with difference in means are only minimax optimal up to first order in $n.$ We derive further results on admissibility of these procedures and discuss the practical implications of our results.

Pawn loans offer borrowers a substantial degree of repayment flexibility in exchange for a harsh penalty in case of default: forfeit of collateral worth more than the loan amount along with any payments made toward recovery. Using a large RCT conducted in Mexico City, we document key stylized facts about pawn lending and explore the merits of replacing flexibility with structured repayment contracts in this important but understudied form of credit. Our experimental design includes a mandatory frequent-payments arm, a (status quo) flexible payments arm, and a choice between the two. This design point-identifies not only the average treatment effect, but also the effects of treatment on the treated and the untreated along with the average selection on gains, allowing a rigorous study of mandates versus choice. Although the average treatment effect of assigning borrowers to structured payments is a 19% decrease in their financial cost and a 17.5% decrease in the probability of default, only 11% of borrowers choose structured repayment contracts voluntarily. We show that structured repayment reduces financial costs for nearly all borrowers, including those who would not freely choose it, and find no evidence of selection on gains in cost savings.

When comparison units may also respond to treatment, panel comparisons reflect both the treatment effect and spillovers. If the interference pattern is unknown, observed outcomes alone do not separate the two. I characterize what can nevertheless be learned from panel outcomes under general restrictions, without requiring an exposure mapping or prior classification of affected donors. The framework scales validity bounds for every convex donor weight by its fit before treatment and combines these bounds with prespecified restrictions tailored to the application. The validity bounds constrain the treatment effect relative to spillovers, while the additional restrictions determine its possible values. Together these restrictions yield a sharp identified set. When the additional restrictions have a finite linear representation, checking whether a proposed treatment effect is compatible with the model reduces exactly to asking whether a finite linear system has a solution. Bootstrap calibration tests this condition. Inverting these tests uniformly controls, in large samples, the probability of falsely excluding each compatible value. In an application to the Legal Arizona Workers Act, the resulting 95 percent inversion sets contain effects of both signs across all reported specifications, leaving the sign of the treatment effect unresolved.

Many questions across the sciences take the same form: several coupled series are observed together, and the analyst wants to know not merely that they move together but which one moves first, and how strongly. This paper sets out a complete method built on one organising idea: the direction of a coupled system is exactly the part of its behaviour that changes when the record is played backwards. Tools built on contemporaneous covariance alone (correlation matrices, distance measures, spanning trees, undirected centralities, principal components) carry no information about direction: a reversible system and a circulating one can share identical covariance at every sampling of the same point-in-time record. Formally, direction is a circulation matrix carried by the lagged covariance. Its vanishing is exactly statistical time reversibility for linear systems, feature maps carry the characterisation to nonlinear ones, and under the Gaussian benchmark its magnitude is an entropy-production functional of the identified circulation, the quadratic component of the divergence per unit time between the forward and reversed records. Around this estimand we build a cross-fitted estimator removing first-order bias, delete-block jackknife standard errors, and a randomisation test exact under its stated block null, with a familywise correction and a nonlinear extension. A sampling theory says when the arrow is measurable at all, and a design layer separates transmission from the ordering of clocks. A laboratory of four systems with known answers compares the method with correlation networks, Granger causality, transfer entropy, and connectedness indices, reporting the failures of each when read as a measure of direction, including our own. Complete algorithms and worked examples in two languages make the paper the base reference for a series of applications.

In organized crime, membership can adjust around enforcement while division norms remain in place. We model this in three stages: division norms fix how every possible coalition divides its proceeds; the authority then attaches deterrence capacity to named members before knowing which coalition will form; membership adjusts last, among those who remain undeterred. Under population-monotone division, where no incumbent loses rent as a coalition expands, the minimum budget needed to block every coalition is the cost of a shortest dismantling path, removing one member at a time. A high-rent hub can be the best first target because her removal lowers the rents of those who remain. Holding coalition proceeds fixed, the Shapley value applied to every coalition uniquely maximizes this budget among population-monotone norms whenever it is itself population monotone. It does so by making every removal order equally costly. Allowing part of the proceeds to be reallocated after capacities are observed weakly raises the budget for any fixed population-monotone norm, although exact Shapley optimality can fail even when the reallocated share is arbitrarily small.

I consider an AR($p$) process that is observed every $q$ periods, either as a snapshot (stock variable) or as a sum over the sampling interval (flow variable). I first characterize the resulting ARMA process followed by observables. Under fairly mild assumptions, I then derive the identified set for general lag lengths $p \in \mathbb{N}$ and sampling frequencies $q \in \mathbb{N}$, I bound its cardinality, and I provide an algorithm to compute all candidate points and determine their membership in the identified set. My exact but implicit characterization supports the following conjecture that I prove in some settings and verify numerically more broadly: (i) the error term-variance is point-identified, (ii) under temporal aggregation, the autoregressive parameters are point-identified, and (iii) under discrete sampling they are point-identified for odd $q$ and identified up to alternating sign for even $q$. My analysis supplements existing inference results that show consistency and asymptotic Normality of the Gaussian Maximum Likelihood estimator conditional on point-identification. Holding the number of observations fixed, I show that its precision does not necessarily decrease with $q$.

We consider the classical additive measurement-error model $X=Y+Z$, where the latent random variable $Y$ has unknown distribution $F_Y$ and the error $Z$ has a known distribution. We develop direct estimators for three functionals of $F_Y$: (i) $F_Y(x)$ at continuity points; (ii) interval probabilities $F_Y(y)-F_Y(x)$ when $x<y$ are continuity points; and (iii) the size of a jump at a prespecified discontinuity. We derive non-asymptotic bias and variance bounds, and establish asymptotic unbiasedness and consistency. Unlike previous work, we do not require $F_Y$ to admit a density, have a mixture representation, or satisfy global Sobolev smoothness assumptions. The framework accommodates arbitrary latent distributions, including those with both discrete and continuous components, and distributions with multiple jumps. These results rely on a link between Fourier inversion theorems and the algebraic structure of a class of estimators proposed in Mynbaev, Martins-Filho and Henderson (2022). A simulation study evaluates feasible tuning procedures and, where available, compares the finite-sample performance of the proposed estimators with existing methods.