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arxiv: 2604.22740 · v1 · submitted 2026-04-24 · 📡 eess.SP · cs.IT· math.IT

Minimax Optimal Procedures for Joint Detection and Estimation

Dominik Reinhard , Michael Fau{\ss} , Abdelhak M. Zoubir This is my paper

Pith reviewed 2026-05-08 10:10 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT
keywords minimax proceduresjoint detection and estimationdistributional uncertaintyf-similarityleast favorable distributionsBayesian formulationNeyman-Pearson formulationband-type uncertainty
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The pith

The optimal policy for joint detection and estimation under distributional uncertainty is obtained by maximizing an induced f-similarity to identify the least favorable distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes minimax optimal procedures for jointly testing a pair of composite hypotheses and estimating a random parameter when the data distribution is subject to uncertainty. It develops both a Bayesian formulation and a Neyman-Pearson-like formulation. A reader would care because the approach yields robust procedures that guarantee performance against the worst-case distributions inside given uncertainty sets. The central step is proving that any optimal policy induces an f-similarity whose maximization locates those least favorable distributions. The theory is made concrete for band-type uncertainty models by adapting existing algorithms to improve convergence while preserving stability.

Core claim

The optimal policy for the joint problem induces an f-similarity that must be maximized to identify the least favorable distributions inside the uncertainty sets. This holds for both the Bayesian and Neyman-Pearson-like formulations. For band-type uncertainty models the resulting minimax procedures are obtained by modifying existing algorithms to increase convergence speed while maintaining numerical stability, with the approach illustrated by numerical results.

What carries the argument

The f-similarity induced by the optimal policy, which is maximized over the uncertainty sets to locate the least favorable distributions.

If this is right

  • Minimax procedures can be designed for both Bayesian and Neyman-Pearson formulations by maximizing the same f-similarity.
  • Band-type uncertainty models admit practical computation after the algorithms are modified for faster convergence.
  • Numerical evaluation confirms that the resulting procedures attain the guaranteed performance under the modeled uncertainties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same f-similarity construction could be tried with other uncertainty models once their least-favorable pairs are characterized.
  • The joint formulation may simplify separate detection-then-estimation pipelines when the uncertainty sets are identical for both tasks.
  • A direct numerical check on small discrete uncertainty sets would verify whether the maximizer of the f-similarity indeed equals the minimax value.

Load-bearing premise

Least-favorable distributions exist inside the given uncertainty sets and maximizing the induced f-similarity produces a well-defined and computable optimal policy.

What would settle it

An explicit uncertainty set and loss function for which the distributions that maximize the f-similarity fail to achieve the minimax risk of the joint detection-estimation problem.

Figures

Figures reproduced from arXiv: 2604.22740 by Abdelhak M. Zoubir, Dominik Reinhard, Michael Fau{\ss}.

Figure 1
Figure 1. Figure 1: Illustrative example of a density band model. The set of feasible distributions P θ depends on the location parameter θ. where J(u, v) is some cost function. In the equation above, u represents the statistical procedure, e.g., it can be a decision rule in the context of hypothesis testing or an estimator in the context of parameter estimation. The variable v on the other hand represents the quantity which … view at source ↗
Figure 2
Figure 2. Figure 2: Least-favorable densities and resulting decision rule under the Bayesian formulation. A. Bayesian Formulation For the Bayesian setup, we manually set the detection costs to λ0 = 0.75, λ1 = 1 and the estimation cost to µ1 = 1.1. Additionally, we chose c = 0.8 and c¯ = 1.2 for the neighborhood. In view at source ↗
Figure 3
Figure 3. Figure 3: Least-favorable densities and resulting decision rules under the NP-like formulation. TABLE II: NP-like formulation: results of Monte Carlo simulation. error probabilities type-I type-II MSE J NP opt. with p 0.050 0.305 0.455 0.378 opt. with qD 0.074 0.357 0.477 0.419 opt. with qD,0/qE 0.074 0.291 0.617 0.459 minimax with p 0.033 0.257 0.511 0.422 minimax with qD 0.050 0.305 0.540 0.470 minimax with qD,0/q… view at source ↗
read the original abstract

We investigate the problem of jointly testing a pair of composite hypotheses and, depending on the test result, estimating a random parameter under distributional uncertainties. Specifically, it is assumed that the distribution of the data given the parameter of interest, is subject to uncertainty. Both, a Bayesian formulation and a Neyman-Pearson-like formulation, are considered. It is shown that the optimal policy induces an $f$-similarity that must be maximized to identify the least favorable distributions. Besides the general results, the implementation is investigated using a band-type uncertainty model. For designing the minimax procedures, existing algorithms are modified to increase convergence speed while maintaining numerical stability. The proposed theory is supplemented by numerical results for both formulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a minimax framework for the joint problem of testing composite hypotheses and estimating a random parameter when the conditional distributions are subject to uncertainty. For both Bayesian and Neyman-Pearson formulations, it shows that the optimal policy is characterized by an f-similarity that must be maximized to identify the least favorable distributions. General results are supplemented by explicit constructions under band-type uncertainty sets, modified algorithms that accelerate convergence while preserving stability, and numerical experiments.

Significance. If the central characterization holds, the work provides a unified theoretical link between joint detection-estimation and f-divergence theory, extending classical minimax results to this composite setting. The explicit band-model constructions and the stable, faster algorithms constitute concrete, implementable contributions with direct relevance to signal-processing applications under uncertainty. Numerical validation supports the claims and demonstrates practical utility.

minor comments (3)
  1. [§2] §2: The definition of f-similarity is introduced via an integral expression but the subsequent use in the optimality condition (around Eq. (12)) would benefit from an explicit statement that the maximizing pair (P0*,P1*) is attained inside the given uncertainty sets.
  2. [§5.1] §5.1, Algorithm 1: The modified step-size rule is stated without a supporting lemma showing that it preserves the monotonicity property of the original iteration; a short proof or reference would clarify why stability is retained.
  3. [Numerical Results] Figure 3: The plotted curves for the Neyman-Pearson formulation lack error bars or indication of the number of Monte-Carlo trials, making it difficult to judge whether the reported gains over the nominal policy are statistically significant.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful review and the positive overall assessment of the manuscript. The recognition of the unified theoretical link to f-divergence theory, the concrete contributions from the band-model constructions, and the stable accelerated algorithms is appreciated. The recommendation for minor revision is noted, and we will incorporate any editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivation shows that the optimal joint detection-estimation policy induces an f-similarity whose maximization identifies the least favorable distributions under the given uncertainty sets. This follows directly from minimax formulations (Bayesian and Neyman-Pearson) without reducing the claimed optimum to a parameter fitted from the target data or to a self-citation whose content is itself unverified. The band-type uncertainty model supplies an explicit construction, and modified algorithms are presented for computation; these steps are independent of the result they support. No load-bearing step equates the output to its inputs by definition, and the framework rests on standard f-divergence and minimax theory rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of least-favorable distributions inside the uncertainty sets and on the validity of the f-similarity characterization; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Least-favorable distributions exist and can be identified by maximizing the f-similarity over the uncertainty sets
    Invoked to guarantee that the optimal policy is attained inside the model class.

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