Minimax Estimation Research Articles

Informativeness orders over ambiguous experiments

We generalize Blackwell's informativeness order to ambiguous experiments. A decision maker might view a statistical experiment as ambiguous if she faces uncertainty about the data generating process for its signals. Formally, an ambiguous experiment is modeled as a mapping from an auxiliary state space to the set of unambiguous experiments. Each auxiliary state corresponds to a possible data generating process. We show that one ambiguous experiment is preferred to another by every decision maker in every decision problem if and only if they are related by a condition called prior-by-prior dominance, which states that for any first-order belief the decision maker entertains on the auxiliary state space, the expected experiment resulting from this belief for the first experiment is Blackwell more informative than that of the second. This equivalence is robust for any class of monotone ambiguity preferences that nests expected utility. We obtain another informativeness order when we restrict attention to decision makers who apply the maxmin criterion to evaluate ambiguous experiments and connect this informativeness order to comparisons of sets of Blackwell experiments.

Minimax estimation for the deconvolution model in random design with long-memory dependent errors

. Deconvolution model is investigated when the design points t i are random and follow a bounded density h. We propose a wavelet hard-thresholding estimator, and show that it achieves asymptotically near-optimal convergence rates for the L 2-risk when the response function f belongs to Besov space. We consider the problem under two noise structure; (1) i . i . d . Gaussian errors and (2) long-memory Gaussian errors. At a first stage, we assume h is known. For the i . i . d . Gaussian errors case, our convergence rates are similar to those found in the literature under regular design. For the long memory case, our convergence rates are new. However, at each elbow, the convergence rates will depend on the long-memory parameter only if it is small enough, otherwise, the rates will be identical to those we obtained under i . i . d . errors. At a second stage, we assume h is unknown, construct the usual hard-thresholding estimator, and derive the upper bounds for its L 2-risk under the new settings. The rates seem to depend on the smoothness parameter, γ, of h when γ is relatively small.

Approximating the shortest path problem with scenarios

This paper discusses the shortest path problem in a general directed graph with n nodes and K cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to approximate within Ω(log1−ϵ⁡K) for any ϵ>0 unless NP⊆DTIME(npolylogn) even for arc series-parallel graphs and within Ω(log⁡n/log⁡log⁡n) unless NP⊆ZPTIME(nlog⁡log⁡n) for acyclic graphs. The best approximation algorithm for the min-max shortest path problem in general graphs, known to date, has an approximation ratio of K. In this paper, an O˜(n) flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed. It is also shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation.

Two-stage robust multimodal hub network design under budgeted demand uncertainty: A Benders decomposition approach and a case study

The widening use of hub networks in urban agglomeration freight systems requires several actual extensions in conventional hub network design problems. For this purpose, we introduce a two-stage robust multimodal hub network design problem for the urban agglomeration freight system by considering incomplete hub network topology, multiple transportation modes, travel time limit and discuss the uncertainty in the constructed network from the demand point of view. Particularly, we model the demand uncertainty for the considered problem in two different ways. The basic model supposes that interval-budgeted uncertainty set is adopted to characterize uncertain demand, while the expanded model additionally considers possible states of the uncertain demand and weights summation of performances over multiple uncertainty sets, namely state-wise budgeted uncertainty set. By using a min–max criterion, we develop the path-based mixed-integer programming formulations for the proposed problem, which can significantly decrease the number of required integer variables and constraints. To handle large-sized problems, we propose an improved Benders decomposition algorithm, in which the master problem is implemented in a branch-and-bound framework and the subproblem is optimality solved by a customized two-step strategy. In addition to evaluating on the standard CAB, TR and AP datasets, we conduct a real-world case study of the Beijing–Tianjin–Hebei urban agglomeration freight system to explore the effect of incorporating uncertainty and showcase the superior performance of the proposed methods.

An Integrated SIMUS–Game Theory Approach for Sustainable Decision Making—An Application for Route and Transport Operator Selection

The choice of management strategy for companies operating in different sectors of the economy is of great importance for their sustainable development. In many cases, companies are in competition within the scope of the same activities, meaning that the profit of one company is at the expense of the other. The choice of strategies for each of the firms in this case can be optimized using game theory for a non-cooperative game case where the two players have antagonistic interests. The aim of this research is to develop a methodology which, in non-cooperative games, accounts for the benefits of different criteria for each of the strategies of the two participants. In this research a new integrated sequential interactive model for urban systems (SIMUS)–game theory technique for decision making in the case of non-cooperative games is proposed. The methodology includes three steps. The first step consists of a determination of the strategies of both players and the selection of criteria for their assessment. In the second step the SIMUS method for multi-criteria analysis is applied to identify the benefits of the strategies for both players according to the criteria. The model formation in game theory is drawn up in the third step. The payoff matrix of the game is formed based on the benefits obtained from the SIMUS method. The strategies of both players are solved by dual linear programming. Finally, to verify the results of the new approach we apply four criteria to make a decision—Laplace’s criterion, the minimax and maximin criteria, Savage’s criterion and Hurwitz’s criterion. The new integrated SIMUS–game theory approach is applied to a real example in the transport sector. The Bulgarian transport network is investigated regarding route and transport type selection for a carriage of containers between a starting point, Sofia, and a destination, Varna, in the case of competition between railway and road operators. Two strategies for a railway operator and three strategies for a road operator are examined. The benefits of the strategies for both operators are determined using the SIMUS method, based on seven criteria representing environmental, technological, infrastructural, economic, security and safety factors. The optimal strategies for both operators are determined using the game model and dual linear programming. It is discovered that the railway operator will apply their first strategy and that the road operator will also apply their first strategy. Both players will obtain a profit if they implement their optimal strategies. The new integrated SIMUS–game theory approach can be used in different areas of research, when the strategies for both players in non-cooperatives games need to be established.

K-means-ten clustering algorithm based on min-max criterion and region division

Aiming at the problem of unstable clustering results and low solution accuracy of -means- + algorithm, an I--means-+ clustering algorithm based on min-max criterion and region division is proposed. Firstly, the min-max criterion is proposed to calculate the distance from each data point to the nearest center, and the data point with the largest distance is preferentially selected as the new cluster center to avoid the situation where multiple initial centers are clustered in the same cluster; secondly, the data points in the split cluster are divided into different regions, and a data point is selected in each region as the candidate center to increase the diversity of the candidate centers; finally, for the clusters that fail to pair, the new split cluster is reselected by gain to pair with the original deleted cluster again to improve the pairing success rate and further reduce the objective function value. Experimental results show that compared with the I means-+ algorithm, the proposed algorithm has an average improvement in solution accuracy −k −and a more stable clustering result kunder the premise of basically equivalent operation efficiency compared with 6.47%-means and kmeans+ algorithms, the proposed algorithm has higher solution accuracy.

Robust fusion filter for networked uncertain descriptor systems with colored noise and cyber-attacks

The robust fusion filtering problem of multi-sensor networked uncertain descriptor systems (NUDSs) with colored noise, uncertain noise variances and cyber-attacks is investigated. During data transmission in unreliable communication networks, the data can be maliciously attacked by attackers. In other words, the local filters (LFs) may receive false data or may not receive data because of the cyber-attacks. By adopting the singular value decomposition (SVD) method, the original NUDSs can be converted into two reduced-order subsystems with uncertain correlated fictitious white noises, and the cyber-attacks are transformed into the fictitious noises. Cross-covariance matrices between local filtering errors are derived. The robust LFs are obtained according to the minimax robust estimation principle. Under the linear unbiased minimum variance criterion, three weighted fusion algorithms are applied to fuse the LFs. For all allowable uncertainties of noise variances and cyber-attacks, the minimal upper bounds of covariance matrices of the local and distributed fusion filters are guaranteed. The proof of their robustness is established through the minimax estimation principle and Lyapunov equation method. Finally, the correctness and effectiveness of the proposed algorithms are verified by a circuit system example.

Space-filling designs on Riemannian manifolds

This paper proposes a new approach to generating space-filling designs over Riemannian manifolds by using a Hilbert curve. Different from ordinary Euclidean spaces, a novel transformation is constructed to link the uniform distribution over a Riemannian manifold and that over its parameter space. Using this transformation, the uniformity of the design points in the sense of Riemannian volume measure can be guaranteed by the intrinsic measure preserving property of the Hilbert curve. It is proved that these generated designs are not only asymptotically optimal under minimax and maximin distance criteria, but also perform well in minimizing the Wasserstein distance from the target distribution and controlling the estimation error in numerical integration. Furthermore, an efficient algorithm is developed for numerical generation of these space-filling designs. The advantages of the new approach are verified through numerical simulations.

Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation

Multi-product formulas (MPFs) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs [A. M. Childs , ] to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling, and hardware noise. We call this method the minimax MPF and we provide a rigorous bound on its error. Published by the American Physical Society 2024

Adaptive minimax estimation of service time distribution in the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from departure data

This article deals with the problem of estimating the service time distribution of the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure.

Efficient Aperture Fill Time Correction for Wideband Sparse Array Using Improved Variable Fractional Delay Filters.

To solve the problem of aperture fill time (AFT) for wideband sparse arrays, variable fractional delay (VFD) FIR filters are applied to eliminate linear coupling between spatial and time domains. However, the large dimensions of the filter coefficient matrix result in high system complexity. To alleviate the computational burden of solving VFD filter coefficients, a novel multi-regultion minimax (MRMM) model utilizing the sparse representation technique has been presented. The error function is constrained by the introduction of L2-norm and L1-norm regularizations within the minimax criterion. The L2-norm effectively resolves the problems of overfitting and non-unique solutions that arise in the sparse optimization of traditional minimax (MM) models. Meanwhile, the use of multiple L1-norms enables the optimal design of the smallest sub-filter number and order of the VFD filter. To solve the established nonconvex model, an improved sequential-alternating direction method of multipliers (S-ADMM) algorithm for filter coefficients is proposed, which utilizes sequential alternation to iteratively update multiple soft-thresholding problems. The experimental results show that the optimized VFD filter reduces system complexity significantly and corrects AFT effectively in a wideband sparse array.

Bayesian and minimax estimators of loss

Bayesian and minimax estimators of loss

Bayes minimax estimator of the mean vector in an elliptically contoured distribution

This paper investigates the Bayes estimator of the mean of an elliptically contoured distribution with unknown scale parameter under the quadratic loss. The Laplace transform and inverse Laplace transform of density facilitate us to obtain the expression of Bayes estimator. Then we prove the minimaxity of the Bayes estimator under certain conditions.

Stochastic resonance noise modified decision solution for binary hypothesis-testing under minimax criterion

In this paper, on the premise that the prior probability is unknown, a noise enhanced binary hypothesis-testing is investigated under the Minimax criterion for a general nonlinear system. Firstly, for lowering the decision risk, an additive noise is intentionally injected to the input and a decision is made under Minimax criterion based on the noise modified output. Then an optimization problem for minimizing the maximum of Bayesian conditional risk under an equality constraint is formulated via analyzing the relationship between the additive noise and the optimal noise modified Minimax decision rule. Furthermore, lemma and theorem are proposed to prove that the optimal noise is a constant vector, which simplifies the optimization problem greatly. An algorithm is also developed to search the optimal constant and the key parameter of detector, and further to determine the decision rule and the Bayes risk. Finally, simulation results about the original (in the absence of additive noise) and the noise-modified optimal decision solutions under Minimax criterion for a sine transform system are provided to illustrate the theoretical results.

Low-rank tensor regression for selection of grouped variables

Low-rank tensor regression (LRTR) problems are widely studied in statistics and machine learning, in which the regressors are generally grouped by clustering strongly correlated variables or variables corresponding to different levels of the same predictive factor in many practical applications. By virtue of the idea of group selection in the classical linear regression framework, we propose an LRTR method for adaptive selection of grouped variables in this article, which is formulated as a group SLOPE penalized low-rank, orthogonally decomposable tensor optimization problem. Moreover, we introduce the notion of tensor group false discovery rate (TgFDR) to measure the group selection performance. The proposed regression method provably controls TgFDR and achieves the asymptotically minimax estimate under the assumption that variable groups are orthogonal to each other. Finally, an alternating minimization algorithm is developed for efficient problem resolution. We demonstrate the performance of our proposed method in group selection and low-rank estimation through simulation studies and real dataset analysis.

Robust and efficient factorial designs under baseline parametrization

Factorial designs, traditionally based on orthogonal parametrization, are widely used to study the effects of factors on responses. While the baseline parametrization has more applicability in many cases compared to the orthogonal parametrization, constructing baseline parametrization design is still challenging. This article presents a method based on approximation theory to search for highly efficient baseline parametrization designs, enhancing the accuracy of factor effect estimations. In addition, this article employs the D-optimal minimax criterion to ensure the robustness of the design. The examples and simulation show that the constructed designs have high efficiency.

Minimax estimation of functional principal components from noisy discretized functional data

AbstractFunctional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.

A New Class of Bayes Minimax Estimators of the Mean Matrix of a Matrix Variate Normal Distribution

Bayes minimax estimation is important because it provides a robust approach to statistical estimation that considers the worst-case scenario while incorporating prior knowledge. In this paper, Bayes minimax estimation of the mean matrix of a matrix variate normal distribution is considered under the quadratic loss function. A large class of (proper and generalized) Bayes minimax estimators of the mean matrix is presented. Two examples are given to illustrate the class of estimators, showing, among other things, that the class includes classes of estimators presented by Tsukuma.

A novel fully distributed discrete-time min–max consensus seeking algorithm with high convergence speed

Abstract In this paper, we propose a novel discrete-time consensus algorithm based on the new min–max criterion for the discrete-time second-order linear agents. To boost the convergence speed, the proposed algorithm allocates three different operating modes to the network agents. Unlike most of the existing consensus algorithms in which the links of the communication graph have specific weights and these weights are required to have positive lower bounds to achieve asymptotic convergence, the communication graph in this paper has unspecific link weights. Accordingly, because of data transmission errors and faults, the weights of the communication links can even go from a positive to a negative value. Thus, the proposed algorithm can bring the agents to consensus under more general conditions. To demonstrate the efficacy of the proposed algorithm, numerical simulations are performed, and their results are compared with those of the minimum consensus algorithm, which shows a significantly faster convergence speed.

Minimax estimation of low-rank quantum states and their linear functionals

Minimax estimation of low-rank quantum states and their linear functionals