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Journal articles on the topic 'Minimax rate'

Author: Grafiati

Published: 26 October 2024

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1

Bose, N. K., and C. Charoenlarpnopparut. "Minimax controller design using rate feedback." Circuits, Systems, and Signal Processing 18, no. 1 (January 1999): 17–25. http://dx.doi.org/10.1007/bf01206542.

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2

Bu, Yuheng, Shaofeng Zou, Yingbin Liang, and Venugopal V. Veeravalli. "Estimation of KL Divergence: Optimal Minimax Rate." IEEE Transactions on Information Theory 64, no. 4 (April 2018): 2648–74. http://dx.doi.org/10.1109/tit.2018.2805844.

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3

Bose, N. K., and C. Charoenlarpnopparut. "Minimax controller using rate feedback: Latest results." IFAC Proceedings Volumes 32, no. 2 (July 1999): 3714–19. http://dx.doi.org/10.1016/s1474-6670(17)56635-5.

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4

Gao, Wei. "Minimax Learning Rate for Multi-dividing Ontology Algorithm." Journal of Information and Computational Science 11, no. 6 (April 10, 2014): 1853–60. http://dx.doi.org/10.12733/jics20103216.

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5

Yuhong Yang. "Minimax rate adaptive estimation over continuous hyper-parameters." IEEE Transactions on Information Theory 47, no. 5 (July 2001): 2081–85. http://dx.doi.org/10.1109/18.930947.

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6

Carpentier, A., O. Collier, L. Comminges, A. B. Tsybakov, and Yu Wang. "Minimax Rate of Testing in Sparse Linear Regression." Automation and Remote Control 80, no. 10 (October 2019): 1817–34. http://dx.doi.org/10.1134/s0005117919100047.

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7

Zhao, Puning, and Lifeng Lai. "Minimax Rate Optimal Adaptive Nearest Neighbor Classification and Regression." IEEE Transactions on Information Theory 67, no. 5 (May 2021): 3155–82. http://dx.doi.org/10.1109/tit.2021.3062078.

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8

Wang, Jane-Ling. "Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate." Annals of Statistics 14, no. 3 (September 1986): 1113–31. http://dx.doi.org/10.1214/aos/1176350053.

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9

Wiest, E. J., and E. Polak. "On the rate of convergence of two minimax algorithms." Journal of Optimization Theory and Applications 71, no. 1 (October 1991): 1–30. http://dx.doi.org/10.1007/bf00940037.

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10

Geraniotis, E., and H. Poor. "Minimax discrimination for observed Poisson processes with uncertain rate functions." IEEE Transactions on Information Theory 31, no. 5 (September 1985): 660–69. http://dx.doi.org/10.1109/tit.1985.1057091.

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11

Efromovich, Sam. "Minimax theory of nonparametric hazard rate estimation: efficiency and adaptation." Annals of the Institute of Statistical Mathematics 68, no. 1 (September 30, 2014): 25–75. http://dx.doi.org/10.1007/s10463-014-0487-4.

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12

Yang, Guowu, and Yuhong Yang. "Minimax-rate adaptive nonparametric regression with unknown correlations of errors." Science China Mathematics 62, no. 2 (January 18, 2019): 227–44. http://dx.doi.org/10.1007/s11425-018-9394-x.

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13

Guerre, Emmanuel, and Pascal Lavergne. "OPTIMAL MINIMAX RATES FOR NONPARAMETRIC SPECIFICATION TESTING IN REGRESSION MODELS." Econometric Theory 18, no. 5 (July 17, 2002): 1139–71. http://dx.doi.org/10.1017/s0266466602185069.

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In the context of testing the specification of a nonlinear parametric regression function, we adopt a nonparametric minimax approach to determine the maximum rate at which a set of smooth alternatives can approach the null hypothesis while ensuring that a test can uniformly detect any alternative in this set with some predetermined power. We show that a smooth nonparametric test has optimal asymptotic minimax properties for regular alternatives. As a by-product, we obtain the rate of the smoothing parameter that ensures rate-optimality of the test. We show that, in contrast, a class of nonsmooth tests, which includes the integrated conditional moment test of Bierens (1982, Journal of Econometrics 20, 105–134), has suboptimal asymptotic minimax properties.

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14

Chen, Xiaohong, and Markus Reiss. "ON RATE OPTIMALITY FOR ILL-POSED INVERSE PROBLEMS IN ECONOMETRICS." Econometric Theory 27, no. 3 (October 11, 2010): 497–521. http://dx.doi.org/10.1017/s0266466610000381.

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In this paper we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and NPIV models under two basic regularity conditions: the approximation number and the link condition. We show that both a simple projection estimator for the NPIR model and a sieve minimum distance estimator for the NPIV model can achieve the minimax risk lower bounds and are rate optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.

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15

Zhu, Yuancheng, and John Lafferty. "Quantized minimax estimation over Sobolev ellipsoids." Information and Inference: A Journal of the IMA 7, no. 1 (June 21, 2017): 31–82. http://dx.doi.org/10.1093/imaiai/iax007.

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Abstract We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker's theorem for non-parametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to quantization in terms of the number of bits, the signal size and the noise level. This establishes the Pareto optimal tradeoff between storage and risk under quantization constraints for Sobolev spaces. Our results and proof techniques combine elements of rate distortion theory and minimax analysis. The proposed quantized estimation scheme, which shows achievability of the lower bounds, is adaptive in the usual statistical sense, achieving the optimal quantized minimax rate without knowledge of the smoothness parameter of the Sobolev space. It is also adaptive in a computational sense, as it constructs the code only after observing the data, to dynamically allocate more codewords to blocks where the estimated signal size is large. Simulations are included that illustrate the effect of quantization on statistical risk.

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16

Cai, T. Tony, and Anru Zhang. "Minimax rate-optimal estimation of high-dimensional covariance matrices with incomplete data." Journal of Multivariate Analysis 150 (September 2016): 55–74. http://dx.doi.org/10.1016/j.jmva.2016.05.002.

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17

Kroll, Martin. "Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems." ESAIM: Probability and Statistics 23 (2019): 524–51. http://dx.doi.org/10.1051/ps/2018027.

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We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the sequence of eigenvalues, we develop a minimax theory for this problem. We propose a truncated series estimator and show that it attains the optimal rate of convergence if the truncation parameter is chosen appropriately. Consequences for testing problems in inverse problems are equally discussed: in particular, the minimax rates of testing for signal detection and goodness-of-fit testing are derived.

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18

Royset, J. O., and E. Y. Pee. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems." Journal of Optimization Theory and Applications 155, no. 3 (July 4, 2012): 855–82. http://dx.doi.org/10.1007/s10957-012-0109-3.

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19

Ding, Litao, and Peter Mathé. "Minimax Rates for Statistical Inverse Problems Under General Source Conditions." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 603–8. http://dx.doi.org/10.1515/cmam-2017-0055.

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AbstractWe describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R. C. Liu, and B. MacGibbon [4]. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.

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20

Galstyan, Tigran, and Arshak Minasyan. "Optimality of the Least Sum of Logarithms in the Problem of Matching Map Recovery in the Presence of Noise and Outliers." Armenian Journal of Mathematics 15, no. 5 (March 30, 2023): 1–9. http://dx.doi.org/10.52737/18291163-2023.15.5-1-9.

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We consider the problem of estimating the matching map between two sets of feature-vectors observed in a noisy environment and contaminated by outliers. It was already known in the literature that in the outlier-free setting, the least sum of squares (LSS) and the least sum of logarithms (LSL) are both minimax-rate-optimal. It has been recently proved that the optimality properties of the LSS continue to hold in the case the data sets contain outliers. In this work, we show that the same is true for the LSL as well. Therefore, LSL has the same desirable properties as the LSS, and, in addition, it is minimax-rate-optimal in the outlier-free setting with heteroscedastic noise.

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21

Ouyang, Liang-Yuh, and Bor-Ren Chuang. "A MINIMAX DISTRIBUTION FREE PROCEDURE FOR STOCHASTIC INVENTORY MODELS WITH A RANDOM BACKORDER RATE." Journal of the Operations Research Society of Japan 42, no. 3 (1999): 342–51. http://dx.doi.org/10.15807/jorsj.42.342.

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22

del Álamo, Miguel, and Axel Munk. "Total variation multiscale estimators for linear inverse problems." Information and Inference: A Journal of the IMA 9, no. 4 (March 2, 2020): 961–86. http://dx.doi.org/10.1093/imaiai/iaaa001.

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Abstract Even though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation ($BV$), meaning $L^1$ functions on a $d$-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of $BV$ functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper, we propose a novel technique for estimating $BV$ functions in an inverse problem setting and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the $L^q$-risk, for any $q\in [1,\infty )$. This is to the best of our knowledge the first convergence result for $BV$ functions in inverse problems in dimension $d\geq 2$, and it extends the results of Donoho (1995, Appl. Comput. Harmon. Anal., 2, 101–126) in $d=1$. Furthermore, our analysis unravels a novel regime for large $q$ in which the minimax rate is slower than $n^{-1/(d+2\beta +2)}$, where $\beta$ is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of $BV$ functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.

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23

Wally Zaher, Jiddah r., and Ali Hameed Yousif. "Proposing Shrinkage Estimator of MCP and Elastic-Net penalties in Quantile Regression Model." Wasit Journal of Pure sciences 1, no. 3 (December 24, 2022): 126–34. http://dx.doi.org/10.31185/wjps.73.

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In some studies, there is a need to estimate the conditional distribution of the response variable at different points, and this is not available in linear regression. The alternative procedure to deal with these problems is quantile regression. In this research, a new estimator for estimating and selecting variables is proposed in the quantile regression model. A new estimator was combines two estimators Minimax Concave Penalty (MCP) and Elastic-Net called shrinkage estimator. It was compared with estimators (Minimax Concave Penalty (MCP) and Elastic-Net) by using simulation and based on Mean Square Error (MSE) and measures of sparsity False Positive Rate (FPR) and False negative rate (FNR ). We concluded that the proposed method is the best in terms of estimation and selection of variables

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24

Xie, Fangzheng, and Yanxun Xu. "Optimal Bayesian estimation for random dot product graphs." Biometrika 107, no. 4 (July 6, 2020): 875–89. http://dx.doi.org/10.1093/biomet/asaa031.

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Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data.

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25

Zhao, Puning, and Zhiguo Wan. "Robust Nonparametric Regression under Poisoning Attack." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 15 (March 24, 2024): 17007–15. http://dx.doi.org/10.1609/aaai.v38i15.29644.

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This paper studies robust nonparametric regression, in which an adversarial attacker can modify the values of up to q samples from a training dataset of size N. Our initial solution is an M-estimator based on Huber loss minimization. Compared with simple kernel regression, i.e. the Nadaraya-Watson estimator, this method can significantly weaken the impact of malicious samples on the regression performance. We provide the convergence rate as well as the corresponding minimax lower bound. The result shows that, with proper bandwidth selection, supremum error is minimax optimal. The L2 error is optimal with relatively small q, but is suboptimal with larger q. The reason is that this estimator is vulnerable if there are many attacked samples concentrating in a small region. To address this issue, we propose a correction method by projecting the initial estimate to the space of Lipschitz functions. The final estimate is nearly minimax optimal for arbitrary q, up to a logarithmic factor.

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26

Liu, Yi, and Sin-Ho Jung. "An analytical study of the critical values of response rate in single-arm phase II clinical trial designs." Biometrics & Biostatistics International Journal 11, no. 5 (December 30, 2022): 178–83. http://dx.doi.org/10.15406/bbij.2023.12.00374.

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A single-arm phase II clinical trial is usually conducted for finding an appropriate dose-level and testing toxicity for an experimental cancer therapy in comparison to some historical controls, and is usually the most doable trial type due to the feasibility under limited budget, patient pool and medical conditions that can be met. We considered the standard setting of a single-arm two-stage phase II clinical trial, and investigated the patterns of critical values and sample sizes at both two stages of minimax and optimal designs under different design parameters, i.e., under different response rates of control and treatment, significant level of the test, and statistical power. We provided analytic derivations to the patterns we are interested under large sample approximation, and investigated them under finite ans small sample via a numerical study by considering extensively possible design parameters over a fine grid. We finally concluded that the critical values at different stages of the test are related to the sample sizes at different stages in a similar way for both optimal and minimax designs, but they also reveal some differences and reflected the nature of these two types of design.

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27

Liu, Yi, and Sin-Ho Jung. "An analytical study of the critical values of response rate in single-arm phase II clinical trial designs." Biometrics & Biostatistics International Journal 11, no. 5 (December 30, 2022): 178–83. http://dx.doi.org/10.15406/bbij.2022.11.00374.

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A single-arm phase II clinical trial is usually conducted for finding an appropriate dose-level and testing toxicity for an experimental cancer therapy in comparison to some historical controls, and is usually the most doable trial type due to the feasibility under limited budget, patient pool and medical conditions that can be met. We considered the standard setting of a single-arm two-stage phase II clinical trial, and investigated the patterns of critical values and sample sizes at both two stages of minimax and optimal designs under different design parameters, i.e., under different response rates of control and treatment, significant level of the test, and statistical power. We provided analytic derivations to the patterns we are interested under large sample approximation, and investigated them under finite ans small sample via a numerical study by considering extensively possible design parameters over a fine grid. We finally concluded that the critical values at different stages of the test are related to the sample sizes at different stages in a similar way for both optimal and minimax designs, but they also reveal some differences and reflected the nature of these two types of design.

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28

Zhou, Yang, and Di-Rong Chen. "Optimal rate for prediction when predictor and response are functions." Analysis and Applications 18, no. 04 (June 6, 2020): 697–714. http://dx.doi.org/10.1142/s0219530520500037.

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In functional data analysis, linear prediction problems have been widely studied based on the functional linear regression model. However, restrictive condition is needed to ensure the existence of the coefficient function. In this paper, a general linear prediction model is considered on the framework of reproducing kernel Hilbert space, which includes both the functional linear regression model and the point impact model. We show that from the point view of prediction, this general model works as well even the coefficient function does not exist. Moreover, under mild conditions, the minimax optimal rate of convergence is established for the prediction under the integrated mean squared prediction error. In particular, the rate reduces to the existing result when the coefficient function exists.

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29

Charoenlarpnopparut, Chalie. "Optimal Minimax Controller for Plants with Four Oscillatory Modes Using Grobner Basis." ECTI Transactions on Electrical Engineering, Electronics, and Communications 7, no. 1 (August 20, 2008): 52–61. http://dx.doi.org/10.37936/ecti-eec.200971.171808.

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Optimal minimax rate feedback controller design problems was proposed and partially solved by R.S. Bucy et al in 1990. The application of the problem have found in the oscillation suppressor design of large space structure with multiple oscillatory/resonance modes. By employing Grobner basis technique, the complete symbolic solution for the case when the cardinality of the plant oscillatory mode is three or fewer was later found by N.K. Bose and the author. In this paper, the case when the cardinality is four is considered based on the use of Grobner bases. In general, the higher order (four or more) problem is analytically intractable and suboptimal solutions based on numerical techniques are then the only recourse.In addition, it is also shown that, for a specified generic plant, by incorporating in rate feedback controller the additional parameter available in the basic design procedure, significant improvement in feedback system performance over what was believed to be possible can be realized . This proposed additional design parameter expands the searching space for optimal solution (i.e. provides higher degree of freedom). Various numerical examples are shown to illustrate the e®ectiveness of the proposed method.

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30

Yan, Xin, Zhouping Xiao, and Zheng Ma. "A Quadratic Surface Minimax Probability Machine for Imbalanced Classification." Symmetry 15, no. 1 (January 13, 2023): 230. http://dx.doi.org/10.3390/sym15010230.

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In this paper, a kernel-free minimax probability machine model for imbalanced classification is proposed. In this model, a quadratic surface is adopted directly for separating the data points into two classes. By using two symmetry constraints to define the two worst-case classification accuracy rates, the model of maximizing both the F1 value of the minority class and the classification accuracy rate of all the data points is proposed. The proposed model corresponds to a fractional programming problem. Since the two worst-case classification accuracy rates are the symmetry, the proposed model can be further simplified. After this, the alternating descent algorithm is adopted for efficiently solving. The proposed method reduces the computational costs by both using the kernel-free technique and adopting the efficient algorithm. Some numerical tests on benchmark datasets are conducted to investigate the classification performance of the proposed method. The numerical results demonstrate that the proposed method performs better when compared with the other state-of-the-art methods, especially for classifying the imbalanced datasets. The better performance for the imbalanced classification is also demonstrated on a Wholesale customers dataset. This method can provide methodological support for the research in areas such as customer segmentation.

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31

Wu, Jong-Wuu, Wen-Chuan Lee, and Chia-Ling Lei. "Optimal Inventory Policy Involving Ordering Cost Reduction, Back-Order Discounts, and Variable Lead Time Demand by Minimax Criterion." Mathematical Problems in Engineering 2009 (2009): 1–19. http://dx.doi.org/10.1155/2009/928932.

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This paper allows the backorder rate as a control variable to widen applications of a continuous review inventory model. Moreover, we also consider the backorder rate that is proposed by combining Ouyang and Chuang (2001) (or Lee (2005)) with Pan and Hsiao (2001) to present a new form. Thus, the backorder rate is dependent on the amount of shortages and backorder price discounts. Besides, we also treat the ordering cost as a decision variable. Hence, we develop an algorithmic procedure to find the optimal inventory policy by minimax criterion. Finally, a numerical example is also given to illustrate the results.

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32

ARAGONE, LAURA S., SILVIA C. DI MARCO, and ROBERTO L. V. GONZÁLEZ. "NUMERICAL ANALYSIS OF A MINIMAX OPTIMAL CONTROL PROBLEM WITH AN ADDITIVE FINAL COST." Mathematical Models and Methods in Applied Sciences 12, no. 02 (February 2002): 183–203. http://dx.doi.org/10.1142/s021820250200160x.

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In this paper we deal with the numerical analysis of an optimal control problem of minimax type with finite horizon and final cost. To get numerical approximations we devise here a fully discrete scheme which enables us to compute an approximated solution. We prove that the fully discrete solution converges to the solution of the continuous problem and we also give the order of the convergence rate. Finally we present some numerical results.

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33

Gupta, N., S. Sivananthan, and B. K. Sriperumbudur. "Convergence analysis of kernel conjugate gradient for functional linear regression." Journal of Applied and Numerical Analysis 1, no. 1 (December 25, 2023): 33–47. http://dx.doi.org/10.30970/ana.2023.1.33.

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In this paper, we discuss the convergence analysis of the conjugate gradient-based algorithm for the functional linear model in the reproducing kernel Hilbert space framework, utilizing early stopping results in regularization against over-fitting. We establish the convergence rates depending on the regularity condition of the slope function and the decay rate of the eigenvalues of the operator composition of covariance and kernel operator. Our convergence rates match the minimax rate available from the literature.

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34

Ma, Don Yitong. "Optimization of Alpha-Beta pruning based on heuristic algorithm." Applied and Computational Engineering 6, no. 1 (June 14, 2023): 1151–55. http://dx.doi.org/10.54254/2755-2721/6/20230498.

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Games have become an important place for testing Artificial Intelligence (AI). Minimax and Alpha-Beta Pruning are two common and basic algorithms implemented in game AIs. However, there are still limitations of the searching time and searching depth. This paper strives to improve the game AI with a Heuristic Algorithm to optimize both the searching time and depth. The experiment consists of three AIs built with Minimax, Alpha-Beta Pruning, and Heuristic Algorithm to evident the improvement. These AIs are built to play a traditional Chinese zero-sum game, Gobang, which can be seen as an enhanced version of tic-tac-toe but more advanced. The data is collected when AIs compete with each other. Comparing the search time of three AIs, there is significant improvement of AI implemented Heuristic Algorithm; the median search time for Heuristic AI is only half of the Alpha-Beta Pruning AI, and only quarter of the Minimax AI. Moreover, because the search time decreases, the searching depth of the Heuristic AI can also be increased. With the larger searching depth, the Heuristic AI also gains a higher winning rate against the other two AIs.

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35

Qi, Xinyu, Jinru Wang, and Jiating Shao. "Minimax perturbation bounds of the low-rank matrix under Ky Fan norm." AIMS Mathematics 7, no. 5 (2022): 7595–605. http://dx.doi.org/10.3934/math.2022426.

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<abstract><p>This paper considers the minimax perturbation bounds of the low-rank matrix under Ky Fan norm. We first explore the upper bounds via the best rank-$ r $ approximation $ \hat{A}_r $ of the observation matrix $ \hat{A} $. Next, the lower bounds are established by constructing special matrix groups to show the upper bounds are tight on the low-rank matrix estimation error. In addition, we derive the rate-optimal perturbation bounds for the left and right singular subspaces under Ky Fan norm $ \sin\Theta $ distance. Finally, some simulations have been carried out to support our theories.</p></abstract>

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36

Yang, Liu, Steve Hanneke, and Jaime Carbonell. "Bounds on the minimax rate for estimating a prior over a VC class from independent learning tasks." Theoretical Computer Science 716 (March 2018): 124–40. http://dx.doi.org/10.1016/j.tcs.2017.11.025.

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37

Zhao, Puning, and Lifeng Lai. "Efficient Classification with Adaptive KNN." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 12 (May 18, 2021): 11007–14. http://dx.doi.org/10.1609/aaai.v35i12.17314.

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In this paper, we propose an adaptive kNN method for classification, in which different k are selected for different test samples. Our selection rule is easy to implement since it is completely adaptive and does not require any knowledge of the underlying distribution. The convergence rate of the risk of this classifier to the Bayes risk is shown to be minimax optimal for various settings. Moreover, under some special assumptions, the convergence rate is especially fast and does not decay with the increase of dimensionality.

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38

Efromovich, Sam. "Missing, Modified, and Large-p-small-n Data in Nonparametric Curve Estimation." Calcutta Statistical Association Bulletin 69, no. 1 (May 2017): 1–34. http://dx.doi.org/10.1177/0008068317695906.

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Nonparametric curve estimation, which makes no assumptions about shape of estimated functions, is one of the main pillars of the modern statistical science. It is used when no adequate parametric or semi-parametric model is available. Asymptotic results on adaptive estimation of nonparametric curves, under both traditional and shrinking minimaxes, are presented. The latter approach allows us to explain the phenomenon of superefficiency when a function can be estimated with a rate faster than the minimax one. Resent results on sequential nonparametric estimation, which yields an assigned risk with minimal average stopping time, are also presented. Then it is explained how the theory can be used in practically important cases of missing data, modified data in survival analysis, and Big Data. Wavelet applications in the analysis of microarrays and fMRI illustrate feasibility of the proposed nonparametric estimation.

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39

CAPONNETTO, ANDREA, and YUAN YAO. "CROSS-VALIDATION BASED ADAPTATION FOR REGULARIZATION OPERATORS IN LEARNING THEORY." Analysis and Applications 08, no. 02 (April 2010): 161–83. http://dx.doi.org/10.1142/s0219530510001564.

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We consider learning algorithms induced by regularization methods in the regression setting. We show that previously obtained error bounds for these algorithms, using a priori choices of the regularization parameter, can be attained using a suitable a posteriori choice based on cross-validation. In particular, these results prove adaptation of the rate of convergence of the estimators to the minimax rate induced by the "effective dimension" of the problem. We also show universal consistency for this broad class of methods which includes regularized least-squares, truncated SVD, Landweber iteration and ν-method.

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40

Brajević, Ivona. "A Shuffle-Based Artificial Bee Colony Algorithm for Solving Integer Programming and Minimax Problems." Mathematics 9, no. 11 (May 27, 2021): 1211. http://dx.doi.org/10.3390/math9111211.

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The artificial bee colony (ABC) algorithm is a prominent swarm intelligence technique due to its simple structure and effective performance. However, the ABC algorithm has a slow convergence rate when it is used to solve complex optimization problems since its solution search equation is more of an exploration than exploitation operator. This paper presents an improved ABC algorithm for solving integer programming and minimax problems. The proposed approach employs a modified ABC search operator, which exploits the useful information of the current best solution in the onlooker phase with the intention of improving its exploitation tendency. Furthermore, the shuffle mutation operator is applied to the created solutions in both bee phases to help the search achieve a better balance between the global exploration and local exploitation abilities and to provide a valuable convergence speed. The experimental results, obtained by testing on seven integer programming problems and ten minimax problems, show that the overall performance of the proposed approach is superior to the ABC. Additionally, it obtains competitive results compared with other state-of-the-art algorithms.

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Levine, Michael. "Minimax rate of convergence for an estimator of the functional component in a semiparametric multivariate partially linear model." Journal of Multivariate Analysis 140 (September 2015): 283–90. http://dx.doi.org/10.1016/j.jmva.2015.05.010.

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Hsieh, Fushing, and Bruce W. Turnbull. "A note on the local asymptotically minimax rate for estimating a crossing point in a diagnostic marker problem." Statistics & Probability Letters 24, no. 2 (August 1995): 181–85. http://dx.doi.org/10.1016/0167-7152(94)00167-7.

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Ke, Zheng Tracy, and Jingming Wang. "Entry-Wise Eigenvector Analysis and Improved Rates for Topic Modeling on Short Documents." Mathematics 12, no. 11 (May 28, 2024): 1682. http://dx.doi.org/10.3390/math12111682.

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Topic modeling is a widely utilized tool in text analysis. We investigate the optimal rate for estimating a topic model. Specifically, we consider a scenario with n documents, a vocabulary of size p, and document lengths at the order N. When N≥c·p, referred to as the long-document case, the optimal rate is established in the literature at p/(Nn). However, when N=o(p), referred to as the short-document case, the optimal rate remains unknown. In this paper, we first provide new entry-wise large-deviation bounds for the empirical singular vectors of a topic model. We then apply these bounds to improve the error rate of a spectral algorithm, Topic-SCORE. Finally, by comparing the improved error rate with the minimax lower bound, we conclude that the optimal rate is still p/(Nn) in the short-document case.

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Haris, Asad, Ali Shojaie, and Noah Simon. "Nonparametric regression with adaptive truncation via a convex hierarchical penalty." Biometrika 106, no. 1 (December 13, 2018): 87–107. http://dx.doi.org/10.1093/biomet/asy056.

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SUMMARY We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size.

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Green, Alden, Sivaraman Balakrishnan, and Ryan J. Tibshirani. "Minimax optimal regression over Sobolev spaces via Laplacian Eigenmaps on neighbourhood graphs." Information and Inference: A Journal of the IMA 12, no. 3 (April 27, 2023): 2423–502. http://dx.doi.org/10.1093/imaiai/iaad034.

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Abstract In this paper, we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for non-parametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\textbf Y} = (Y_1,\ldots ,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighbourhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) and goodness-of-fit testing ($n^{-4s/(4s + d)}$). We also consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence.

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Prędki, Artur. "Estymacja zbioru możliwości produkcyjnych w ramach formalnego modelu statystycznego." Przegląd Statystyczny. Statistical Review 2010, no. 4 (December 31, 2010): 3–18. http://dx.doi.org/10.59139/ps.2010.04.1.

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In the paper some general statistical model is presented and one of its particular version is exploited to formal estimate of the production set within the DEA and FDH methods. Properties of the FDH and DEA estimators are presented and their realizations for a finite sample are illustrated. Elements of the minimax approach are introduced and the rate of convergence is exploited to express the definition of asymptotic optimality of the estimators.

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Kieser, M., and C. U. Kunz. "Optimal Two-stage Designs for Single-arm Phase II Oncology Trials with Two Binary Endpoints." Methods of Information in Medicine 50, no. 04 (2011): 372–77. http://dx.doi.org/10.3414/me10-01-0037.

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SummaryObjectives: In phase II clinical trials in oncology, the potential efficacy of a new treatment regimen is assessed in terms of anticancer activity. The standard approach consists of a single-arm two-stage design where a single binary endpoint is compared to a specified target value. However, a new drug would still be considered promising if it showed a lower tumor response rate than the target level but would lead, for example, to disease stabilization.Methods: We present an analytical solution for the calculation of the type I and type II error rate for a two-stage design where the hypothesis test considers two endpoints and provide optimal and minimax solutions. Furthermore, the problem of inference about the two single endpoints following rejection of the global null hypothesis is addressed by deriving a multiple test procedure that controls the experimentwise type I error rate in the strong sense.Results: The proposed methods are illustrated with a real data example, and the new design is tabulated for a wide range of parameter values. Similar to two-stage designs with a single endpoint, the characteristics of optimal and minimax designs with two endpoints with respect to expected and maximum sample size can be quite different. Therefore, the choice of an admissible design may be a valuable compromise.Conclusions: The new procedure extends Simon’s two-stage design to two endpoints. This approach allows a more comprehensive assessment of the overall picture of antitumor efficacy of a new treatment than restriction to a single outcome.

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Shen, Shuqi. "The development of computer-based algorithms based on gobang game." Applied and Computational Engineering 4, no. 1 (June 14, 2023): 284–88. http://dx.doi.org/10.54254/2755-2721/4/20230472.

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Gobang is a worldwide two-player strategy board game, which is popular especially in the Asian region. Recently, the science and technology has been developing fast and artificial intelligence was applied in numerous fields like the board game. Gobang, a strategy game with moderate difficulty, is a suitable example for people to test algorithms and solve board game problems. As a result, many related algorithms were appearing with different advantages and disadvantages. In this work, these already existed algorithms, game tree, minimax search, alpha-beta pruning, genetic algorithm and monte carlo tree search, were discussed and compared. The results and comparison showed that game tree and minimax search had a large number of nodes to calculate to reach a suitable search depth, about 1.00E+12 and 1.29E+14 respectively, which meant they need a long calculating time, while the alpha-beta pruning need to calculate about 2.2E+07 nodes and genetic algorithm only need to calculate about 1.00E+04 nodes, which cost 0.6 seconds for every move. Plus, the monte carlo tree search could reach nearly 100% win rate through self-play, which making gobang algorithm become more refined. Additionally, these algorithms had already made gobang AI powerful with fast move and high win rate, so they also had been applied in many different fields to develop and spread gobang.

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Horowitz, Joel L. "Optimal Rates of Convergence of Parameter Estimators in the Binary Response Model with Weak Distributional Assumptions." Econometric Theory 9, no. 1 (January 1993): 1–18. http://dx.doi.org/10.1017/s0266466600007301.

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The smoothed maximum score estimator of the coefficient vector of a binary response model is consistent and, after centering and suitable normalization, asymptotically normally distributed under weak assumptions [5]. Its rate of convergence in probability is N−h/(2h+1), where h ≥ 2 is an integer whose value depends on the strength of certain smoothness assumptions. This rate of convergence is faster than that of the maximum score estimator of Manski [11,12], which converges at the rate N−1/3 under assumptions that are somewhat weaker than those of the smoothed estimator. In this paper I prove that under the assumptions of smoothed maximum score estimation, N−h/(2h+1) is the fastest achievable rate of convergence of an estimator of the coefficient vector of a binary response model. Thus, the smoothed maximum score estimator has the fastest possible rate of convergence. The rate of convergence is defined in a minimax sense so as to exclude superefficient estimators.

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Wang, Shuaiwen, Haolei Weng, and Arian Maleki. "Does SLOPE outperform bridge regression?" Information and Inference: A Journal of the IMA 11, no. 1 (November 15, 2021): 1–54. http://dx.doi.org/10.1093/imaiai/iaab025.

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Abstract A recently proposed SLOPE estimator [6] has been shown to adaptively achieve the minimax $\ell _2$ estimation rate under high-dimensional sparse linear regression models [25]. Such minimax optimality holds in the regime where the sparsity level $k$, sample size $n$ and dimension $p$ satisfy $k/p\rightarrow 0, k\log p/n\rightarrow 0$. In this paper, we characterize the estimation error of SLOPE under the complementary regime where both $k$ and $n$ scale linearly with $p$, and provide new insights into the performance of SLOPE estimators. We first derive a concentration inequality for the finite sample mean square error (MSE) of SLOPE. The quantity that MSE concentrates around takes a complicated and implicit form. With delicate analysis of the quantity, we prove that among all SLOPE estimators, LASSO is optimal for estimating $k$-sparse parameter vectors that do not have tied nonzero components in the low noise scenario. On the other hand, in the large noise scenario, the family of SLOPE estimators are sub-optimal compared with bridge regression such as the Ridge estimator.

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