We use the methods of “The Welch-Satterthwaite test”, “The Cochran-Cox test”, “The Generalized p-value test”, “Computational Approach test” to structure different Confidence Distributions, and use the Confiden...We use the methods of “The Welch-Satterthwaite test”, “The Cochran-Cox test”, “The Generalized p-value test”, “Computational Approach test” to structure different Confidence Distributions, and use the Confidence Distributions to give an new solution the confidence interval of the difference between two population means where the populations are assumed to be normal with unknown and unequal variances. Finally, we find the most effective solution through the numerical simulation.展开更多
Based on the Confidence Distribution method to the Behrens-Fisher problem, we consider two approaches of combining Confidence Distributions: P Combination and AN Combination to solve the Behrens-Fisher problem. Firstl...Based on the Confidence Distribution method to the Behrens-Fisher problem, we consider two approaches of combining Confidence Distributions: P Combination and AN Combination to solve the Behrens-Fisher problem. Firstly, we provide some Confidence Distributions to the Behrens-Fisher problem, and then we give the Confidence Distribution method to the Behrens-Fisher problem. Finally, we compare the “combination” and the “single” through the numerical simulation.展开更多
In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley's paradox. We have p...In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley's paradox. We have provided a general proof that for any prior which yields a linear combination of two independent t random variables as posterior distribution of the di erence of means, the new Bayesian measure of evidence given that prior will solve Lindleys' paradox thereby serving as a general proof for the works of Yin and Li (2014, Journal of Applied Mathematics, 2014(978691)) and Goltong?and Doguwa (2018, Open Journal of Statistics, 8: 902-914).?Using the Pareto prior as an example, we have shown by the use of?simulation results that the new Bayesian measure of evidence solves?Lindley's paradox.展开更多
Yin [1] has developed a new Bayesian measure of evidence for testing a point null hypothesis which agrees with the frequentist p-value thereby, solving Lindley’s paradox. Yin and Li [2] extended the methodology of Yi...Yin [1] has developed a new Bayesian measure of evidence for testing a point null hypothesis which agrees with the frequentist p-value thereby, solving Lindley’s paradox. Yin and Li [2] extended the methodology of Yin [1] to the case of the Behrens-Fisher problem by assigning Jeffreys’ independent prior to the nuisance parameters. In this paper, we were able to show both analytically and through the results from simulation studies that the methodology of Yin?[1] solves simultaneously, the Behrens-Fisher problem and Lindley’s paradox when a Gamma prior is assigned to the nuisance parameters.展开更多
Testing the equality of means of two normally distributed random variables when their variances are unequal is known in the statistical literature as the “Behrens-Fisher problem”. It is well-known that the posterior...Testing the equality of means of two normally distributed random variables when their variances are unequal is known in the statistical literature as the “Behrens-Fisher problem”. It is well-known that the posterior distributions of the parameters of interest are the primitive of Bayesian statistical inference. For routine implementation of statistical procedures based on posterior distributions, simple and efficient approaches are required. Since the computation of the exact posterior distribution of the Behrens-Fisher problem is obtained using numerical integration, several approximations are discussed and compared. Tests and Bayesian Highest-Posterior Density (H.P.D) intervals based upon these approximations are discussed. We extend the proposed approximations to test of parallelism in simple linear regression models.展开更多
For high-dimensional nonparametric Behrens-Fisher problem in which the data dimension is larger than the sample size,the authors propose two test statistics in which one is U-statistic Rankbased Test(URT)and another i...For high-dimensional nonparametric Behrens-Fisher problem in which the data dimension is larger than the sample size,the authors propose two test statistics in which one is U-statistic Rankbased Test(URT)and another is Cauchy Combination Test(CCT).CCT is analogous to the maximumtype test,while URT takes into account the sum of squares of differences of ranked samples in different dimensions,which is free of shapes of distributions and robust to outliers.The asymptotic distribution of URT is derived and the closed form for calculating the statistical significance of CCT is given.Extensive simulation studies are conducted to evaluate the finite sample power performance of the statistics by comparing with the existing method.The simulation results show that our URT is robust and powerful method,meanwhile,its practicability and effectiveness can be illustrated by an application to the gene expression data.展开更多
Convex feasibility problems are widely used in image reconstruction,sparse signal recovery,and other areas.This paper is devoted to considering a class of convex feasibility problem arising from sparse signal recovery...Convex feasibility problems are widely used in image reconstruction,sparse signal recovery,and other areas.This paper is devoted to considering a class of convex feasibility problem arising from sparse signal recovery.We rst derive the projection formulas for a vector onto the feasible sets.The centralized circumcentered-reection method is designed to solve the convex feasibility problem.Some numerical experiments demonstrate the feasibility and e ectiveness of the proposed algorithm,showing superior performance compared to conventional alternating projection methods.展开更多
During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive...During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive distance)to a moving target as quickly as possible,resulting in the extended minimum-time intercept problem(EMTIP).Existing research has primarily focused on the zero-distance intercept problem,MTIP,establishing the necessary or sufficient conditions for MTIP optimality,and utilizing analytic algorithms,such as root-finding algorithms,to calculate the optimal solutions.However,these approaches depend heavily on the properties of the analytic algorithm,making them inapplicable when problem settings change,such as in the case of a positive effective range or complicated target motions outside uniform rectilinear motion.In this study,an approach employing a high-accuracy and quality-guaranteed mixed-integer piecewise-linear program(QG-PWL)is proposed for the EMTIP.This program can accommodate different effective interception ranges and complicated target motions(variable velocity or complicated trajectories).The high accuracy and quality guarantees of QG-PWL originate from elegant strategies such as piecewise linearization and other developed operation strategies.The approximate error in the intercept path length is proved to be bounded to h^(2)/(4√2),where h is the piecewise length.展开更多
Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(...Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(3)+p_(5)^(5)+_6^(5)is solvable with being an almost-prime P_(6) and the other variables primes.This result constitutes an enhancement upon the previous result of Hooley[Recent Progress in Analytic Number Theory,Vol.1(Durham,1979),London:Academic Press,1981,127-191].展开更多
Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2))...Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.展开更多
Automatically answer math word problems is a challenging task in artificial intelligence.Previous solvers constructed mathematical expressions in sequence or binary tree.However,these approaches may suffer from the fo...Automatically answer math word problems is a challenging task in artificial intelligence.Previous solvers constructed mathematical expressions in sequence or binary tree.However,these approaches may suffer from the following issues:Models relying on such structures exhibit fixed-order reasoning(e.g.,left-to-right),limiting flexibility and increasing error susceptibility;prior models rely on autoregressive reasoning in a single pass,accumulating minor errors(e.g.,incorrect math symbols)during generation,resulting in reduced accuracy.To address the above issues,we emulate the human“check and modify”process in reasoning and propose a unified M-tree self-correction solver(UTSCSolver)by iterative inference with self-correction mechanism.First,we use an iterative,non-autoregressive process for generating mathematical expressions,free from fixed generation orders to handle complex and diverse problems.Additionally,we design a self-correction mechanism based on alternating execution between a generator and a discriminator.This module iteratively detects and rectifies errors in generated expressions,leveraging previous iteration information for subsequent generation guidance.Experimental results show that our UTSC-Solver outperforms traditional models in accuracy on two popular datasets,while it improves the interpretability of mathematical reasoning.展开更多
文摘We use the methods of “The Welch-Satterthwaite test”, “The Cochran-Cox test”, “The Generalized p-value test”, “Computational Approach test” to structure different Confidence Distributions, and use the Confidence Distributions to give an new solution the confidence interval of the difference between two population means where the populations are assumed to be normal with unknown and unequal variances. Finally, we find the most effective solution through the numerical simulation.
文摘Based on the Confidence Distribution method to the Behrens-Fisher problem, we consider two approaches of combining Confidence Distributions: P Combination and AN Combination to solve the Behrens-Fisher problem. Firstly, we provide some Confidence Distributions to the Behrens-Fisher problem, and then we give the Confidence Distribution method to the Behrens-Fisher problem. Finally, we compare the “combination” and the “single” through the numerical simulation.
文摘In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley's paradox. We have provided a general proof that for any prior which yields a linear combination of two independent t random variables as posterior distribution of the di erence of means, the new Bayesian measure of evidence given that prior will solve Lindleys' paradox thereby serving as a general proof for the works of Yin and Li (2014, Journal of Applied Mathematics, 2014(978691)) and Goltong?and Doguwa (2018, Open Journal of Statistics, 8: 902-914).?Using the Pareto prior as an example, we have shown by the use of?simulation results that the new Bayesian measure of evidence solves?Lindley's paradox.
文摘Yin [1] has developed a new Bayesian measure of evidence for testing a point null hypothesis which agrees with the frequentist p-value thereby, solving Lindley’s paradox. Yin and Li [2] extended the methodology of Yin [1] to the case of the Behrens-Fisher problem by assigning Jeffreys’ independent prior to the nuisance parameters. In this paper, we were able to show both analytically and through the results from simulation studies that the methodology of Yin?[1] solves simultaneously, the Behrens-Fisher problem and Lindley’s paradox when a Gamma prior is assigned to the nuisance parameters.
文摘Testing the equality of means of two normally distributed random variables when their variances are unequal is known in the statistical literature as the “Behrens-Fisher problem”. It is well-known that the posterior distributions of the parameters of interest are the primitive of Bayesian statistical inference. For routine implementation of statistical procedures based on posterior distributions, simple and efficient approaches are required. Since the computation of the exact posterior distribution of the Behrens-Fisher problem is obtained using numerical integration, several approximations are discussed and compared. Tests and Bayesian Highest-Posterior Density (H.P.D) intervals based upon these approximations are discussed. We extend the proposed approximations to test of parallelism in simple linear regression models.
基金supported by Beijing Natural Science Foundation under Grant No.Z180006the National Nature Science Foundation of China under Grant No.11722113。
文摘For high-dimensional nonparametric Behrens-Fisher problem in which the data dimension is larger than the sample size,the authors propose two test statistics in which one is U-statistic Rankbased Test(URT)and another is Cauchy Combination Test(CCT).CCT is analogous to the maximumtype test,while URT takes into account the sum of squares of differences of ranked samples in different dimensions,which is free of shapes of distributions and robust to outliers.The asymptotic distribution of URT is derived and the closed form for calculating the statistical significance of CCT is given.Extensive simulation studies are conducted to evaluate the finite sample power performance of the statistics by comparing with the existing method.The simulation results show that our URT is robust and powerful method,meanwhile,its practicability and effectiveness can be illustrated by an application to the gene expression data.
基金Supported by the Natural Science Foundation of Guangxi Province(Grant Nos.2023GXNSFAA026067,2024GXN SFAA010521)the National Natural Science Foundation of China(Nos.12361079,12201149,12261026).
文摘Convex feasibility problems are widely used in image reconstruction,sparse signal recovery,and other areas.This paper is devoted to considering a class of convex feasibility problem arising from sparse signal recovery.We rst derive the projection formulas for a vector onto the feasible sets.The centralized circumcentered-reection method is designed to solve the convex feasibility problem.Some numerical experiments demonstrate the feasibility and e ectiveness of the proposed algorithm,showing superior performance compared to conventional alternating projection methods.
基金supported by the National Natural Sci‐ence Foundation of China(Grant No.62306325)。
文摘During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive distance)to a moving target as quickly as possible,resulting in the extended minimum-time intercept problem(EMTIP).Existing research has primarily focused on the zero-distance intercept problem,MTIP,establishing the necessary or sufficient conditions for MTIP optimality,and utilizing analytic algorithms,such as root-finding algorithms,to calculate the optimal solutions.However,these approaches depend heavily on the properties of the analytic algorithm,making them inapplicable when problem settings change,such as in the case of a positive effective range or complicated target motions outside uniform rectilinear motion.In this study,an approach employing a high-accuracy and quality-guaranteed mixed-integer piecewise-linear program(QG-PWL)is proposed for the EMTIP.This program can accommodate different effective interception ranges and complicated target motions(variable velocity or complicated trajectories).The high accuracy and quality guarantees of QG-PWL originate from elegant strategies such as piecewise linearization and other developed operation strategies.The approximate error in the intercept path length is proved to be bounded to h^(2)/(4√2),where h is the piecewise length.
基金Supported by NSFC (Nos.12471009,12301006,12001047,11901566)Beijing Natural Science Foundation (No.1242003)National Training Program of Innovation and Entrepreneurship for Undergraduates(No.202307011)。
文摘Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(3)+p_(5)^(5)+_6^(5)is solvable with being an almost-prime P_(6) and the other variables primes.This result constitutes an enhancement upon the previous result of Hooley[Recent Progress in Analytic Number Theory,Vol.1(Durham,1979),London:Academic Press,1981,127-191].
基金supported by the Talent Fund of Beijing Jiaotong University(No.2020RC012)NSFC(No.11871295),supported by NSFC(No.11971476),supported by NSFC(No.12071421)。
文摘Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.
基金supported by the National Natural Science Foundation of China(62106244)the Fundamental Research Funds for the Central Universities(WK2150110021)the University Synergy Innovation Program of Anhui Province(GXXT-2022-042).
文摘Automatically answer math word problems is a challenging task in artificial intelligence.Previous solvers constructed mathematical expressions in sequence or binary tree.However,these approaches may suffer from the following issues:Models relying on such structures exhibit fixed-order reasoning(e.g.,left-to-right),limiting flexibility and increasing error susceptibility;prior models rely on autoregressive reasoning in a single pass,accumulating minor errors(e.g.,incorrect math symbols)during generation,resulting in reduced accuracy.To address the above issues,we emulate the human“check and modify”process in reasoning and propose a unified M-tree self-correction solver(UTSCSolver)by iterative inference with self-correction mechanism.First,we use an iterative,non-autoregressive process for generating mathematical expressions,free from fixed generation orders to handle complex and diverse problems.Additionally,we design a self-correction mechanism based on alternating execution between a generator and a discriminator.This module iteratively detects and rectifies errors in generated expressions,leveraging previous iteration information for subsequent generation guidance.Experimental results show that our UTSC-Solver outperforms traditional models in accuracy on two popular datasets,while it improves the interpretability of mathematical reasoning.
