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Steady-state probability density function in wave turbulence under large volume limit
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作者 Yeontaek Choi Sang Gyu Jo 《Chinese Physics B》 SCIE EI CAS CSCD 2011年第5期139-145,共7页
We investigate the possibility for two-mode probability density function (PDF) to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown... We investigate the possibility for two-mode probability density function (PDF) to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown that in this limit all the steady-state twoor higher-mode PDFs are the product of one-mode PDFs. The flux of this steady-state solution turns out to be zero for any finite mode PDF. 展开更多
关键词 wave turbulence probability flux multi-mode probability density function Larguerreequation
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The Probability Density Function Related to Shallow Cumulus Entrainment Rate and Its Influencing Factors in a Large-Eddy Simulation 被引量:3
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作者 Lei ZHU Chunsong LU +5 位作者 Xiaoqi XU Xin HE Junjun LI Shi LUO Yuan WANG Fan WANG 《Advances in Atmospheric Sciences》 SCIE CAS CSCD 2024年第1期173-187,共15页
The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameteri... The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameterization within the overall cumulus parameterization scheme.In this study,an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculateλof cumulus clouds in a large-eddy simulation.The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method,becauseλ,as calculated from the improved method,falls within the range ofλvalues obtained from the traditional method using different conserved variables.The probability density functions ofλfor all data,different times,and different heights can be well-fitted by a log-normal distribution,which supports the assumed stochastic entrainment process in previous studies.Further analysis demonstrate that the relationship betweenλand the vertical velocity is better than other thermodynamic/dynamical properties;thus,the vertical velocity is recommended as the primary influencing factor for the parameterization ofλin the future.The results of this study enhance the theoretical understanding ofλand its influencing factors and shed new light on the development ofλparameterization. 展开更多
关键词 large-eddy simulation cumulus clouds entrainment rate probability density functions spatial and temporal distribution
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Robustness of the octupole collectivity in 144Ba within the cranking covariant density functional theory in 3D lattice
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作者 Ze‑Kai Li Yuan‑Yuan Wang 《Nuclear Science and Techniques》 SCIE EI CAS CSCD 2024年第8期124-131,共8页
The octupole deformation and collectivity in octupole double-magic nucleus 144Ba are investigated using the Cranking covariant density functional theory in a three-dimensional lattice space.The reduced B(E3)transition... The octupole deformation and collectivity in octupole double-magic nucleus 144Ba are investigated using the Cranking covariant density functional theory in a three-dimensional lattice space.The reduced B(E3)transition probability is implemented for the first time in semiclassical approximation based on the microscopically calculated electric octupole moments.The available data,including the I-ωrelation and electric transitional probabilities B(E2)and B(E3)are well reproduced.Furthermore,it is shown that the ground state of 144Ba exhibits axial octupole and quadrupole deformations that persist up to high spins(I≈24h). 展开更多
关键词 Octupole collectivity Cranking covariant density functional theory Rotational spectrum Electric transitional probabilities
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Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space
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作者 Chol Jong Jin-Sim Kim +1 位作者 Shin-Hyok Jon Son-Il Jo 《World Journal of Mechanics》 2023年第1期20-72,共53页
The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamenta... The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics. 展开更多
关键词 Quantum Ensemble Theory Quantum Mechanics In Phase Space Wave function probability density Schrödinger Equation
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Probability Density Analysis of Nonlinear Random Ship Rolling 被引量:1
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作者 CHEN Jia YANG Jianming +2 位作者 SHEN Kunfan CHANG Zongyu ZHENG Zhongqiang 《Journal of Ocean University of China》 SCIE CAS CSCD 2023年第5期1227-1242,共16页
Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equ... Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms. 展开更多
关键词 ship rolling homoclinic rolling heteroclinic rolling finite element method Fokker Planck equation probability density function
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On the Probability Density Function of the Real and Imaginary Parts in WFRFT Signals 被引量:7
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作者 Xiaolu Wang Lin Mei +1 位作者 Zhenduo Wang Naitong Zhang 《China Communications》 SCIE CSCD 2016年第9期44-52,共9页
Recently a Hybrid Carrier (HC) scheme based on Weighted-type Fractional Fourier Transform (WFRFT) was proposed and developed, which contains Single Carrier (SC) and Multi-Carrier (MC) synergetie transmission. ... Recently a Hybrid Carrier (HC) scheme based on Weighted-type Fractional Fourier Transform (WFRFT) was proposed and developed, which contains Single Carrier (SC) and Multi-Carrier (MC) synergetie transmission. The wide interest is primarily due to its appealing characteristics, such as the robust performances in different types of selective fading channels and a great deal of potential for secure communications. According to the literatures, the HC signal and SC or MC signal probability distributions are different. In particular, some benefits of this HC scheme are brought by the quasi-Gaussian distribution of WFRFT signals. However, until now researchers have only presented statistic properties through computer simulations, and the accurate expressions of signals are not derived yet. In this paper, we derive the accu- rate and rigorously established closed-form expressions of Probability Density Function (PDF) of WFRFT signal real and imaginary parts with a large number of QPSK subcarriers, and this PDF can describe the behavior of data modulated by WFRFT, avoiding the complex computation for extensive computer simulations. Furthermore, the components of PDF expression are described and analyzed, and it is revealed that the tendency of signal quasi-Gaussian changes with the increasing of the parameter a (a in (0,1]). To validate the analytical results, extensive simulations have been conducted, showing a very good match between the analytical results and the real situations. The contribution of this paper may be useful to deduce the closed form expressions of Bit Error Ratio (BER), the Complementary Cumulative Distribution Function (CCDF) of Peak to Average Power Ratio (PAPR), and other analytical studies which adopt the PDF. 展开更多
关键词 weighted-type fractional Fourier transform probability density function single carrier MULTI-CARRIER hybrid carrier
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Shape control on probability density function in stochastic systems 被引量:4
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作者 Lingzhi Wang Fucai Qian Jun Liu 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2014年第1期144-149,共6页
A novel strategy of probability density function (PDF) shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimiza... A novel strategy of probability density function (PDF) shape control is proposed in stochastic systems. The control er is designed whose parameters are optimal y obtained through the improved particle swarm optimization algorithm. The parameters of the control er are viewed as the space position of a particle in particle swarm optimization algorithm and updated continual y until the control er makes the PDF of the state variable as close as possible to the expected PDF. The proposed PDF shape control technique is compared with the equivalent linearization technique through simulation experiments. The results show the superiority and the effectiveness of the proposed method. The control er is excellent in making the state PDF fol ow the expected PDF and has the very smal error between the state PDF and the expected PDF, solving the control problem of the PDF shape in stochastic systems effectively. 展开更多
关键词 stochastic systems probability density function (PDF) shape control improved particle swarm optimization.
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Probability Density Function Method for Observing Reconstructed Attractor Structure 被引量:2
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作者 陆宏伟 陈亚珠 卫青 《Journal of Shanghai University(English Edition)》 CAS 2004年第1期75-79,共5页
Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important in... Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6-6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system. 展开更多
关键词 probability density function (PDF) RR intervals correlation dimension (CD) phase space reconstruction chaos.
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Joint probability density function of the stochastic responses of nonlinear structures 被引量:1
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作者 陈建兵 李杰 《Earthquake Engineering and Engineering Vibration》 SCIE EI CSCD 2007年第1期35-47,共13页
The joint probability density fimction (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution ... The joint probability density fimction (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation. 展开更多
关键词 stochastic response NONLINEARITY joint probability density function probability density evolution method number theoretical method finite difference method
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Linearized Controller Design for the Output Probability Density Functions of Non-Gaussian Stochastic Systems 被引量:1
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作者 Pousga Kabore Husam Baki 《International Journal of Automation and computing》 EI 2005年第1期67-74,共8页
This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density fun... This paper presents a linearized approach for the controller design of the shape of output probability density functions for general stochastic systems. A square root approximation to an output probability density function is realized by a set of B-spline functions. This generally produces a nonlinear state space model for the weights of the B-spline approximation. A linearized model is therefore obtained and embedded into a performance function that measures the tracking error of the output probability density function with respect to a given distribution. By using this performance function as a Lyapunov function for the closed loop system, a feedback control input has been obtained which guarantees closed loop stability and realizes perfect tracking. The algorithm described in this paper has been tested on a simulated example and desired results have been achieved. 展开更多
关键词 Dynamic stochastic systems probability density function B splines neural networks Lyapunov stability theory
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ON DOUBLE PEAK PROBABILITY DENSITY FUNCTIONS OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND RANDOM EXCITATIONS
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作者 戎海武 王向东 +2 位作者 孟光 徐伟 方同 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2006年第11期1569-1576,共8页
The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude an... The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results. 展开更多
关键词 Duffing oscillator double peak probability density function multiple scale method linearization method
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Developing theory of probability density function for stochastic modeling of turbulent gas-particle flows
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作者 Lixing ZHOU 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2018年第7期1019-1030,共12页
Turbulent gas-particle flows are studied by a kinetic description using a prob- ability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the... Turbulent gas-particle flows are studied by a kinetic description using a prob- ability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are di- rectly solved either using a finite-difference method for two-dimensional (2D) problems or using a Monte-Carlo (MC) method for three-dimensional (3D) problems. The proposed differential stress model together with the PDF (DSM-PDF) is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment (SOM) two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results. 展开更多
关键词 probability density function(PDF)modeling turbulent flow gas-particleflow
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Age-Related Changes in Probability Density Function of Pairwise Euclidean Distances between Multichannel Human EEG Signals
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作者 Mikhail Trifonov Vladimir Rozhkov 《Journal of Biosciences and Medicines》 2014年第4期19-23,共5页
The probability density functions (pdf’s) and the first order structure functions (SF’s) of the pairwise Euclidean distances between scaled multichannel human EEG signals at different time lags under hypoxia and in ... The probability density functions (pdf’s) and the first order structure functions (SF’s) of the pairwise Euclidean distances between scaled multichannel human EEG signals at different time lags under hypoxia and in resting state at different ages are estimated. It is found that the hyper gamma distribution is a good fit for the empirically derived pdf in all cases. It means that only two parameters (sample mean of EEG Euclidean distances at a given time lag and relevant coefficient of variation) may be used in the approximate classification of empirical pdf’s. Both these parameters tend to increase in the first twenty years of life and tend to decrease as healthy adults getting older. Our findings indicate that such age-related dependence of these parameters looks like as age- related dependence of the total brain white matter volume. It is shown that 15 min hypoxia (8% oxygen in nitrogen) causes a significant (about 50%) decrease of the mean relative displacement EEG value that is typical for the rest state. In some sense the impact of the oxygen deficit looks like the subject getting older during short-term period. 展开更多
关键词 EEG Development HYPOXIA probability density functions HYPER GAMMA Distribution
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Probability density function and estimation for error of digitized map coordinates in GIS
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作者 童小华 刘大杰 《Journal of Central South University of Technology》 SCIE EI CAS 2004年第1期69-74,共6页
Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in... Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in GIS does not obey the normal distribution but the p-norm distribution with a determinate parameter. Assuming that the error is random and has the same statistical properties, the probability density function of the normal distribution, Laplace distribution and p-norm distribution are derived based on the arithmetic mean axiom, median axiom and p-median axiom, which means that the normal distribution is only one of these distributions but not the least one. Based on this ideal distribution fitness tests such as Skewness and Kurtosis coefficient test, Pearson chi-square chi(2) test and Kolmogorov test for digitized data are conducted. The results show that the error in map digitization obeys the p-norm distribution whose parameter is close to 1.60. A least p-norm estimation and the least square estimation of digitized data are further analyzed, showing that the least p-norm adjustment is better than the least square adjustment for digitized data processing in GIS. 展开更多
关键词 probability density function distribution fitness test least p-norm estimation
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A Lagrangian-based flame index for the transported probability density function method
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作者 Zhen Lu Hua Zhou +2 位作者 Zhuyin Ren Yue Yang Hong G.Im 《Theoretical & Applied Mechanics Letters》 CSCD 2022年第1期30-34,共5页
We propose a new flame index for the transported probability density function(PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as t... We propose a new flame index for the transported probability density function(PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as the metrics to identify the combustion mode, with the Burke-Schumann solution as a reference. A priori validation of the flame index is conducted with a series of constructed turbulent partially premixed reactors. It indicates that the proposed flame index is able to identify the combustion mode based on the subgrid mixing information. The flame index is then applied the large eddy simulation/PDF datasets of turbulent partially premixed jet flames. Results show that the flame index separate different combustion modes and extinction correctly. The proposed flame index provides a promising tool to analyze and model the partially premixed flames adaptively. 展开更多
关键词 Flame index Transported probability density function Partially premixed combustion
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Reconstruction of Probability Density Function for Gamma Distribution
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作者 FAN Jinwei LI Zhongsheng TIAN Bin 《Journal of Donghua University(English Edition)》 EI CAS 2020年第4期327-333,共7页
The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the... The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types. 展开更多
关键词 small sample probability density function(PDF) gamma distribution early failure mean time between failures(MTBF)
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Auxiliary error and probability density function based neuro-fuzzy model and its application in batch processes
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作者 贾立 袁凯 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2015年第12期2013-2019,共7页
This paper focuses on resolving the identification problem of a neuro-fuzzy model(NFM) applied in batch processes. A hybrid learning algorithm is introduced to identify the proposed NFM with the idea of auxiliary erro... This paper focuses on resolving the identification problem of a neuro-fuzzy model(NFM) applied in batch processes. A hybrid learning algorithm is introduced to identify the proposed NFM with the idea of auxiliary error model and the identification principle based on the probability density function(PDF). The main contribution is that the NFM parameter updating approach is transformed into the shape control for the PDF of modeling error. More specifically, a virtual adaptive control system is constructed with the aid of the auxiliary error model and then the PDF shape control idea is used to tune NFM parameters so that the PDF of modeling error is controlled to follow a targeted PDF, which is in Gaussian or uniform distribution. Examples are used to validate the applicability of the proposed method and comparisons are made with the minimum mean square error based approaches. 展开更多
关键词 Batch process Auxiliary error model probability density function Neuro-fuzzy model
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Identification of Neuro-Fuzzy Hammerstein Model Based on Probability Density Function
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作者 方甜莲 贾立 《Journal of Donghua University(English Edition)》 EI CAS 2016年第5期703-707,共5页
A new identification method of neuro-uzzy Hammerstein model based on probability density function(PDF) is presented,which is different from the idea that mean squared error(MSE) is employed as the index function in tr... A new identification method of neuro-uzzy Hammerstein model based on probability density function(PDF) is presented,which is different from the idea that mean squared error(MSE) is employed as the index function in traditional identification methods.Firstly,a neuro-fuzzy based Hammerstein model is constructed to describe the nonlinearity of Hammerstein process without any prior process knowledge.Secondly,a kind of special test signal is used to separate the link parts of the Hammerstein model.More specifically,the conception of PDF is introduced to solve the identification problem of the neuro-fuzzy Hammerstein model.The antecedent parameters are estimated by a clustering algorithm,while the consequent parameters of the model are identified by designing a virtual PDF control system in which the PDF of the modeling error is estimated and controlled to converge to the target.The proposed method not only guarantees the accuracy of the model but also dominates the spatial distribution of PDF of the model error to improve the generalization ability of the model.Simulated results show the effectiveness of the proposed method. 展开更多
关键词 probability clustering guarantees separate converge prior generalization conception squared nonlinearity
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A New Maximum Entropy Probability Function for the Surface Elevation of Nonlinear Sea Waves 被引量:21
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作者 张立振 徐德伦 《China Ocean Engineering》 SCIE EI 2005年第4期637-646,共10页
Based on the maximum entropy principle a new probability density function (PDF) f(x) for the surface elevation of nonlinear sea waves, X, is derived through performing a coordinate transform of X and solving a var... Based on the maximum entropy principle a new probability density function (PDF) f(x) for the surface elevation of nonlinear sea waves, X, is derived through performing a coordinate transform of X and solving a variation problem subject to three constraint conditions of f( x ). Compared with the maximum entropy PDFs presented previously, the new PDF has the following merits: (1) it has four parameters to be determined and hence can give more refined fit to observed data and has wider suitability for nonlinear waves in different conditions; (2) these parameters are expressed in terms of distribution moments of X in a relatively simple form and hence are easy to be determined from observed data; (3) the PDF is free of the restriction of weak nonlinearity and possible to be used for sea waves in complicated conditions, such as those in shallow waters with complicated topography; and (4) the PDF is simple in form and hence convenient for theoretical and practical uses. l.aboratory wind-wave experiments have been conducted to test the competence of the new PDF for the surface elevation of nonlinear waves. The experimental results manifest that the new PDF gives somewhat better fit to the laboratory wind-wave data than the well-known Gram-Charlier PDF and beta PDF. 展开更多
关键词 maximum entropy probability density function (PDF) Gram-Charlier PDF beta PDF
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Strong Consistency for the Kernal Estimates of the Random Window Width of the Density Function and its Derivatives Under Φ-Mixing Samples
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作者 樊家琨 《Chinese Quarterly Journal of Mathematics》 CSCD 1993年第3期52-56,共5页
In the paper,we study the strong uniform consistency for the kernal estimates of random window w■th of density function and its derivatives under the condition that the sequence{X_n}of the ■ are the identically Φ-m... In the paper,we study the strong uniform consistency for the kernal estimates of random window w■th of density function and its derivatives under the condition that the sequence{X_n}of the ■ are the identically Φ-mixing random variabks. 展开更多
关键词 Φ-mixing sample probability density function random window width kemal estimate strng uniform consistency
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