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Journal of Applied Mathematics and Physics 2016年第1期4卷 130-139页

作者： Farrin Payandeh Touraj Mohammadpour Department of Physics Payame Noor University (PNU) Tehran Iran

In this paper, pursuing a new advised method called delta method which is basically similar to variational method, we find the ground and excited states, according to a typical quantum Hamiltonian. Moreover, applying this method, the upper bound values for the eigenenergies of the so-called ground and excited states are estimated. We will show that this new method, is as beneficial as the traditional variational method which is common in deriving eigenenergies of some of the quantum Hamiltonians. This method helps physics students to broaden their knowledge about the possible mathematical ways;they can use to obtain eigenenergies of some quantum Hamiltonians. The advantage of delta method to variational method is in its simplicity and reduction of the calculation procedures.

关键词： Quantum Mechanics Eigenenergies Alternative methods delta method

Using Pearson’s System of Curves to Approximate the Distributions of the Difference between Two Correlated Estimates of Signal-to-Noise Ratios: The Cases of Bivariate Normal and Bivariate Lognormal Distributions

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Open Journal of Statistics 2024年第3期14卷 207-227页

作者： Mohamed M. Shoukri Department of Epidemiology and Biostatistics Schulich School of Medicine and Dentistry University of Western Ontario London Ontario Canada

Background: The signal-to-noise ratio (SNR) is recognized as an index of measurements reproducibility. We derive the maximum likelihood estimators of SNR and discuss confidence interval construction on the difference between two correlated SNRs when the readings are from bivariate normal and bivariate lognormal distribution. We use the Pearsons system of curves to approximate the difference between the two estimates and use the bootstrap methods to validate the approximate distributions of the statistic of interest. methods: The paper uses the delta method to find the first four central moments, and hence the skewness and kurtosis which are important in the determination of the parameters of the Pearsons distribution. Results: The approach is illustrated in two examples;one from veterinary microbiology and food safety data and the other on data from clinical medicine. We derived the four central moments of the target statistics, together with the bootstrap method to evaluate the parameters of Pearsons distribution. The fitted Pearsons curves of Types I and II were recommended based on the available data. The R-codes are also provided to be readily used by the readers.

关键词： Signal-to-Noise Ratio Bivariate Distributions Bootstrap methods delta method Pearson System of Curves

Performance and uncertainty analysis of a short-term climate reconstruction based on multi-source data in the Tianshan Mountains region,China

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Journal of Arid Land 2020年第3期12卷 374-396页

作者： LI Xuemei Slobodan P SIMONOVIC LI Lanhai ZHANG Xueting QIN Qirui Faculty of Geomatics Lanzhou Jiaotong UniversityLanzhou 730070China National-Local Joint Engineering Research Center of Technologies and Applications for National Geographic State Monitoring Lanzhou 730070China Gansu Provincial Engineering Laboratory for National Geographic State Monitoring Lanzhou 730070China Facility for Intelligent Decision Support Department of Civil and Environmental EngineeringWestern UniversityLondon N6A 3K7Canada State Key Laboratory of Desert and Oasis Ecology Xinjiang Institute of Ecology and GeographyChinese Academy of SciencesUrumqi 830011China Qilian Alpine Ecology&Hydrology Research Station Key Laboratory of Ecohydrology of Inland River BasinNorthwest Institute of Eco-Environment and ResourcesChinese Academy of SciencesLanzhou 730000China University of Chinese Academy of Sciences Beijing 100049China

Short-term climate reconstruction,i.e.,the reproduction of short-term(several decades)historical climatic time series based on the relationship between observed data and available longer-term reference data in a certain area,can extend the length of climatic time series and offset the shortage of *** can be used to assess regional climate change over a much longer time *** on monthly grid climate data from a Coupled Model Inter-comparison Project phase 5(CMIP5)dataset for the period of 1850–2000,the Climatic Research Unit(CRU)dataset for the period of 1901–2000 and the observed data from 53 meteorological stations located in the Tianshan Mountains region(TMR)of China during the period of 1961–2011,we calibrated and validated monthly average temperature(MAT)and monthly accumulated precipitation(MAP)in the TMR using the delta,physical scaling(SP)and artificial neural network(ANN)*** and uncertainty during the calibration(1971–1999)and verification(1961–1970)periods were assessed and compared using traditional performance indices and a revised set pair analysis(RSPA)*** calibration and verification processes were subjected to various sources of uncertainty due to the influence of different reconstructed variables,different data sources,and/or different methods *** to traditional performance indices,both the CRU and CMIP5 datasets resulted in satisfactory calibrated and verified MAT time series at 53 meteorological stations and MAP time series at 20 meteorological stations using the delta and SP methods for the period of 1961–***,the results differed from those obtained by the RSPA *** showed that the CRU dataset produced a low degree of uncertainty(positive connection degree)during the calibration and verification of MAT using the delta and SP methods compared to the CMIP5 ***,the calibrated and verified MAP had a high degree of uncertainty(negative connection degree)regardless of the

关键词： climate reconstruction climate change delta method physical scaling method artificial neural network(ANN) CRU dataset CMIP5 dataset

A note on the estimation of variance for big BAF sampling

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Forest Ecosystems 2020年第4期7卷 819-832页

作者： Jeffrey H.Gove Timothy G.Gregoire Mark J.Ducey Thomas B.Lynch USDA Forest Service Northern Research Station271 Mast RoadDurham NH03824USA. School of Forestry&Environmental Studies Yale University360 Prospect StNew Haven CT06511USA. Department of Natural Resources and the Environment University of New Hampshire114 James HallDurham NH03824USA. Professor Emeritus Department of Natural Resource Ecology and ManagementOklahoma State UniversityAg Hall Room 008CStillwater OK74078USA.

Background:The double sampling method known as“big BAF sampling”has been advocated as a way to reduce sampling effort while still maintaining a reasonably precise estimate of volume.A well-known method for variance determination,Bruce’s method,is customarily used because the volume estimator takes the form of a product of random ***,the genesis of Bruce’s method is not known to most foresters who use the method in ***:We establish that the Taylor series approximation known as the delta method provides a plausible explanation for the origins of Bruce’s *** were conducted on two different tree populations to ascertain the similarities of the delta method to the exact variance of a ***,two alternative estimators for the variance of individual tree volume-basal area ratios,which are part of the estimation process,were compared within the overall variance estimation ***:The simulation results demonstrate that Bruce’s method provides a robust method for estimating the variance of inventories conducted with the big BAF *** simulations also demonstrate that the variance of the mean volume-basal area ratios can be computed using either the usual sample variance of the mean or the ratio variance estimators with equal accuracy,which had not been shown previously for Big BAF ***:A plausible explanation for the origins of Bruce’s method has been set forth both historically and mathematically in the delta *** most settings,there is evidently no practical difference between applying the exact variance of a product or the delta method—either can be used.A caution is articulated concerning the aggregation of tree-wise attributes into point-wise summaries in order to test the correlation between the two as a possible indicator of the need for further covariance augmentation.

关键词： Bitterlich sampling delta method Double sampling Forest inventory Horizontal point sampling Variance of a product

An approximate point-based alternative for the estimation of variance under big BAF sampling

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Forest Ecosystems 2021年第3期8卷 439-457页

作者： Thomas B.Lynch Jeffrey H.Gove Timothy G.Gregoire Mark J.Ducey Professor Emeritus Department of Natural Resource Ecology and ManagementOklahoma State UniversityAg Hall Room 008CStillwater 74078OKUSA USDA Forest Service Northern Research Station271 Mast RoadDurham 03824NHUSA Yale School of Environment Yale University360 Prospect StNew Haven 06511CTUSA Department of Natural Resources and the Environment University of New Hampshire114 James HallDurham 03824NHUSA

Background:A new variance estimator is derived and tested for big BAF(Basal Area Factor)sampling which is a forest inventory system that utilizes Bitterlich sampling(point sampling)with two BAF sizes,a small BAF for tree counts and a larger BAF on which tree measurements are made usually including DBHs and heights needed for volume ***:The new estimator is derived using the delta method from an existing formulation of the big BAF estimator as consisting of three sample *** new formula is compared to existing big BAF estimators including a popular estimator based on Bruce’s ***:Several computer simulation studies were conducted comparing the new variance estimator to all known variance estimators for big BAF currently in the forest inventory *** simulations the new estimator performed well and comparably to existing variance ***:A possible advantage of the new estimator is that it does not require the assumption of negligible correlation between basal area counts on the small BAF factor and volume-basal area ratios based on the large BAF factor selection trees,an assumption required by all previous big BAF variance estimation *** this correlation was negligible on the simulation stands used in this study,it is conceivable that the correlation could be significant in some forest types,such as those in which the DBH-height relationship can be affected substantially by density perhaps through *** derived a formula that can be used to estimate the covariance between estimates of mean basal area and the ratio of estimates of mean volume and mean basal *** also mathematically derived expressions for bias in the big BAF estimator that can be used to show the bias approaches zero in large samples on the order of 1n where n is the number of sample points.

关键词： Bitterlich sampling delta method Double sampling Estimator bias Forest inventory Horizontal point sampling Variance of a product Volume basal area ratio Covariance estimation

Quasi-Negative Binomial: Properties, Parametric Estimation, Regression Model and Application to RNA-SEQ Data

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Open Journal of Statistics 2022年第2期12卷 216-237页

作者： Mohamed M. Shoukri Maha M. Aleid Department of Epidemiology and Biostatistics Schulich School of Medicine and Dentistry University of Western Ontario London Canada Department of Biostatistics Epidemiology and Scientific Computing King Faisal Specialist Hospital and Research Center Riyadh Saudi Arabia

Background: The Poisson and the Negative Binomial distributions are commonly used to model count data. The Poisson is characterized by the equality of mean and variance whereas the Negative Binomial has a variance larger than the mean and therefore both models are appropriate to model over-dispersed count data. Objectives: A new two-parameter probability distribution called the Quasi-Negative Binomial Distribution (QNBD) is being studied in this paper, generalizing the well-known negative binomial distribution. This model turns out to be quite flexible for analyzing count data. Our main objectives are to estimate the parameters of the proposed distribution and to discuss its applicability to genetics data. As an application, we demonstrate that the QNBD regression representation is utilized to model genomics data sets. Results: The new distribution is shown to provide a good fit with respect to the “Akaike Information Criterion”, AIC, considered a measure of model goodness of fit. The proposed distribution may serve as a viable alternative to other distributions available in the literature for modeling count data exhibiting overdispersion, arising in various fields of scientific investigation such as genomics and biomedicine.

关键词： Queuing Models Overdispersion Moment Estimators delta method Bootstrap Maximum Likelihood Estimation Fisher’s Information Orthogonal Polynomials Regression Models RNE-Seq Data

Computing Confidence Intervals for the Postal Service’s Cost-Elasticity Estimates

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Open Journal of Statistics 2021年第5期11卷 607-619页

作者： Bzhilyanskaya Y. Lyudmila Margaret M. Cigno Soiliou D. Namoro US Postal Regulatory Commission Washington DC USA

This paper provides methods for assessing the precision of cost elasticity estimates when the underlying regression function is assumed to be polynomial. Specifically, the paper adapts two well-known methods for computing confidential intervals for ratios: the delta-method and the Fieller method. We show that performing the estimation with mean-centered explanatory variables provides a straightforward way to estimate the elasticity and compute a confidence interval for it. A theoretical discussion of the proposed methods is provided, as well as an empirical example based on publicly available postal data. Possible areas of application include postal service providers worldwide, transportation and electricity.

关键词： Volume Variability Confidence Interval Ratio Parameter delta method Fieller method