This Book is intended to be textbook studied for undergraduate course in multivariate analysis. This book is designed to be used in semester system. In order to achieve the goals of the book, it is divided into the following chapters (as done in the first edition 2019). Chapter One introduces matrix algebra. Chapter Two devotes to Linear Equation System Solution with quadratic forms, Characteristic roots & vectors. Chapter Three discusses Partitioned Matrices and how to get Inverse, Jacobi and Hessian matrices. Chapter Four deals with Multivariate Normal Distribution (MVN). Chapter Five concern with Joint, Marginal and Conditional Normal Distribution, independency and correlations. While the revised new chapters have been added (as the current second edition 2024). Chapter six introduces mean vector estimation and covariance matrix estimation. Chapter seven devotes to testing concerning mean: one sample mean, and two sample mean. Chapter eight discusses special case of factorial analysis which is principal components analysis. Chapter nine deals with discriminant analysis. While chapter ten deals with cluster analysis. Many solved examples are intended in this book, in addition to a variety of unsolved relied problems at the end of each chapter to enrich the statistical knowledge of the readers.
Elastic electron scattering form factors, charge density distributions and charge,
neutron and matter root mean square (rms) radii for P
24
PMg, P
28
PSi and P
32
PS nuclei are
studied using the effect of occupation numbers. Single-particle radial wave functions
of harmonic-oscillators (HO) potential are used. In general, the results of elastic
charge form factors showed good agreement with experimental data. The occupation
numbers are taken to reproduce the quantities mentioned above. The inclusion of
occupation numbers enhances the form factors to become closer to the data. For the
calculated charge density distributions, the results show good agreement with
experimental data except the fail to
The experimental proton resonance data for the reaction P+48Ti have been used to calculate and evaluate the level density by employed the Gaussian Orthogonal Ensemble, GOE version of RMT, Constant Temperature, CT and Back Shifted Fermi Gas, BSFG models at certain spin-parity and at different proton energies. The results of GOE model are found in agreement with other, while the level density calculated using the BSFG Model showed less values with spin dependence more than parity, due the limitation in the parameters (level density parameter, a, Energy shift parameter, E1and spin cut off parameter, σc). Also, in the CT Model the level density results depend mainly on two parameters (T and ground state back shift energy, E0), which are app
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The estimation of the stressÙ€ strength reliability of Invers Kumaraswamy distribution will be introduced in this paper based on the maximum likelihood, moment and shrinkage methods. The mean squared error has been used to compare among proposed estimators. Also a Monte Carlo simulation study is conducted to investigate the performance of the proposed methods in this paper.
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In this paper, Bayes estimators of the parameter of Maxwell distribution have been derived along with maximum likelihood estimator. The non-informative priors; Jeffreys and the extension of Jeffreys prior information has been considered under two different loss functions, the squared error loss function and the modified squared error loss function for comparison purpose. A simulation study has been developed in order to gain an insight into the performance on small, moderate and large samples. The performance of these estimators has been explored numerically under different conditions. The efficiency for the estimators was compared according to the mean square error MSE. The results of comparison by MSE show that the efficiency of Bayes est
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In this paper, a new class of ordinary differential equations is designed for some functions such as probability density function, cumulative distribution function, survival function and hazard function of power function distribution, these functions are used of the class under the study. The benefit of our work is that the equations ,which are generated from some probability distributions, are used to model and find the solutions of problems in our lives, and that the solutions of these equations are a solution to these problems, as the solutions of the equations under the study are the closest and the most reliable to reality. The existence and uniqueness of solutions the obtained equations in the current study are dis
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The fuzzy assignment models (FAMs) have been explored by various literature to access classical values, which are more precise in our real-life accomplishment. The novelty of this paper contributed positively to a unique application of pentagonal fuzzy numbers for the evaluation of FAMs. The new method namely Pascal’s triangle graded mean (PT-GM) has presented a new algorithm in accessing the critical path to solve the assignment problems (AP) based on the fuzzy objective function of minimising total cost. The results obtained have been compared to the existing methods such as, the centroid formula (CF) and centroid formula integration (CFI). It has been demonstrated that operational efficiency of this conducted method is exquisitely deve
... Show MoreThe fuzzy assignment models (FAMs) have been explored by various literature to access classical values, which are more precise in our real-life accomplishment. The novelty of this paper contributed positively to a unique application of pentagonal fuzzy numbers for the evaluation of FAMs. The new method namely Pascal's triangle graded mean (PT-GM) has presented a new algorithm in accessing the critical path to solve the assignment problems (AP) based on the fuzzy objective function of minimising total cost. The results obtained have been compared to the existing methods such as, the centroid formula (CF) and centroid formula integration (CFI). It has been demonstrated that operational efficiency of this conducted method is exquisitely develo
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This article discusses the estimation methods for parameters of a generalized inverted exponential distribution with different estimation methods by using Progressive type-I interval censored data. In addition to conventional maximum likelihood estimation, the mid-point method, probability plot method and method of moments are suggested for parameter estimation. To get maximum likelihood estimates, we utilize the Newton-Raphson, expectation -maximization and stochastic expectation-maximization methods. Furthermore, the approximate confidence intervals for the parameters are obtained via the inverse of the observed information matrix. The Monte Carlo simulations are used to introduce numerical comparisons of the proposed estimators. In ad
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The Khor Mor gas-condensate processing plant in Iraq is currently facing operational challenges due to foaming issues in the sweetening tower caused by high-soluble hydrocarbon liquids entering the tower. The root cause of the problem could be liquid carry-over as the separation vessels within the plant fail to remove liquid droplets from the gas phase. This study employs Aspen HYSYS v.11 software to investigate the performance of the industrial three-phase horizontal separator, Bravo #2, located upstream of the Khor Mor sweetening tower, under both current and future operational conditions. The simulation results, regarding the size distribution of liquid droplets in the gas product and the efficiency gas/liquid separation, r
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