Parameters estimation methods for different probability density functions.
Mkolesia, Andrew Chikondi Peter
Mkolesia, Andrew Chikondi Peter
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Abstract
Natural and physical phenomena are modeled by non-linear mathematical models. The non-linearity of the models poses a challenge in the estimation of parameters that model such phenomena. Current methods of estimation use iterative routines which require provision of initial guess values (IGV) of the unknown parameters in order to optimize the model parameters. To provide IGVs one should have expert knowledge in the eld being studied, and this is always more of an art than science. Providing IGVs that are not near the optimal solution may result into, long computation time and also failure to converge to the required optimal solution in some instances. The objective of the study was to develop parameter estimation methods which do not require initialization of the non-linear models' parameters in order to provide optimal estimates. Non-linear statistical models that can be formulated as rst order linear ordinary differential equations are considered. For parameter estimation, the models are rstly linearized using the differential techniques. Formulated models are linear in the unknown parameters, that are estimated using ordinary least-squares methods. In this work the three mixture Gaussian models, one-parameter Rayleigh, two-parameter Rayleigh distributions are considered in the one-dimensional parameter estimation framework. In the two-dimensional estimation framework, a multivariate Gaussian is considered. Three methods and theorems are proposed and proven using appropriate numerical examples. The methods are also tested on real life data to check their accuracy and statistical properties. A novel method for the recognition of multiple Gaussian patterns (RoMGP) is formulated and a theorem is developed to estimate the multiple Gaussian patterns. Another method is developed for the parameter estimation for the Rayleigh (one and two) parameter distributions. The technique is to linearize the density function via differential methods. The estimation is done using the ordinary least-squares method through the minimization of the formulated goal function. Two data sets one simulated and the other real data, are used to assess the performance of the proposed differential least-squares method (DLSM). Graphical methods are also used to compare the DLSM and the maximum likelihood estimator (MLE) on the real data. It is shown that the proposed DLSM works well when the sample size is n 15. A method for estimating the parameters of a multivariate Gaussian using the principle of n-cross sections (PCS) is proposed and tested using numerical analysis. The PCS uses hyperplanes that splice the multivariate Gaussian distribution, the generated one-dimensional distributions are in turn estimated using the minimization of the goal function and the estimation of the
parameters are done by the generated systems of equations from the hyperplanes. The proposed methods can be adopted to compute the parameters of mixed density function models and data that follows a Rayleigh distribution. The methods can be used or adopted to compute the initial guess values for the maximum likelihood estimator, method of moments estimator (MME) or other iterative routines, for both procedural and theoretical Statistics, since the RoMGP, DLSM and PCS uses nontrivial assumptions non the data.
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Submitted in partial ful filment of the requirements for the degree, Doctor of Philosophy in Science in the Department of Mathematics and Statistics, Faculty of Science at the Tshwane University of Technology.
Date
2017-03-02
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Tshwane University of Technology
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Keywords
Non-linear, Mathematical, Non-linearity, Equations, Two-dimensional, Density