Advances and Applications in Statistics
Volume 49, Issue 1, Pages 1 - 20
(July 2016) http://dx.doi.org/10.17654/AS049010001 |
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A METHOD OF MOMENTS APPROACH TO PARAMETER ESTIMATION IN INTRINSICALLY NONLINEAR REGRESSION MODELS
Trijya Singh
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Abstract: In an intrinsically nonlinear model, parameters occur nonlinearly and the model cannot be made linear in parameters by any transformation. In view of this, normal equations obtained by equating the partial derivatives of the sums of squares of errors (SSE), with respect to parameters, do not yield solutions in close explicit form. So, we have to use iterative procedures for obtaining the least squares estimates. Alternatively, one can directly minimize SSE for a choice of parameters by using nonlinear optimization algorithms. However, in both cases, we need good initial estimates to start the iterations. In nonlinear regression, if many parameters are involved, then the surface of SSE is often quite rough and full of spikes and troughs. Therefore, unless the initial estimates are close to the true least squares estimates, we may face serious problems. The optimization algorithm may end up with a local minimum or we may need a lot of iterations for convergence to a global minimum. It is also possible in some ill-conditioned situations that convergence does not occur at all. Consequently, we need at least two sets of good initial estimates to ensure global convergence. Some methods that exist in literature are only applicable to equally spaced observations which are a serious limitation. To circumvent the above difficulties, we have developed a methodology based on the method of moments for finding the initial estimates. The methodology can be applied to both equally and unequally spaced observations. Using a published data set, we have compared the performance of the proposed methods with that of an existing method. We have demonstrated that the proposed methods perform better than the existing method. Moreover, the proposed methods are shown to work well in the case of unequally spaced observations as well. |
Keywords and phrases: nonlinear regression, theory of least squares, method of moments, nonlinear optimization. |
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