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Misra1b
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Misra1b
Results
Certification Method & Definitions



Model:


The general statistical model assumed for the nonlinear least squares regression problems is



where y denotes the response (dependent) variable, x denotes predictor (independent) variables and (unsubscripted) denotes the vector of p unknown parameters to be estimated. The specific functional form for each dataset is given in the header information provided on each page.



Methodology:


The certified values for the nonlinear least squares regression problems were obtained using 128-bit precision, with the reported results confirmed by at least two different algorithms and software packages using analytic derivatives.



Definitions:

Estimates of , ,... ,

The certified values for the estimates, b = (b1, b2,..., bp)T, of the true model parameters, , ,... , are those that produced the smallest residual sum of squares, i.e.



where n denotes the number of observations. Under the assumption that

,

it follows that these are the maximum likelihood estimators.


Standard Deviation of the Estimates of , ,... ,

The certified values for the standard deviations of the estimates of the model parameters are the square roots of the diagonal elements of the asymptotic covariance matrix,



where



J denotes the Jacobian matrix with ijth element



evaluated at the current values of the parameters, b1, b2,..., bp, and n and p denote the number of observations and number of parameters, respectively.


Residual Sum of Squares

The certified value of the residual sum of squares is defined by



where n denotes the number of observations.


Residual Standard Deviation

The certified value of the residual standard deviation is defined by



where n and p denote the number of observations and number of parameters, respectively.


Residual Degrees of Freedom

The certified value of the residual degrees of freedom is defined by n-p, where n and p denote the number of observations and number of parameters, respectively.