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Certification Method & Definitions


Model:

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



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


Methodology:

For all datasets, multiple precision calculations (accurate to 500 digits) were made using the preprocessor and FORTRAN subroutine package of Bailey (1995, available from NETLIB). Data were read in exactly as multiple precision numbers and all calculations were made with this very high precision. The results were output in multiple precision, and only then rounded to fifteen significant digits. These multiple precision results are an idealization. They represent what would be achieved if calculations were made without roundoff or other errors. Any typical numerical algorithm (i.e. not implemented in multiple precision) will introduce computational inaccuracies, and will produce results which differ slightly from these certified values.


Definitions:

Estimates of

The certified values for the estimates, , of the true parameters, , are defined by

,

the ordinary least squares estimates.


Residual Sum of Squares and Degrees of Freedom

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


with the certified value for the residual degrees of freedom defined by n-p.


Residual Mean Square

The certified value for the residual mean square is defined by



Residual Standard Deviation

The certified value for the residual standard deviation is defined by



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 variance-covariance matrix



R-Squared

When the intercept term     is included in the model, the certified value for R-squared is defined by



When the intercept term     is not included in the model, the certified value for R-squared is defined by



Regression Sum of Squares and Degrees of Freedom

When the intercept term     is included in the model, the certified value for the regression sum of squares is defined by


with the certified value for the regression degrees of freedom defined by p-1.

When the intercept term     is not included in the model, the certified value for the regression sum of squares is defined by


with the certified value for the regression degrees of freedom defined by p.


Regression Mean Square

The certified value for the regression mean square is defined by



F Statistic

The certified value for the F statistic is defined by