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Certification Method and Definitions

Model:

The general statistical model assumed for the univariate datasets is


where

denotes the population mean,
denotes the population standard deviation; and
denotes the population lag-1 autocorrelation coefficient.

Methodology:

For all datasets multiple precision calculations (accurate to 500 digits) were made using the post-processor 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 (without error) 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 roundoff error, and will produce results which differ slightly from these certified values.

Definitions:

For sake of testing computational stability, we set aside statistical estimation efficiency issues and estimate by the sample mean ; estimate by the sample standard deviation ; and estimate by the sample lag-1 autocorrelation coefficient .

Sample Mean

The sample mean is defined as




Sample Standard Deviation

The sample standard deviation is defined as




Sample Lag-1 Autocorrelation Coefficient

The sample lag-1 autocorrelation coefficient may have several definitions; we choose to use the definition popular in time series analysis: