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: