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: