[R] comparing segments of a time series

Thomas W Blackwell tblackw at umich.edu
Thu Aug 21 23:04:36 CEST 2003

Tony  -

I happen to have a copy of Erich L. Lehmann and H.J.M D'Abrera (1975)
Nonparametrics: Statistical Methods based on Ranks.  Holden-Day, SF,
sitting beside me on my desk this afternoon.  What you want is covered
in Section 2.7L on pp. 104-105, titled "Scale tests with unknown location".

Lehmann says:  "As was pointed out in Ch. 1 Sec. 6, the assumption
made there -- and in the preceding Secs. 7I and 7J -- that the two
[samples] being compared measure the same quantity, often cannot be
trusted.  If xi and eta denote the quantities measured [in the two
samples], it is then natural to estimate xi and eta by estimates,
say, hat{xi} and hat{eta} and to apply the scale test to the adjusted
observations  X_1 - hat{xi} [etc].  The significance level of the
resulting test will be affected by the substitution of hat{xi} and
hat{eta} for xi and eta, but one may hope that the effect will not
be serious and that asymptotically the significance level will retain
its value.  Conditions under which this is the case are given by
Sukhatme (1958), Raghavachari (1965a) and Gross (1966)."

I don't think the situation has changed much in the ensuing 30 years.
There probably isn't any exact test for your situation of unknown
location.  I highly recommend Lehmann as a reference, especially
Section 1.6, pp. 32-34.

-  tom blackwell  -  u michigan medical school  -  ann arbor  -

On Thu, 21 Aug 2003, Tony Marlboro wrote:

> 	I have a time series of 38 wintertime average snow depths
> measured at a particular meteorological station.  The data appear to
> undergo a "climate shift" in the early 1980s:  before the shift the
> mean and sd are 152 +/- 58, after the shift 92 +/- 36.  The
> distribution is not normal; there's a hard limit at zero of course and
> there are outlier years with very high snowfall.  I don't feel
> justified making a log transformation on the data, so I'd rather use
> distribution-free methods.
> 	I would like to have statistical justification for the
> statement that the snow depth in the second period is less than in the
> first half, and that the variability decreased as well.  For the
> difference in central measures, I am using the (unpaired) wilcox.test,
> but I really have no idea how to address the question of changes in
> variability using nonparametric means.  Any ideas?
> 	Thanks,
> 		Tony

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