GEOSTATS: Sampling and independence
- I would like to pose an exercise for the group to investigate the role of
the assumption of statistical independence in statistical inference. I pose
the basic problem of estimating the mean of a random function over a finite
space. There are two types of means we may be interested in: the finite
population mean (i.e. the mean of the single realization from which we
sample) vs superpopulation mean (i.e. the mean of the process from which the
single realization came).
A) consider estimation of the finite population mean. Standard statistical
methods say select a random sample from the population, calculate x_bar and
(s^2 /n) and a 95% confidence interval is given by x_bar +/- t_(0.975, df=n-1).
The question of sampling is do e get 95% coverage if we follow this
procedure when the underlying data are spatially correlated. To investigate
this do the following:
Generate one simulation from GSLIB or any algorithm for generation of
correlated random fields, say a 128 by 128 image with dx=dy=1.0 and
correlation function with range of influence 32 and nugget = 0.0.
Calculate the finite population mean (i.e. average of the 128*128 numbers).
Now draw a sample of size say 20 from this finite population and build a 95%
CI. Check to see if the CI contains the true mean of the image.
Repeat this step say 10,000 times and keep track of the number of times the
CIcontains the true mean. (Sample from the same image each time) If
sampling provides independence this should work properly and if not we
should not get the correctcoverage of 95% intervals.
B) Now to investigate the super population issue, repeat the above
experiment except with each resampling event, also simulate a new image.
Check the nominal coverage by comparing CI to the "theortical" mean from the
population you are simulating from.
Does random sampling form the basis for valid statistical inferences about
the super population mean, finite population mean, both or neither.
I would love to see players speculate to the list about how these
experiments will turn out before conducting them.
Finally, if generating 10000 images simulations takes too much time to make
the investigation, Consider using FFT methods to do the trick.
This sounds like homework!!!
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