hello,

I have a question about what is/should be typically done when kriging is

used for spatial interpolation of a process X(z) where z gives spatial

location (e.g. z=(x,y) with cartesian coordinates x,y) and X(z) has a

skewed continuous distribution with nonnegative support. For instance

lognormal.

Now,

if all data are in the form of point samples, X(z)'s can obviously be

transformed by taking logs to Y(z)=log(X(z)) which are exactly (with

lognormal X's) or approximately Gaussian, so that kriging can be done

comfortably (and the result backtransformed with easy correction for the

fact that E f(X) is generally not equal to f(E X), based on the formula

for lognormal expected value or Taylor expansion).

If at least some data are not point samples, but correspond to the

regional averages, then problem occurs due to the facts that: i) sum

of lognormals is not lognormal, ii) the log of the sum (or average)

of lognormals is not normal.

Obviously, one can do:

i) the kriging on logs anyway with some hand-waving (effectively

replacing sums by products based on delta method),

ii) or one can (quite inefficiently) work with original data without log

transformation and argue that at least method of moments estimators

are invoked (with proper weighting),

iii)or one can use some kind of Monte Carlo computationally-intensive

approach to compute likelihood (or posterior) based on sums of

lognormals.

At this point, I am not interested in either of the three. My question

is whether people used some other parametric family (it cannot be

lognormal) of marginal distributions with positive support, positive

skew, that is closed under convolution (or under taking weighted

averages, to be more general) - so that the regional averages and point

values will have distribution of the same type, differing only in

parameters (just like in normal case and real support case). One

possibility would be gamma, what about others?

Thanks in advance for any suggestions.

Best Regards

Ing. Marek Brabec, PhD