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I am going to do a project based on kriging. The targeted area would be as big as United States and my grid would be 10km*10km. I am wondering the differences of efficiencies for the methods implemented in geoR package and ArcGIS. Have anyone done any research on the performance yet. I would prefer geoR if it works well.

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Usually, the big issue is capabilities rather than performance, unless you are doing extensive compute-intensive simulation. A 10 km grid in the US would be fairly small (it should need only about 75,000 nodes). geoR (along with geoRglm) implements a wider variety of models (such as generalized linear stochastic fields and Bayes priors) than anything available for ArcGIS; by means of its integration with R, it also supports far more exploratory and diagnostic tools (which are essential for defensible modeling). geoR is harder to use than Geostatistical Analyst. –  whuber Feb 12 '12 at 0:00
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up vote 3 down vote accepted

I would also suggest to consider the gstat package (either standalone or the R package), which -in my experience - is a bit between geoR and Geostatistical Analyst: you do not have the very advanced models of geoR but it's more efficient from a computing perspective (its core is in C).

Also, you might be interested in the automap and intamap packages.

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