Here is the 10 year study 8" vs 4" gauge.
http://www.cocorahs.org/Media/docs/SMOI%202.2%20Ten-Year%20Gauge%20Comparison.pdfSummary I'll post:
4. CONCLUSIONS
Based on ten years of side by side
measurements of the NWS 8" diameter Standard
Rain Gauge and the similar design but small 4"
diameter clear plastic gauge, the 4" gauge
consistently showed more precipitation than the 8"
gauge. Most daily difference were 0.00” or 0.01”
but when summed over all precipitation events
each year, the differences amounted to 0.48" or
3% of the annual average of 15.96 inches. Annual
differences ranged from 0.8% to 5.8%. The
majority of the differences accrued during spring
and summer months. Large daily differences of
0.05" or greater were uncommon but typically
occurred during snow, heavy rain, hail, or high
wind. In rare instances of high winds and snow, or
heavy rains with hail, the 4" gauge would show
less precipitation than the 8" gauge.