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hark just went through a situation where she wound up canning her Sigma 70-200mm because it wasn't performing well. She opted for the 70-200mm f/2.8 as she has specific low light needs. Perhaps she can add details. You're talking 1 stop, which in
certain situations is a big deal. But high ISO performance is getting better and better, and lets just say I've yet to run into a situation where I cursed myself for having an f/4 - at least on longer zooms.
Nikon f4, but you really seem hung up on 2.8. You need to buy what you really want otherwise will always question whether you should've got the other. The tamron 2.8 VC is rated closest to the nikon
As Jake mentioned, I had the latest version of Sigma's 70-200mm f/2.8 OS lens. Before purchasing, I tested 3 different copies and bought the one that was sharpest (yes, the one I picked was noticeably sharper especially wide open). It worked well except in low light because its focus was off slightly then. It back focused on my D90 and D600 but front focused on my D610. Sigma would have calibrated it if I chose to do so, but after renting a Nikon 70-200mm f/2.8, the SOOC shots were noticeably sharper. There was some type of odd depth of field issue with the Sigma--as though the Sigma only had a few inches of depth of field although when using a DOF calculator, the depth should have been much more. I didn't have that issue with the Nikon.
As Rick suggested, you should look into Tamron's 70-200mm f/2.8 as the DXO Mark scores are closer to Nikon's than the Sigma's scores. I've seen some amazing shots here taken with the Sigma, but my copy didn't meet my expectations. Even my Nikon 70-300mm f/4.5-5.6 VR is sharper although not suitable for low light situations.
If my only choice was between a Sigma 70-20mm f/2.8 and the Nikon 70-200mm f/4, I'd choose the Nikon. If you really NEED the extra stop, then consider the Tamron. All three lenses are close in price.