"Better" must be rigorously defined.
And then it can be rigorously argued :-)

For camera comparisons it is usual to make lens variations and effects as small as possible so that you are actually comparing the cameras. There will be some cases where a lens performs so terribly on an FX sensor and so well on a DX sensor that the D7000 will win BUT that is not the sort of lens that you will usually see in a test. Also, it is not usually the sort of lens a D800 user would probably care about when attempting to take good photos. And it is notionally possible that the best equivalent DX lens to a FF lens gives a better DX result - but this would usually be rare, as the FF cameras are the flagship models on which camera makers aim to produce superior camera + lens results.

One means by which you might "win" with the DX camera, which meets you criteria, is to use the same lens in each case under ideal conditions and to adjust subject distance etc to produce visibly identical images. In this case the DX camera is using the centre portion of the lens while the DX camera uses the whole lens. The effect on the actual image of factors such as dynamic range and given ISO signal to noise ratio are minimised by being able to use optimum lighting levels. Under such conditions the improved quality usually achieved by using the lens centre may well offset the DX sensors overall performance. [Example: I have an APSC Sony A77 with a 24 megapixel sensor and a DX Nikon D700 with a 12 megapixel sensor. In studio conditions with "good" lenses, for practical purposes the Sony would usually win. Take them to the circus and photograph a trapeze artist at work and the D700 would be utterly superior.]

Three measures of "better" which can be numerically expressed are

• The ISO setting at which a camera can achieve a specified signal to noise ratio

• The dynamic range achievable (in bits)

• Colour depth (also in bits)

each under specified conditions.

Figures for most DSLRs and large format cameras are available from
the "DxOMark Sensor" sensor comparison site..

The above link takes you to the ISO evaluation table - which is where my book mark is set, as high-ISO noise performance is what I most care about when looking at new results, but the other tabs cover color-depth, dynamic range and overall result.

The D7000 has an 1167 ISO rating and the D800 has a 2853 rating. The difference of 2853/1167 = 2.44, which is 1.27 stops - making the D7000 an extremely good APSC performer.
[Stops = log_base2(ISO ratio) = log_base10(ISO ratio) / log_base10(2) ]

I personally find these evaluations suspect as the reality seems to be slightly different in most cases than the table suggests. They compare sensors with different numbers of pixels by scaling the sensor noise readings DOWN by a factor of square root(megapixels high/ 12). ie they use 12 megapixels as their standard sensor size and reduce the noise measured with large megapixel sensors to scale the results. This is equivalent to downsizing the large images with an algorithm that improves the signal to noise ratio by sqrt(mp/12).

My suspicion - based on both overall image appearance and on extensive pixel peeping - is that this is too generous to noisy high mp count sensors, i.e. when you scale your 36 mP D800 image down to their reference 12 mp size I suspect that it does not get sqrt(36/12) = 1.7 times less noisy as they deem.

So if I am correct (which is moot) the D7000 may be closer noise wise to the D800 than this chart suggests.

DxO weight their 3 scores to provide an overall rating. Some "overall" results are

• D800 = 95
• D4 = 89
• D3s = 82 !!
• D3200 = 81 !!!!!!!!!!!!!!!!!
• 5D MkIII = 81
• D7000 = 80
• D700 = 80

I recently bought a D700.
I would not have considered buying a D3200.
That may say more about me than about the cameras :-).

I rather feel DxO's weightings may need looking at, but their overall rating is at least one available measure of "better" on which discussion can be based.
On this basis the D800 stands very clearly ahead of the pack.
But, the D7000 is still a very good camera.