Aperture is specified as an f-number which is actually a ratio between the diameter of the aperture opening to the lens diameter. So a lower f-number is larger than a higher f-number in the same way that 1/2 is larger than 1/4. For a more in depth explanation, have a look at this short video:
Is it true f/1.8 – f/2.5 has the faster lens speed than f/1.8 – f/5.6 because the former has the smaller minimum aperture f/2.5 than the latter f/5.6?
I see that the Olympus XZ-2 and Pentax MX-1 list their maximum apertures as f/1.8-f/2.5, so I'm guessing this is where you got that figure. This means that the camera has a variable maximum aperture of f/1.8 at the wide end (zoom out) and f/2.5 at the telephoto end (zoomed in).
If one camera lists f/1.8-f/2.5 and another lists f/1.8-f/5.6 under aperture, this is merely indicating that they have variable maximum apertures. Between the two, f/2.5 and f/5.6 refer to each camera's maximum aperture at the telephoto end. Your use of "minimum aperture" is incorrect.
Looking at the Pentax MX-1 it's maximum aperture range is f/1.8-f/2.5 with a minimum aperture of f/8. The minimum aperture is often times not listed on compacts for some reason. I had to do a couple of searches to find that.
To specifically answer your question, it is true that the lens with maximum apertures of f/1.8-f/2.5 is faster than a lens with maximum apertures of f/1.8-f/5.6. However, focal length is also a factor. Usually the longer the focal length gets the smaller the maximum aperture will get. A compact/bridge like the Panasonic FZ200 has a constant aperture of f/2.8, which means that it has the same maximum aperture at the wide and telephoto ends.
If so, what compact zoom camera has the smallest minimum aperture (highest f-number) in the market? Are they Olympus XZ-2 and Pentax MX-1? Is there any other compact zoom camera has the minimum aperture smaller than f/2.5 in the market? I am not talking about the maximum but only minimum aperture (highest f-number) on the compact zoom camera. Thank you for your answer.
Your question is confusing because taking the figures you've given, what you are asking seems to be "what camera has a wider maximum aperture than f/2.5 at the telephoto end?". It's hard to answer because the maximum aperture range by itself isn't a good judge of a camera.
For instance, the Pentax MX-1 has a maximum aperture range of f/1.8-f/2.5 and a focal length range of 28-112mm. The Panasonic FZ200 has a constant aperture of f/2.8 and a focal length range of 25-600mm. While the maximum apertures of the MX-1 are wider than the FZ200's constant aperture, the FZ200 has a much larger focal length range. However the MX-1 has a larger sensor.
The Sony RX100 has a maximum aperture range of f/1.8-f/4.9 and a focal length range of 28-100mm. If you look at the maximum aperture and focal length range of the MX-1 it would seem to be the better camera. However the RX100 has a much larger sensor which means it will perform better in low light despite having a smaller maximum aperture at the telephoto end.
My point is, when looking at cameras you have to consider several factors not just aperture ranges and balance them with your needs.