Compact Fluorescent Plug In Light Bulbs are not like normal light bulbs or even normal screw in compact fluorescent light bulbs. CFL light bulbs contain a gas that ignites which lights up. Screw in type cfls have a ballast on the end (the part that screws) that ignites this gas. Plug in cfl bulbs do not contain their own ballast. The ballast is build in to the lighting fixture.
Most often than not, it's the ballast on these that goes bad. And since plug in compact fluorescent bulbs don't have a ballast, you'll need to look at the ballast that is connected to your lighting fixture. Only 1 in several thousand plug in compact fluorescent plug in light bulbs do we see that it's the bulb that is the problem and that's because the bulb has concealed damage.