# difference between limiting and special case

In Mathematics and Statistics we see generalized distributions having a number of parameters. varying the values of these parameters we get special or limiting distributions of the generalized distribution. what I am not clear about is the difference between special and limiting case? could anyone explain the difference.

• Consider some family indexed by $\theta$ in $(0,+\infty)$. A special case could be $\theta=1$. Limiting cases would be limits when $\theta\to0^+$ and/or when $\theta\to\infty$. Example: define the PDF $f_\theta(x)=\theta x^{\theta-1}\mathbf 1_{(0,1)}(x)$ for every $\theta$ in $(0,+\infty)$, can you identify the limits when $\theta\to0^+$ and when $\theta\to\infty$ and show they are not in the original family $(f_\theta)_{\theta>0}$?
– Did
Jul 21, 2015 at 8:01
• @Did I get your point clearly, thanks. Jul 21, 2015 at 8:04
• Great. So... which limits did you find?
– Did
Jul 21, 2015 at 8:05
• @Did ,I just needed to clear my understanding of limiting and special case. which you did nicely just now. Jul 21, 2015 at 8:08
• I know that--and I am asking you if you solved the example I proposed as an illustration of the concepts. If you did not, I will regret it, but you could just say so.
– Did
Jul 21, 2015 at 8:11