A concentrated Cauchy distribution with finite moments

作者：Ben Israel Adi^*

来源：Annals of Operations Research, 2013, 208(1): 147-153.

DOI：10.1007/s10479-011-0995-z

• Cauchy distribution
• Lorentz distribution
• Moments

The Cauchy distribution
(sic)(a, b) (x) = 1/pi b(1 + (x a/b)(2), -infinity < x < infinity,
with a,b real, b > 0, has no moments (expected value, variance, etc.), because the defining integrals diverge. An obvious way to "concentrate" the Cauchy distribution, in order to get finite moments, is by truncation, restricting it to a finite domain. An alternative, suggested by an elementary problem in mechanics, is the distribution
(sic)(g)(a, b)(x) = root 1+2bg/pi b(1 + x-a/b)(2)root 1 - 2bg(x-a/b)(2) a-root b/2g < x < a + root b/2g,
with a,b as above and a third parameter ga parts per thousand yen0. It has the Cauchy distribution C(a,b) as the special case with g=0, and for any g > 0, a" (g) (a,b) has finite moments of all orders, while keeping the useful "fat tails" property of a"(a,b).

• 出版日期2013-9