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Short-Term Actuarial Math Old Exam C Forum

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  #1  
Old 09-24-2016, 04:24 PM
msmnCasualty msmnCasualty is offline
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Default Shortcut kth central moment

I remember taking statistics course in college and my professor taught us shortcut on how to calculate kth central moments where k = 3,4,....n

Could anyone explain the shortcut to me? I really appreciate your time and help!

Thanks!
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  #2  
Old 09-26-2016, 02:19 AM
Demo99 Demo99 is offline
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I'm not aware of any shortcut, but I will say I haven't seen many third or higher moments on this exam. Are you talking about taking the nth derivative of the moment generating function at 0?
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Old 09-26-2016, 09:25 PM
msmnCasualty msmnCasualty is offline
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Quote:
Originally Posted by Demo99 View Post
I'm not aware of any shortcut, but I will say I haven't seen many third or higher moments on this exam. Are you talking about taking the nth derivative of the moment generating function at 0?
Nope, I'm talking about calculating skewness and kurtosis which involve third and fourth central moment.. such that E(X_i - X_bar)^k

Common approach is to expand the term through algebra but the equation will get messy sometimes so I just want to find the alternative to save time.
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Old 09-27-2016, 08:45 PM
Z3ta Z3ta is offline
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The third central moment is the 3rd derivative of the natural log of the moment generating function at 0...

ie. the third central moment is

The larger central moments aren't quite as easy.

For instance the fourth central moment is or equivalently

Is this what you were thinking of?

There are nice shortcuts for certain distributions of course (like the normal). Could you have been remembering a shortcut that only works with certain distributions?

Last edited by Z3ta; 09-27-2016 at 08:51 PM..
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Old 10-01-2016, 02:44 AM
Demo99 Demo99 is offline
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I just saw something in Mahler today (only for Poisson). The second central moment of a Compound Poisson Count and Gamma Severity distribution was E[N]*E[X^2] and the 3rd central moment was E[N]*E[X^3].

Last edited by Demo99; 10-01-2016 at 02:47 PM.. Reason: correction
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Old 10-01-2016, 02:46 AM
BuddhaWilliams BuddhaWilliams is offline
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Quote:
Originally Posted by Demo99 View Post
I just saw something in Mahler today. The second central moment of a Compound Poisson Count and Gamma Severity distribution was E[N]*E[X^2] and the 3rd central moment was E[N]*E[X^3].
Thats not for finding skewness/kutosis though. Thats for finding the variance of aggregate losses
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Old 10-01-2016, 02:54 AM
Demo99 Demo99 is offline
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He used it for finding the skewness of the aggregate distribution. Probably won't work for just severity or counts alone.
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Old 10-01-2016, 05:02 AM
BuddhaWilliams BuddhaWilliams is offline
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You right
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Old 10-01-2016, 11:28 AM
Academic Actuary Academic Actuary is offline
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Quote:
Originally Posted by Demo99 View Post
He used it for finding the skewness of the aggregate distribution. Probably won't work for just severity or counts alone.
Formula based upon Poisson frequency. Does not hold in general.
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Old 10-01-2016, 11:54 AM
BuddhaWilliams BuddhaWilliams is offline
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You right
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