Mixture distribution

[ellenlcy]

Hi,

In Chapter 4 of Miller, it says

If two normal distributions have the same mean, they combine (mix) to produce mixture
distribution with leptokurtosis (heavy-tails). Otherwise, mixtures are infinitely flexible.


what does that 'otherwise' mean? Does it mean if those two normals have different mean? Or the mixture is not leptokurtic? Then mixtures are infinitely flexible.

And do the same mean and leptokurtosis are neccessarily true for both direction?

Thanks!

 

[ShaktiRathore]

Hi,
Yes Miller i think mean two normal distributions have different mean only.Yes its referring to both directions left and right tail both.
When two normal distributions have the same mean: mixture distribution with leptokurtosis (heavy-tails)
otherwise if two normal distributions different mean then mixtures are infinitely flexible.
thanks

 

[ellenlcy]

Thanks !

 

[David Harper CFA FRM]

@ellenlcy

I agree with the inference that Miller is saying if the means are different (in addition to different variances), then the mixture is infinitely flexible. The normal mixture is sort of an amazing combination of convenience (e.g., it is a natural for modeling two regimes: stable and crisis) and flexibility. If you like, here is a simple spreadsheet with the five inputs (i.e., two means, two variances and one weighting param), this allows you to visualize the variety: http://trtl.bz/mra-vol1-normal-mixture

This XLS is from Carol Alexander in MRA vol 1 who writes (please note she suggests that heavy tails are only implied by a variance mixture distribution; ie, when the means are identical):

"A variance mixture distribution is a mixture of distributions with the same expectations but different variances. The variance normal mixture density will be a symmetric but heavy-tailed density ...
The kurtosis is always greater than 3 in any mixture of two zero-mean normal components, so the mixture of two zero-mean normal densities always has a higher peak and heavier tails than the normal density of the same variance. The same comment applies when we take any number of component normal densities in the mixture, provided they all have the same expectation. More generally, taking several components of different expectations and variances in the mixture can lead to almost any shape for the density,12 whilst retaining the tractability of the analysis under normality assumptions. Even a mixture of just two normal densities can have positive or negative skewness, and light or heavy tails."