Domain of Attraction (Notes pg.107)

[PDEM200]

Hi David,
Can you please clarify how "domain of attraction" is different from Central Limit Theorem? Also, I thought the power/value of CLT is that it applies to all distributions as long as the sample size is sufficiently large. Therefore, would all distributions be considered to have a domain of attraction? Thank you.

 

[David Harper CFA FRM]

Hi Paul,

You are roughly correct about CLT: "the power/value of CLT is that it applies to all distributions [with finite variance] as long as the sample size is sufficiently large." So here the difference between the underlying distribution of random variables (X)--which as you say can be anything provided they have finite 1st/2nd moment--as opposed to the distribution to which the summation (& average by extension) of these variables converge. So, CLT says they (underlying with distribution of ????) will have a summation that *coverges* to normal: sum of X(?) ~ N(0,1) as sample increases.

The "domain of attraction" I need to lookup because I frankly don't know. But K. Dowd to the rescue (in fact, all of A2.8 is very helpful, p 51), sorry for long extract:

"A third helpful notion is that of domains of attraction. To say that a distribution has a domain of attraction means that any distribution ‘close’ to it will have similar properties. This is important because it indicates that small departures from an assumed process (e.g., because of errors in our data) should not produce large errors in any inferenceswemake.More formally, suppose that Sn is the sum of n random variables from a chosen distribution. The distribution of Sn will typically converge as n gets large, so if certain conditions are met (independence, etc.), the distribution of Sn will converge to a normal distribution. This is what the central limit theorem means. But put another way, the distribution of Sn is in the domain of attraction of the normal, and the domain of attraction itself consists of the set of all distributions of Sn that converge to the normal as n gets large. This particular domain of attraction is therefore a counterpart of the standard central limit theorem. There are also other domains of attraction: for example, the set of all distributions of Sn that converge in the limit to the Levy is the domain of attraction of the Levy, and is associated with the central limit theorem as it applies to the Levy. There are also domains of attraction associated with highest and lowest observations in a sample of size n, and these are the domains of attraction encountered in extreme-value theory. These latter domains of attraction are associated with extreme-value theorems."

In summary, my translation:

* Per Rachev, there are (only) three (closed-form) stable distributions: Cauchy, Levy and normal. Or, Dowd refers to Levy distributions as the super-class; i.e., Levy distributions are the only stable distributions, where normal is a subclass of Levy (Levy is stable, normal is subclass of Levy, so normal is stable)

* It is the stable property (stability) is what allows us to scale i.i.d returns per square root rule: n-period variance = n*1-period variance. It's not normality per se, but the broader super-class of Levy that can scale i.i.d this way.

* The domain of attraction might be grossly (imprecisely) summarized as "converges to;" i.e., distributions that tend toward normal as n gets larger are in the normal's domain of attraction.

* In this way, as Dowd suggest, CLT is one application (one instance of domain of attraction) but only as concerns the central body of the distribution. The extreme tails have a domain of attraction such that they converge to the EVT distribution (GPD).

In other words, attraction (convergence) at the body gives us CLT like convergence in the extreme tail gives us GPD.

David

 

[ajsa]

Hi David,

The CLT definition on Quantitative Note p37 is "sampling distribution of sample means tends to be normal (i.e., converges toward a normally shaped distributed) regardless of the shape of the underlying distribution". It does not meantion "summation". Could you please clarify?

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

The CLT applies to both the sampling distribution of the sample mean and the sample *summation* (it must, since the summation is simply scaled by sample size n). So, summation is impled by CLT...

...however, it practice, we don't tend to use the summation much, we tend to use the sample distribution of the sample mean. Specifically, when we use the student's t distribution to test the significance of regression coefficients (e.g., is the slope significant), we are relying on the CLT as it pertains to a sample mean.

Thanks, David