This week, we will start to work on multivariate models, and non-independence. The first idea to discuss non-independence will be to use the concept ofexchangeability. A sequence of random variable Image may be NSFW.
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is said to be exchangeable if for all Image may be NSFW.
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of Image may be NSFW.Clik here to view.
. A standard example is the case whereImage may be NSFW.Clik here to view.
, with
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, a necessary condition is that
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Since this inequality should hold for all Image may be NSFW.
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it comes that necessarily Image may be NSFW.Clik here to view.
.de Finetti (1931): Let Image may be NSFW.
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be a sequence of random variables with values in Image may be NSFW.Clik here to view.
. Image may be NSFW.Clik here to view.
is exchangeable if and only if there exists a distribution function Image may be NSFW.Clik here to view.
on Image may be NSFW.Clik here to view.
such that
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. Note that Image may be NSFW.Clik here to view.
is the distribution function of random variable
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From the exchangeability condition, for any permutation Image may be NSFW.
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of Image may be NSFW.Clik here to view.
,
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, i.e. for all Image may be NSFW.Clik here to view.
, define
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,
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, all possible rearrangements of the ones among the Image may be NSFW.Clik here to view.
elements are equally likely, we can write
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to a binomial distribution Image may be NSFW.Clik here to view.
, when Image may be NSFW.Clik here to view.
becomes large. Then
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and let Image may be NSFW.Clik here to view.
denote the cumulative distribution function of Image may be NSFW.Clik here to view.
.
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, i.e.
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Hewitt & Savage (1955): Let Image may be NSFW.
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be a sequence of random variables with values in Image may be NSFW.Clik here to view.
. Image may be NSFW.Clik here to view.
is exchangeable if and only if there exists a measure Image may be NSFW.Clik here to view.
on Image may be NSFW.Clik here to view.
such that
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is the measure associated to the empirical measure
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, the Image may be NSFW.Clik here to view.
are conditionally independent, with distribution Image may be NSFW.Clik here to view.
. The proof can be found in Kingman (1978) and is based on martingale arguments.Note that in the Gaussian case, Image may be NSFW.
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where Image may be NSFW.Clik here to view.
are i.i.d. random variables. To go further on exchangeability and related topics, see Aldous (1985) (see also here).This construction can be used in credit risk, to model defaults in an homogeneous portfolio, see e.g. Frey (2001),
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(either the company defaults, or not),
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> proba=function(s,a,m,n){ + b=a/m-a + choose(n,s)*integrate(function(t){t^s*(1-t)^(n-s)* + dbeta(t,a,b)},lower=0,upper=1)$value + }
Based on that function, it is possible to plot the probability distribution over Image may be NSFW.
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. In the upper corner is plotted the density of the Beta distribution.
> a=2 > m=.2 + n=10 + V=rep(NA,n+1) + for(i in 0:n){ + V[i+1]=proba(i,a,m,n)} > barplot(V,names.arg=0:10)
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Those two theorems are extremely close,
De Finetti’s theorem: a random sequence Image may be NSFW.
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of Image may be NSFW.
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random variables is exchangeable if and only if Image may be NSFW.
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‘s are conditionnally independent, conditionnally on some random variable Image may be NSFW.
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.
Hewitt-Savage’s theorem: a random sequence Image may be NSFW.
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Olshen (1974), proposed an interesting discussion about those theorems, see also in the Encyclopedia of Statistical Science,
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The subtle difference between those two theorem is also discussed in Freedman (1965)
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