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The basic Azzalini skew-normal model is:
Adding location and scale parameters we get
Where denotes the standard normal density and denotes the corresponding distribution function.
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Genesis: Begin with (X,Y) with a bivariate normal distribution.
But, only keep X if Y is above average.
More generally, keep X if Y exceeds a given threshold, not necessarily its mean.
This model is discussed in some detail in Arnold, Beaver,Groeneveld and Meeker (1993)
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We call these hidden truncation models, because we don’t get to observe the truncating variable Y.
We just see X.
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Thus our simple model is
With bells and whistles (i.e. with location and scale parameters) we have:
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A more general model of the same genre is of the form
In such a model it may be necessary to evaluate the required normalizing constant numerically.
E.G. Cauchy, Laplace, logistic, uniform, etc.
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Multivariate extension: Beginwith a (k+m) dimensional r.v.(X,Y), but only keep X if Y>c
Often (X,Y) is assumed to havea classical multivariate normal
distribution.
The “closed skew-normal model”.
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Back to the case where X and Y are univariate.
The distribution of the observed X’s is
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with corresponding density:
“parameterized” by the choice ofmarginal distribution for Y, the choice of conditional distributionof X given Y and the criticalvalue 0y
.
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Instead of writing the joint densityof (X,Y) as
we can write it as
The model then looks a little different
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It now is of the form:
So that the “hidden truncation” version of the density of X, is clearly displayed as
a weighted version of the original density of X.
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The weight function is:
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This weighted form of hidden truncation densities appears in Arellano-Valle et al. (2002) with 0 0y
.But perhaps someone in the audience knowsan earlier reference.
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In this formulation our density is “parameterized”by the marginal density of X and the weight function which is determined by the conditional density of Y given X and the critical value 0y
.
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In fact the weight function, by a judiciouschoice of conditional distribution of Y given Xand a convenient choice of 0y
can be any weight function bounded above by 1.
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General hidden truncation models ( also called selection models by
Arelleno-Valle, Branco and Genton (2006) )are of the form:
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We focus on 3 special cases
We really only need to consider cases 1 and 3.Case 2 becomes case 1 if we redefine Y to be –Y.
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Life will be smoothest if these conditionalsurvival functions are available in analytic
or at least in tabulated form.
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These may be troublesome to deal with.Exception when (X,Y) is bivariate normal.
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Note that a very broad class of densities can be obtained from a given density via hidden truncation.
Suppose we wish to generate g(x) from f(x).
If g(x)/f(x) is bounded above by c, then we can take a joint density for (X,Y) such that P(Y<0|X=x) = g(x)/cf(x) and thus obtain g(x).
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Suppose that
And
And we consider two-sided hidden truncation
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More generally, we may consider
to get:
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Included in such models as a limiting case, we
find
which has arisen as a marginal of
a bivariate distribution with skew-normal conditionals
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In fact we can obtain just about any weighted normal density in this way .
To get:
We :
and
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We can apply hidden truncation to other bivariate models.
(i) The normal conditionals density
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(ii) Distributions with exponential components:
i.e.
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the corresponding two sided truncation model is
and the lower truncation model is
again an exponential density
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A similar phenomenon occurs with the exponential conditionals distribution
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If the conditional failure rate depends on x in a non-linear manner we can get more interesting distributions via hidden truncation.
E.G.
in particular consider
which yields a truncated normal distribution:
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(iii) Pareto components
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Multivariate cases:
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Classical multivariate normal case:
So that:
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Notation
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The distribution of
will then be given by
Let us define:
The corresponding density of will be
Z
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a.k.a.
closed skew-normal distribution
fundamental skew-normal distribution
multiple constraint skew-normal distribution
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Densities corresponding to two sided truncation have received less attention
though such truncation may be more common in practice than one sided.
They look a bit more ugly
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Thank you for your attention.
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