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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.

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)

We call these hidden truncation models, because we don’t get to observe the truncating variable Y.

We just see X.

Thus our simple model is

With bells and whistles (i.e. with location and scale parameters) we have:

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.

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”.

Back to the case where X and Y are univariate.

The distribution of the observed X’s is

with corresponding density:

“parameterized” by the choice ofmarginal distribution for Y, the choice of conditional distributionof X given Y and the criticalvalue 0y

.

Instead of writing the joint densityof (X,Y) as

we can write it as

The model then looks a little different

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.

The weight function is:

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.

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

.

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.

General hidden truncation models ( also called selection models by

Arelleno-Valle, Branco and Genton (2006) )are of the form:

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.

Life will be smoothest if these conditionalsurvival functions are available in analytic

or at least in tabulated form.

These may be troublesome to deal with.Exception when (X,Y) is bivariate normal.

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).

Suppose that

And

And we consider two-sided hidden truncation

More generally, we may consider

to get:

Included in such models as a limiting case, we

find

which has arisen as a marginal of

a bivariate distribution with skew-normal conditionals

In fact we can obtain just about any weighted normal density in this way .

To get:

We :

and

We can apply hidden truncation to other bivariate models.

(i) The normal conditionals density

(ii) Distributions with exponential components:

i.e.

the corresponding two sided truncation model is

and the lower truncation model is

again an exponential density

A similar phenomenon occurs with the exponential conditionals distribution

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:

(iii) Pareto components

Multivariate cases:

Classical multivariate normal case:

So that:

Notation

The distribution of

will then be given by

Let us define:

The corresponding density of will be

Z

a.k.a.

closed skew-normal distribution

fundamental skew-normal distribution

multiple constraint skew-normal distribution

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

Thank you for your attention.