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slide3

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.

slide4

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)

slide5

We call these hidden truncation models, because

we don’t get to observe the truncating variable Y.

We just see X.

slide6

Thus our simple model is

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

slide7

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.

slide8

Multivariate extension: Begin

with a (k+m) dimensional r.v.

(X,Y), but only keep X if Y>c

Often (X,Y) is assumed to have

a classical multivariate normal

distribution.

The “closed skew-normal model”.

back to the case where x and y are univariate
Back to the case where X and Y are univariate.

The distribution of the

observed X’s is

slide10

with corresponding density:

“parameterized” by the choice of

marginal distribution for Y, the

choice of conditional distribution

of X given Y and the critical

value

.

slide11

Instead of writing the joint density

of (X,Y) as

we can write it as

The model then looks a little different

slide12

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.

slide14

This weighted form of hidden truncation

densities appears in Arellano-Valle et al.

(2002) with

.

But perhaps someone in the audience knows

an earlier reference.

slide15

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

.

slide16

In fact the weight function, by a judicious

choice of conditional distribution of Y given X

and a convenient choice of

can be any weight function bounded above by 1.

slide17

General hidden truncation models

( also called selection models by

Arelleno-Valle, Branco and Genton (2006) )

are of the form:

slide18

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.

slide19

Life will be smoothest if these conditional

survival functions are available in analytic

or at least in tabulated form.

slide20

These may be troublesome to deal with.

Exception when (X,Y) is bivariate normal.

slide23
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).

slide24

Suppose that

And

And we consider two-sided hidden truncation

slide27

Included in such models as a limiting case, we

find

which has arisen as a marginal of

a bivariate distribution with skew-normal conditionals

slide28

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

To get:

We :

and

slide31

the corresponding two sided truncation model is

and the lower truncation model is

again an exponential density

slide33

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:

slide38

The distribution of

will then be given by

Let us define:

The corresponding density of will be

Z

slide39

a.k.a.

closed skew-normal distribution

fundamental skew-normal distribution

multiple constraint skew-normal distribution

slide40

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