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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 don’t get to observe the truncating variable Y.
We just see X.
With bells and whistles (i.e. with location and scale parameters) we have:
In such a model it may be necessary to evaluate the required normalizing constant numerically.
E.G. Cauchy, Laplace, logistic, uniform, etc.
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
The “closed skew-normal model”.
The distribution of the
observed X’s is
“parameterized” by the choice of
marginal distribution for Y, the
choice of conditional distribution
of X given Y and the critical
of (X,Y) as
we can write it as
The model then looks a little different
So that the “hidden truncation” version of
the density of X, is clearly displayed as
a weighted version of the original
density of X.
densities appears in Arellano-Valle et al.
But perhaps someone in the audience knows
an earlier reference.
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
choice of conditional distribution of Y given X
and a convenient choice of
can be any weight function bounded above by 1.
( also called selection models by
Arelleno-Valle, Branco and Genton (2006) )
are of the form:
We really only need to consider cases 1 and 3.
Case 2 becomes case 1 if we redefine Y to be –Y.
survival functions are available in analytic
or at least in tabulated form.
Exception when (X,Y) is bivariate normal.
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).
And we consider two-sided hidden truncation
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 .
(i) The normal conditionals density
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.
in particular consider
which yields a truncated normal distribution:
will then be given by
Let us define:
The corresponding density of will be
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