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3. 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.

4. 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)

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

6. Thus our simple model is With bells and whistles (i.e. with location and scale parameters) we have:

7. 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.

8. 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”.

9. Back to the case where X and Y are univariate. The distribution of the observed X’s is

10. 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 .

11. Instead of writing the joint density of (X,Y) as we can write it as The model then looks a little different

12. 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.

13. The weight function is:

14. 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.

15. 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 .

16. 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.

17. General hidden truncation models ( also called selection models by Arelleno-Valle, Branco and Genton (2006) ) are of the form:

18. 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.

19. Life will be smoothest if these conditional survival functions are available in analytic or at least in tabulated form.

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

23. 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).

24. Suppose that And And we consider two-sided hidden truncation

26. More generally, we may consider to get:

27. Included in such models as a limiting case, we find which has arisen as a marginal of a bivariate distribution with skew-normal conditionals

28. In fact we can obtain just about any weighted normal density in this way . To get: We : and

29. We can apply hidden truncation to other bivariate models. (i) The normal conditionals density

30. (ii) Distributions with exponential components: i.e.

31. the corresponding two sided truncation model is and the lower truncation model is again an exponential density

32. A similar phenomenon occurs with the exponential conditionals distribution

33. 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:

34. (iii) Pareto components

35. Multivariate cases:

36. Classical multivariate normal case: So that:

37. Notation

38. The distribution of will then be given by Let us define: The corresponding density of will be Z

39. a.k.a. closed skew-normal distribution fundamental skew-normal distribution multiple constraint skew-normal distribution

40. 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

42. Thank you for your attention.