4 - Heavy and adaptive tails

Summary

Motivating remarks

The skew-normal density has very short tails. In fact, the rate of decay to 0 of the density φ(x; α) as |x| → ∞ is either the same as the normal density or even faster, depending on whether x and α have equal or opposite sign, as specified by Proposition 2.8. This behaviour makes the skew-normal family unsuitable for a range of application areas where the distribution of the observed data is known to have heavier tails than the normal ones, sometimes appreciably heavier.

To construct a family of distributions of type (1.2) whose tails can be thicker than a normal ones, a solution cannot be sought by replacing the term Φ;(αx) in (2.1) with some other term G₀{w(x)}, since essentially the same behaviour of the SN tails would be reproduced. The only real alternative is to adopt a base density f₀ in (1.2) with heavier tails than the normal density.

For instance, we could select the Laplace density exp(−|x|)/2, whose tails decrease at exponential rate, to play the role of base density and proceed along lines similar to the skew-normal case. This is a legitimate program, but it is preferable that f₀ itself is a member of a family of symmetric density functions, depending on a tail weight parameter, v say, which allows us to regulate tail thickness. For instance, one such choice for f₀ is the Student's t family, where v is represented by the degrees of freedom.