On scale mixtures of normal distributions

Suppose that Y has a standard normal distribution and that ohas some distribution on (0, oo) with a continuous or discrete density h(o-) (o-> 0). Then the distribution of X = Yois referred to as a scale mixture of normals, and with a scale mixing density h(o-). A wide class of continuous, unimodal and symmetric distributions on the real line may be constructed as scale mixtures of normals. Many examples, such as discrete mixtures or contaminated normals, the Student t family, logistic, Laplace or double-exponential, and the stable family, are well known; see, for example, Andrews & Mallows (1974) and West (1984). Their properties have been useful in several areas. In theoretical studies distributional properties such as moments are often easily derived by exploiting the special structure. In practice, robustness studies have often used these distributions for simulation and in the analysis of outlier models; see West (1984) and references therein. In this note, a new family of normal scale mixtures is identified. This class of distributions, the exponential power family, has been used widely in robustness studies; it was introduced and popularized by Box & Tiao (1973) in the context of Bayesian modelling for robustness. However, the normal scale mixture property and an interesting relationship with the class of stable distributions have not, so far, been discussed. That exponential power distributions are normal scale mixtures may be proved in two ways. The first method uses the characterization result of Andrews & Mallows (1974). The second employs a direct method that explicitly identifies the scale mixing distribution. The characterization result referred to above is as follows. Suppose that X is a real-valued random quantity with a continuous, unimodal and symmetric distribution having density p(X) (-oo < X < oo). Without loss of generality, suppose that the mode is at zero. Then the symmetry assumption implies that p(X) =f(IX2) for some positive and decreasing function f(U) (U ? 0). Chu (1973) shows that p(.) has the form