3. Model: A Generalised Azzalini Skew Normal Distribution

Next, we illustrate the power of mathStatica and Mathematica in financial economics by implementing a generalised version of Azzalini's SN density in the context of our example. Azzalini [1] proved that if a random variable has a pdf which is symmetric about 0, and cdf , then is also a pdf, for parameter . If , then is Azzalini's SN density. Unfortunately, the latter does not contain sufficient flexibility for our financial application which will require both a scale parameter (for large variance) and a location parameter.

To start our generalisation, we first introduce a scale parameter hence, let with pdf .

The cdf evaluated at is:

Then, our generalised Azzalini skew density, , is simply:

with domain of support:

Next, we introduce a location parameter into the model by transforming such that . This is facilitated by mathStatica's Transform[ ] function to yield pdf .

with domain of support:

Figure 2 plots our generalised location-scale SN density for three values of the key parameter to verify that its negative values lead to negative skewness in a graph. Note that if a mutual fund has (other things being equal), it is mostly losing money. In the sequel, we shall estimate by maximizing the likelihood function.

Figure 2.