Pearson Type 6 distribution

Format: Pearson6(a1, a2, b)

The Pearson Type 6 distribution corresponds in the Pearson system to the case when the roots of c0 + c1x + c2x2 = 0 are real and of the same sign.

Examples of the Pearson Type 6 distribution are given below:

Uses

We don't find much use for this distribution: its underlying model is unlikely to reflect any of the processes that the analyst may come across, but its three parameters (giving it flexibility), sharp peak and long tail make it a possibly candidate to be fitted to a very large (so you know the pattern is real) data set that other distributions won't fit to well.

Like the Pearson Type 5 distribution , the Pearson(1) Type 6 distribution has proven to be of limited use in risk analysis.

There exists the identity:

Pearson6(a1,a2,b) = b*X/(1-X), where X = Beta(a1,a2)

Setting b = 1 we have

Pearson6(a1,a2,1) = X/(1-X), where X = Beta(a1,a2)

In logistic regression and other methods focusing on comparing probabilities one often transforms a probability p to the variable p/(1-p) as this extends over the range [0, ∞). If p follows a Beta(a1,a2) distribution then p/(1-p) follows a Pearson6(a1,a2,1) distribution.

Rewriting the density function in the form

we can see a similarity with the beta distribution which has a density function of the form

Thus, the Pearson6 distribution is sometimes known as the Beta Distribution of the Second Kind.

ModelRisk functions added to Microsoft Excel for the Pearson Type 6 distribution

VosePearson6 generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VosePearson6Object constructs a distribution object for this distribution.

VosePearson6Prob returns the probability density or cumulative distribution function for this distribution.

VosePearson6Prob10 returns the log10 of the probability density or cumulative distribution function.

VosePearson6Fit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VosePearson6FitObject constructs a distribution object of this distribution fitted to data.

VosePearson6FitP returns the parameters of this distribution fitted to data.