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Format: VosePearson6(a1, a2, b,
U)
The Pearson(1) 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:
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 hasn't proven to be very useful 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.
VosePearson6 generates values from this distribution or calculates a percentile.
VosePearson6Object constructs a distribution object for this distribution. Professional and Industrial editions only.
VosePearson6Prob returns the probability density or cumulative distribution function for this distribution. Professional and Industrial editions only.
VosePearson6Prob10 returns the log10 of the probability density or cumulative distribution function. Professional and Industrial editions only.
VosePearson6Fit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution. Professional and Industrial editions only.
VosePearson6FitObject constructs a distribution object of this distribution fitted to data. Professional and Industrial editions only.
VosePearson6FitP returns the parameters of this distribution fitted to data. Professional and Industrial editions only.
(1) Karl Pearson (1857 - 1936)
