# Excel formula for the inverse Pearson IV distribution?

#### ddurant100

##### New Member
Hello Everyone,

What is the excel formula to generate the inverse Pearson IV distribution? I have mean, standard deviation, skew and kurtosis and I plan to generate random outputs given those stats. How do I write the formula?

My question is very direct and simple. I've looked everywhere for an excel equation for the Pearson IV (4) distribution. I've found software and random formulas, but not one I can figure out how to build in excel. I found and use a great equation for the Pearson III (3) (see: http://gergs.net/2014/07/log-pearson-type-3-excel/), but not for #4. I could use your help.

Thank you,

Dave

#### Dragan

##### Super Moderator
I would suggest you look at the article by Rieck and Nedelman (1991) published in the journal Technometrics, Vol. 33, pp. 51-60.

Basically, you could use:

$$Z=\gamma +\delta \sinh \left ( \frac{X-\xi }{\lambda } \right )$$

where Z is unit normal and X is distributed as a linear function of:

$$\sinh^{-1}\left ( \frac{Z-\gamma }{\delta } \right )$$.

See the article for further details on the parameters associated with the equations above.

#### ddurant100

##### New Member
Hello Dragan,

Thank you so much for responding. How do I write this equation in excel? You had posted this a while ago when Vinux was looking for the same equation: http://www.talkstats.com/showthread.php/5853-Random-numbers

Where do I put mean, standard deviation, skew, and kurtosis in a formula that allows a random probability to generate x(P)?

#### GretaGarbo

##### Human
I asked myself, what on earth is a Pearson type IV distribution?

(I had a rough memory that Student (1908) mentioned a Pearson type III distribution and used that to develop what later became known as the t-test.)

I found this link about the Pearson type IV distribution. I hope it helps.

Also, if it is any good you can expect to find it in R, don't you agree?
I found this package (Package ‘PearsonDS’) about these distributions. Type IV on page 15.

It also says:
"The Pearson Type 0 (aka Normal) Distribution"
"The Pearson Type I (aka Beta) Distribution"
"The Pearson Type II (aka Symmetric Beta) Distribution"
"The Pearson Type III (aka Gamma) Distribution"
"The Pearson Type IV Distribution"
"The Pearson Type V (aka Inverse Gamma) Distribution"
"The Pearson Type VI (aka Beta Prime) Distribution"
"The Pearson Type VII (aka Student’s t) Distribution"

So now maybe these distributions (with nomenclature from 1900) are a little bit less mysterious to us.

The inverse, the quantile values, can be found with the qpearsonIV() function (page 15).

Maybe the OP must do the computations in excel. But it could be good to check that the results corresponds to what you can get from the R-package. (Also, many of us would trust the results much more from an R package.)

#### ddurant100

##### New Member
I think you solved part of the mystery. So how do I translate the inverse into excel?

#### CB

##### Super Moderator
Why do you need to use Excel? It's really not a good piece of software for doing statistical work. R is free and open source :tup:

#### ddurant100

##### New Member
Unfortunately, excel is how my client needs the analysis done. Back office systems and such.

#### qrsplt

##### New Member
Updating this thread in light of new developments. My company has developed a new Office add-in called QRS Toolbox which provides several custom functions to work with the Pearson distributions, including calculating the inverse CDF of the Pearson Type IV. See the following:
The calculations are done numerically on our servers using our own implementation. The results have been validated against R.

#### katxt

##### Active Member
I have mean, standard deviation, skew and kurtosis
Did these values come from real world data?
Is there some theoretical reason for using Pearson IV?