Jerry,

Thank you for the clarifications. I am going to dig into it a bit and see
how I can apply it to my application.

Frank


"Jerry W. Lewis" wrote in message
...
You are welcome. Moreover, clarifications are not a burden.

L1 is a location parameter, but is equal to the mean only if L3=L4, since
A=0 when L3=L4. More Generally,
mean = L1+A/L2

L2 is a spread parameter, but the value of the variance is given by the
formula in my previous post.

L3 and L4 are shape parameters, but the usual coefficient of skewness is
a3, whose formula is given in my previous post. In particular, note that
when L3=L4, the coefficient of skewness is zero (the distribution is
symmetric), regardless of the actual value of L3

Similarly, the usual coefficient of kurtosis is either a4 or a4-3,
depending on whether you consider the kurtosis of the Normal distribution
to be 3 or 0. The formula for a4 is given in my previous post.

Also, note that for positive values of L3 and L4 that
GLDinv(p,L1,L2,L3,L4) is finite for p=0 and p=1. That means that the
distribution function has a finite domain. Thus, the generalized lambda
distribution may poorly fit the extreme tails of distributions (such as
Normal, Gamma, Chi-Square, etc.) that have have an infinite domain, even
though it may be a good approximation elsewhere.

Jerry

Frank & Pam Hayes wrote:

Jerry,

First of all, thank you very much for putting so much thought and effort
into your posting. I appreciate your efforts. I am not a Statistics
whiz, so forgive me if some of my questions are a bit basic.

Am I correct in thinking that L1 is the Mean, L2 is the Variance, L3 is
the Skewness and the L4 is the Kurtosis? Given that Excel can calculate
all of these from my data, is the answer as simple as calculating those
values and plugging them into the formula you provided?

Your formula shows the variable p, but it is not defined in the rest of
the posting. Does this stand for the probability of the occurance from 0
to 1? How would I produce the trend line that reflects the data I am
interested in ... perhaps calculate GLDinv for 100 points (p) from 0 to 1
and then plot that line?

Thank you,

Frank


"Jerry W. Lewis" wrote in message
...

The Generalized Lambda Distribution is the 4-parameter distribution with
inverse
GLDinv(p,L1,L2,L3,L4) = L1 + (p^L3+(1-p)^L4)/L2

This represents a valid distribution if and only if
L3*p^(L3-1)+L4*(1-p)^(L4-1)
has the same sign (positive or negative) for all p in [0,1], as long as
L2 takes that sign also (which in particular is true if L2, L3, and L4
all have the same sign). See the Karian & Dudewicz book for extensive
discussion of valid and invalid regions.

When L3-1/4 and L4-1/4, then the first four moments are
mean = L1+A/L2
var = (B-A^2)/L2^2
a3 = (C-3*A*B+2*A^3)/(L2*SQRT(var))^3
a4 = (D-4*A*C+6*A^2*B-3*A^4)/(L2*SQRT(var))^4
for
A = 1/(1+L3) -1/(1+L4)
B = 1/(1+2*L3) +1/(1+2*L4) -2*beta(1+L3,1+L4)
C = 1/(1+3*L3) -1/(1+3*L4) -3*beta(1+2*L3,1+L4) +3*beta(1+L3,1+2*L4)
D = 1/(1+4*L3) +1/(1+4*L4) -4*beta(1+3*L3,1+L4)
+6*beta(1+2*L3,1+2*L4) -4*beta(1+L3,1+3*L4)

where
a3 = E(X-mean)^3/sigma^3
a4 = E(X-mean)^4/sigma^4

You can use the method of moments to estimate the parameters
(L1,L2,L3,L4) from data. Alternately, you can estimate the parameters
from 4 sample quantiles. Karian & Dudewicz provide tables and Maple code
for fitting GLD from either approach. They also discuss the bivariate
extension.

(L1,L2,L3,L4) = (0, 0.1975, 0.1349, 0.1349) approximates the standard
normal distribution.
(L1,L2,L3,L4) = (0.5, 1/12, 0, 9/5) is Uniform(0,1).
Karian & Dudewicz discusses approximations to other standard
distributions.

Jerry

Jerry W. Lewis wrote:


I presume the Freimer, Mudholkar et al paper you saw was Comm.Stat.A
17:3547-3567, 1988. If you have direct access to the Comm.Stat. series,
you might also look at a couple of Karian & Dudewicz papers from
Comm.Stat.B 25:611-642,1996 and 28:793-819,1999. Another reference
would be Oeztuerk & Dale's Technometrics 27:81-84,1985 paper.

I have access to the Karian & Dudewicz book and Technometrics CDs at the
office. I will bring them home tonight to follow up if the question is
still open.

Jerry

David J. Braden wrote:


Frank (and Pam?)
I want to get the generalized version first; it is not at hand,
unfortunately, and unless I get help from Jerry or someone else in the
community, it will take me a day or so to retrieve it. Once I get it, I
will be happy to walk you through how to use Excel to fit it. Remember,
it works off of the *inverse* cumulative.

Do you know how to set it up? You also need to determine what you mean
by "closeness of fit". Jerry's CRC suggestion might well do the trick;
I haven't seen it yet, so I don't know how the distribution is
generalized, nor how easy the CRC version is to fit. But we'll get
there.

Regards,
dave braden

"Frank & Pam Hayes" wrote in message
news:7Otad.3294$Rp4.15@trnddc01...


David,

The Tukey-lambda fit looks like it has promise for my cumulative
probability curve, but a google search on Tukey-lambda and Excel was
pretty sparse. Searching on Tukey-lambda alone brought many more
results, most of which were beyond my statistical competance. The
cumulative distribution function shown at :
http://www.itl.nist.gov/div898/handb...n3/eda366f.htm
looks to be exactly what I am trying to produce.

Can you point me in the right direction on how I would use
Tukey-lambda in Excel to calculate the cumulative probabilty curve?

Frank

"David J. Braden" wrote in message
. ..


Another idea:
Generalized inverse Tukey-lambda fit, which requires but 4
parameters, and is very well behaved at endpoints. The fit is on the
inverse cumulative, and seems to be very stable wrt Excel.

"Jerry W. Lewis" wrote in message
...


And if the data can meaningfully be fitted to an 8th order
polynomial, I would still worry about numerical problems unless you
were using Excel 2003 and no coefficients were estimated to be
exactly zero
http://groups.google.com/groups?selm...0no_e-mail.com

Jerry

Bernard Liengme wrote:



Use LINEST to generate coefficients - see
www.stfx.ca/people/bliengme/ExcelTips
Use the coefficients to generate trendline data

Do your really have data that can meaningfully be fitted to 8th
order?