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#11
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See the CRC book "Fitting Statistical Distributions: The Generalized
Lambda Distribution and Generalized Bootstrap Methods" by Zaven A. Karian and Edward J. Dudewicz It develops the math and provides Maple code for fitting. You will have to adapt to Excel. Jerry Frank & Pam Hayes wrote: 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? |
#12
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"Long time, no see." Good to have you back.
Jerry David J. Braden wrote: .... |
#13
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Thanks. Finally, there's a stat question I can add something to before *you*
get too far into it vbg. I didn't know of the CRC source you cited. I know only of a neat article by Freimer, Mudhoker at al. that really digs into it, and, AFAIK, was the one to generalize the distribution, though how I don't recall. I'll look into it, though it will take me a few days yet. Could you plz post the generalization from the CRC, or at least point us towards it? Then we can walk the OP through how to fit it to his data using Excel. What brought it to mind is its simplicity and flexibility. If the OP is willing to work with cumulatives (my preference as well) then, short of working directly with the empirical cumulative, I cannot think of a better proximal distribution in this case for data I haven't seen. Regards, dave braden "Jerry W. Lewis" wrote in message ... "Long time, no see." Good to have you back. Jerry David J. Braden wrote: ... |
#14
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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? |
#15
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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? |
#16
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Jerry,
The question is still open. Thanks for the continuing help. Frank "Jerry W. Lewis" wrote in message ... 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? |
#17
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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? |
#18
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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? |
#19
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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? |
#20
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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? |
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