| Title: | Generalize Lambda Distribution and Generalized Bootstrapping |
|---|---|
| Description: | A collection of algorithms and functions for fitting data to a generalized lambda distribution via moment matching methods, and generalized bootstrapping. |
| Authors: | Bin Wang <[email protected]>. |
| Maintainer: | Bin Wang <[email protected]> |
| License: | Unlimited |
| Version: | 2.3.3 |
| Built: | 2024-11-07 06:29:37 UTC |
| Source: | CRAN |
Compute average run length for control chart.
ARL1(x,K,pm1,pI1) ARL0(x,ARL0=370,gridsize=20)ARL1(x,K,pm1,pI1) ARL0(x,ARL0=370,gridsize=20)
x |
An R object generate using |
K |
a vector of the levels |
ARL0 |
in-control average run length |
pm1, pI1
|
out-of-control parameters for the control chart. |
gridsize |
Gridsize of countour levels to search for ARL. |
B. Wang [email protected]
Yang, S.F. and Wang, B. “Using A Kernel Control Region to Monitor Both the Process Location and Dispersion”.
To compute the density, distribution, quantile, and to generate random sample for RS-GLD.
## Default S3 method: degld(x,lambda) pegld(x,lambda) qegld(p,lambda) regld(n,lambda)## Default S3 method: degld(x,lambda) pegld(x,lambda) qegld(p,lambda) regld(n,lambda)
x |
a numeric value or a vector. |
p |
a probability or a vector of probabilities. |
n |
sample size. |
lambda |
a vector of four parameters for RS-GLD. |
B. Wang [email protected]
Karian, Z.A., Dudewicz, E.J., McDonald, P., 1996. The Extended Generalized Lambda Distribution System for Fitting Distributions to Data: history,completion of theory, tables, applications, the “final word” on moment fits, Comm. in Statist.- Simul. \& Comput. 25(3), 611-642.
Karian, Z.A., Dudewicz, E.J., 2000. Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, Chapman and Hall/CRC.
fit.egld,
qrsgld,prsgld,
rrsgld,drsgld.
lambdas = c(2,4,3,4) shape=3;scale=4 x0 = rbeta(5,shape,scale) x1 = x0*lambdas[2]+lambdas[1] qegld(c(0,.1,.5,.7,1),lambdas) qbeta(c(0,.1,.5,.7,1),shape,scale)*lambdas[2]+lambdas[1] pegld(x1,lambdas) pbeta(x0,shape,scale) degld(x1,lambdas) dbeta(x0,shape,scale)/lambdas[2] x0 = sort(rbeta(1000,shape,scale)) y = x0*lambdas[2]+lambdas[1] plot(dbeta(x0,shape,scale)/lambdas[2]~y,type='l') lines(degld(y,lambdas)~y,lty=2,col=2) lines(density(y),col=4,lty=3)lambdas = c(2,4,3,4) shape=3;scale=4 x0 = rbeta(5,shape,scale) x1 = x0*lambdas[2]+lambdas[1] qegld(c(0,.1,.5,.7,1),lambdas) qbeta(c(0,.1,.5,.7,1),shape,scale)*lambdas[2]+lambdas[1] pegld(x1,lambdas) pbeta(x0,shape,scale) degld(x1,lambdas) dbeta(x0,shape,scale)/lambdas[2] x0 = sort(rbeta(1000,shape,scale)) y = x0*lambdas[2]+lambdas[1] plot(dbeta(x0,shape,scale)/lambdas[2]~y,type='l') lines(degld(y,lambdas)~y,lty=2,col=2) lines(density(y),col=4,lty=3)
To compute the density, distribution, quantile, and to generate random sample for RS-GLD.
## Default S3 method: drsgld(x,lambda) prsgld(x,lambda) qrsgld(p,lambda) rrsgld(n,lambda)## Default S3 method: drsgld(x,lambda) prsgld(x,lambda) qrsgld(p,lambda) rrsgld(n,lambda)
x |
a numeric value or a vector. |
p |
a probability or a vector of probabilities. |
n |
sample size. |
lambda |
a vector of four parameters for RS-GLD. |
B. Wang [email protected]
Karian, Z.A., Dudewicz, E.J., McDonald, P., 1996. The Extended Generalized Lambda Distribution System for Fitting Distributions to Data: history,completion of theory, tables, applications, the “final word” on moment fits, Comm. in Statist.- Simul. \& Comput. 25(3), 611-642.
Karian, Z.A., Dudewicz, E.J., 2000. Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, Chapman and Hall/CRC.
fit.gld,
qegld,pegld,
regld,degld.
lambdas = c(0, 0.1975, 0.1349,0.1349) qrsgld(c(0,.1,.5,.7,1),lambdas) prsgld(c(-10,0,1,3,20),lambdas) drsgld(c(-10,0,1,3,20),lambdas) x = sort(rrsgld(100,lambdas)) plot(dnorm(x)~x,type='l') lines(drsgld(x,lambdas)~x,lty=2,col=2) lines(density(x),col=4,lty=3)lambdas = c(0, 0.1975, 0.1349,0.1349) qrsgld(c(0,.1,.5,.7,1),lambdas) prsgld(c(-10,0,1,3,20),lambdas) drsgld(c(-10,0,1,3,20),lambdas) x = sort(rrsgld(100,lambdas)) plot(dnorm(x)~x,type='l') lines(drsgld(x,lambdas)~x,lty=2,col=2) lines(density(x),col=4,lty=3)
To fit a EGLD or generalize beta distribution with the maximum likelihood methods.
fit.egld(x,xmin=NULL,xmax=NULL)fit.egld(x,xmin=NULL,xmax=NULL)
x |
A sample. 'NA' values will be automatically removed. |
xmin |
The lower limit of the underlying distribution. Default: NULL. |
xmax |
The upper limit of the underlying distribution. Default: NULL. |
B. Wang [email protected]
Karian, Z.A., Dudewicz, E.J., McDonald, P., 1996. The Extended Generalized Lambda Distribution System for Fitting Distributions to Data: history,completion of theory, tables, applications, the “final word” on moment fits, Comm. in Statist.- Simul. \& Comput. 25(3), 611-642.
Karian, Z.A., Dudewicz, E.J., 2000. Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, Chapman and Hall/CRC.
fit.gld,
qrsgld,prsgld,
rrsgld,drsgld.
b3=4;b4=4; b1=1;b2=5; # EGLD(b1,b2,b3,b4) b1=0;b2=1; # equivalently beta(b3,b4) b1=-3;b2=5; xr = rbeta(100,b3,b4) x = xr * b2 + b1 min(x); range(x) sum(dbeta(xr,b3,b4,1)) x0 = seq(min(x),max(x),length=100) x1 = (x0-b1)/b2 plot(dbeta(x1,b3,b4)/b2~x0,type='l',lwd=2,col=2) lines(density(x),lty=2, col=2) ## no prior information on min and max (out0 = fit.egld(x)) lines(out0,col=1) ## xmin known (out1 = fit.egld(x,xmin=-3)) lines(out1,col=3,lwd=2) ## xmax known (out2 = fit.egld(x,xmax=2)) lines(out2, col=4) ## both known (out3 = fit.egld(x,xmin=-3,xmax=2)) lines(out3, col=5)b3=4;b4=4; b1=1;b2=5; # EGLD(b1,b2,b3,b4) b1=0;b2=1; # equivalently beta(b3,b4) b1=-3;b2=5; xr = rbeta(100,b3,b4) x = xr * b2 + b1 min(x); range(x) sum(dbeta(xr,b3,b4,1)) x0 = seq(min(x),max(x),length=100) x1 = (x0-b1)/b2 plot(dbeta(x1,b3,b4)/b2~x0,type='l',lwd=2,col=2) lines(density(x),lty=2, col=2) ## no prior information on min and max (out0 = fit.egld(x)) lines(out0,col=1) ## xmin known (out1 = fit.egld(x,xmin=-3)) lines(out1,col=3,lwd=2) ## xmax known (out2 = fit.egld(x,xmax=2)) lines(out2, col=4) ## both known (out3 = fit.egld(x,xmin=-3,xmax=2)) lines(out3, col=5)
To fit a Ramberg-Schmeiser-Tukey (RST) lambda distribution with the three moment-matching methods.
fit.gld(x,method='LMoM')fit.gld(x,method='LMoM')
x |
A sample of size at least 6. 'NA' values will be automatically removed. |
method |
Choose GLD fitting method. Default: 'LMoM'. Other options: 'MoM'– method of moments; "MoP", method of percentiles; "LMoM", method of L-moments. 'best' chooses the best fit from the above three methods, which takes a while. |
B. Wang [email protected]
Karian, Z.A., Dudewicz, E.J., McDonald, P., 1996. The Extended Generalized Lambda Distribution System for Fitting Distributions to Data: history,completion of theory, tables, applications, the “final word” on moment fits, Comm. in Statist.- Simul. \& Comput. 25(3), 611-642.
Karian, Z.A., Dudewicz, E.J., 2000. Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, Chapman and Hall/CRC.
fit.egld,
qrsgld,prsgld,
rrsgld,drsgld.
mu = 34.5; sig=1.5 y = rnorm(1000,mu,sig) x = round(y) ### rounding errors x0 = seq(min(y),max(y),length=100) f0 = dnorm(x0,mu,sig) plot(f0~x0,type='l') lines(density(y),col=4) ## fit with method of moments (out1 = fit.gld(x, method='MoM')) lines(out1,col=2) ## Method of percentile (out2 = fit.gld(x, method='mop')) lines(out2, col=3) ## Method of L-moments (out3 = fit.gld(x, method='lmom')) lines(out3, col=5) ## Fitting EGLD (out0 = fit.egld(x)) lines(out0,col=6) legend(max(x0), max(f0), xjust=1,yjust=1, legend=c("true","kde","MoM","MoP","LMoM","egld"), lty=c(1,1,1,1,1,1), col=c(1,4,2,3,5,6))mu = 34.5; sig=1.5 y = rnorm(1000,mu,sig) x = round(y) ### rounding errors x0 = seq(min(y),max(y),length=100) f0 = dnorm(x0,mu,sig) plot(f0~x0,type='l') lines(density(y),col=4) ## fit with method of moments (out1 = fit.gld(x, method='MoM')) lines(out1,col=2) ## Method of percentile (out2 = fit.gld(x, method='mop')) lines(out2, col=3) ## Method of L-moments (out3 = fit.gld(x, method='lmom')) lines(out3, col=5) ## Fitting EGLD (out0 = fit.egld(x)) lines(out0,col=6) legend(max(x0), max(f0), xjust=1,yjust=1, legend=c("true","kde","MoM","MoP","LMoM","egld"), lty=c(1,1,1,1,1,1), col=c(1,4,2,3,5,6))
Estimate Asymptotic Joint Distribution of EWMA variables for control chart.
fkde(n=5, pm0=0.5, pI0=0.2, lambda=0.05, gridsize=100,B=10000,T=10000)fkde(n=5, pm0=0.5, pI0=0.2, lambda=0.05, gridsize=100,B=10000,T=10000)
n |
sample size. |
lambda |
a parameter to compute EWMA |
pm0, pI0
|
in-control parameters for the control chart. |
gridsize |
gridsize to evalue the joint PDF values |
B, T
|
iteration times and maximum time of |
B. Wang [email protected]
Yang, S.F. and Wang, B. “Using A Kernel Control Region to Monitor Both the Process Location and Dispersion”.
Generalized bootstrapping
gboot(x,gldobj,statistic,...)gboot(x,gldobj,statistic,...)
x |
A random sample. |
gldobj |
Either an object fitting a GLD or EGLD to data 'x'. |
statistic |
User defined function to resample from 'x'. 'fun' could be parametric or non-parametric. |
... |
Controls |
Wang, B., Mishra, S.N., Mulekar, M., Mishra, N.S., Huang, K., (2010). Generalized Bootstrap Confidence Intervals for High Quantiles, In: Karian ZA, Dudewicz, EJ eds. The Handbook on Fitting Statistical Distributions with R. CRC Press. 2010: 877-913.
Wang, B., Mishra, S.N., Mulekar, M., Mishra, N.S., Huang, K., (2010). Comparison of bootstrap and generalized bootstrap methods for estimating high quantiles, Journal of Statistical Planning and Inferences, 140. 2926-2935. DOI: 10.1016/j.jspi.2010.03.016.
Karian, Z.A., Dudewicz, E.J., McDonald, P., 1996. The Extended Generalized Lambda Distribution System for Fitting Distributions to Data: history,completion of theory, tables, applications, the “final word” on moment fits, Comm. in Statist.- Simul. \& Comput. 25(3), 611-642.
Karian, Z.A., Dudewicz, E.J., 2000. Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, Chapman and Hall/CRC.
Dudewicz, E.J., 1992. The Generalized Bootstrap, Bootstrapping and Related Techniques, In: K.H., G. Rothe, W. Sendler, eds., V. 376 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 31-37.
data(ofc) X = ofc$x0 Ta = function(x) mean(x<31) gld0 = fit.gld(X) (out = gboot(X,gld0,statistic=Ta,R=100)) gld1 = fit.egld(X) (out = gboot(X,gld1,statistic=Ta,R=100))data(ofc) X = ofc$x0 Ta = function(x) mean(x<31) gld0 = fit.gld(X) (out = gboot(X,gld0,statistic=Ta,R=100)) gld1 = fit.egld(X) (out = gboot(X,gld1,statistic=Ta,R=100))
Simulated head size data of new borns.
data(ofc)data(ofc)
A data frame with 1000 observations on 2 variables.
x0 |
numeric | Original OFC values |
x |
numeric | OFC values rounded to centimeters |
Wang, CSDA and JSS papers.