| Title: | Goodness-of-Fit Test for Weibull Distribution (Weibullness) |
| Version: | 2.26.9 |
| Date: | 2026-9-21 |
| Author: | Chanseok Park 
[aut, cre] |
| Maintainer: | Chanseok Park <statpnu@gmail.com> |
| Depends: | R (≥ 4.0), graphics, stats |
| Description: | Conducts a goodness-of-fit test for the Weibull distribution (referred to as the weibullness test) and furnishes parameter estimations for both the two-parameter and three-parameter Weibull distributions. Notably, the threshold parameter is derived through correlation from the Weibull plot. Additionally, this package conducts goodness-of-fit assessments for the exponential, Gumbel, and inverse Weibull distributions, accompanied by parameter estimations. For more details, see Park (2017) <doi:10.23055/ijietap.2017.24.4.2848>, Park (2018) <doi:10.1155/2018/6056975>, and Park (2023) <doi:10.3390/math11143156>. This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (No. 2022R1A2C1091319). |
| License: | GPL-2 | GPL-3 |
| Encoding: | UTF-8 |
| URL: | https://AppliedStat.GitHub.io/R/ |
| BugReports: | https://github.com/AppliedStat/R/issues |
| LazyData: | yes |
| LazyDataCompression: | xz |
| Suggests: | bsgof |
| NeedsCompilation: | no |
| Packaged: | 2026-09-21 04:24:08 UTC; cp |
| Repository: | CRAN |
| Date/Publication: | 2026-09-21 04:50:02 UTC |
Exponential quantile values
Description
Quantiles for the exponential goodness-of-fit test. They are obtained
from the sample correlation from ANOVA test for exponential distribution.
The number of Monte Carlo iterations is 1.0E08.
Dataset representing the quantiles and the associated critical values for the Weibullness
test. They were obtained by conducting Monte Carlo simulations where the
sample correlation coefficients were calculated based on ANOVA test for exponential distribution.
We used 1.0E08 Monte Carlo iterations in the simulation.
Usage
Exponential.ANOVA.QuantilesFormat
This data frame contains 998 rows and 1001 columns.
Gumbel quantile values
Description
Quantiles for the Gumbel goodness-of-fit test. They are obtained
from the sample correlation from the Gumbel probability plot.
The number of Monte Carlo iterations is 1.0E08.
Dataset representing the quantiles and the associated critical values for the Gumbel
goodness-of-fit test. They were obtained by conducting Monte Carlo simulations where the
sample correlation coefficients were calculated based on the Gumbel probability plot.
We used 1.0E08 Monte Carlo iterations in the simulation.
Usage
Gumbel.Plot.QuantilesFormat
This data frame contains 998 rows and 1001 columns.
Inverse Weibull quantile values
Description
Quantiles for the inverse Weibullness Test. They are obtained
from the sample correlation from the inverse Weibull plot.
The number of Monte Carlo iterations is 1.0E08.
Dataset representing the quantiles and the associated critical values for the inverse Weibullness
test. They were obtained by conducting Monte Carlo simulations where the
sample correlation coefficients were calculated based on the inverse Weibull plot.
We used 1.0E08 Monte Carlo iterations in the simulation.
Usage
IW.Plot.QuantilesFormat
This data frame contains 998 rows and 1001 columns.
References
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Dataset
Description
bearing: It is from Lieblein and Zelen (1956).
These data are deep-groove ball bearing failure times (number of millions of revolutions) in eudurance tests.
glassfiber1.5 and glassfiber15: They are from Smith and Naylor (1987).
These datasets are from experimental data for the strength of glass fiber of length 1.5cm and 15cm, respectively.
radiotherapy and radio.chemotherapy: They are from Finkelstein, D. M. (1986)
and Lindsey, J. C. and L. M. Ryan (1998).
These data are interval-censored observations from a study of patients with breast cancer.
The measurement is the time to cosmetic deterioration of the breast for women who received radiotherapy
and women who received radio-chemotherapy.
shelflife: It is from Acula and Kubala (1975).
It is about shelflife (in days) of a food product.
urinary: It is from Santiago and Smith (2013).
It is about the days in between discharge of males in nosocomial urinary tract infections in patients.
Usage
Wdata
References
Finkelstein, D. M. (1986). A proportional hazards model for interval-censored failure time data. Biometrics, 42, 845-865.
Lieblein, J. and M. Zelen (1956). Statistical Investigation of the Fatigue Life of Deep-Groove Ball Bearings. Journal of Research of the National Bureau of Standards, 57(5), 273-316.
Lindsey, J. C. and L. M. Ryan (1998). Tutorial in biostatistics: Methods for interval-censored data. Stat. Med., 17, 219-238.
Gacula, M. C., JR and Kubala, J. J. (1975). Statistical models for shelf life failures. Journal of Food Science, 40, 404-409.
Santiago, E. and J. Smith (2013). Control Charts Based on the Exponential Distribution: Adapting Runs Rules for the t Chart. Quality Engineering, 25(2), 85-96.
Smith, R. L. and J. C. Naylor (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Journal of the Royal Statistical Society. Series C (Applied Statistics), 36(3), 358-369.
Examples
# Attach datasets
attach(Wdata)
bearing
glassfiber1.5
glassfiber15
radiotherapy
radio.chemotherapy
shelflife
urinary
Weibull quantile values
Description
Quantiles for the Weibullness Test. They are obtained
from the sample correlation from the Weibull plot.
The number of Monte Carlo iterations is 1.0E08.
Dataset representing the quantiles and the associated critical values for the Weibullness
test. They were obtained by conducting Monte Carlo simulations where the
sample correlation coefficients were calculated based on the Weibull plot.
We used 1.0E08 Monte Carlo iterations in the simulation.
Usage
Weibull.Plot.QuantilesFormat
This data frame contains 998 rows and 1001 columns.
References
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Finite-sample breakdown point for the power-weighted repeated median (PWRM) estimators
Description
Calculates the lower and upper finite-sample breakdown points for the PWRM estimators of the regression coefficients.
Usage
breakdown.pwrm(x, power = 1)
Arguments
x |
a numeric vector of observations. |
power |
the power used in the weights of the PWRM method. The default value is 1. |
Details
For details on the lower and upper finite-sample breakdown points, see the Appendix of Park et al. (2024).
Value
A numeric vector containing the lower and upper finite-sample breakdown points.
Author(s)
Chanseok Park
References
Park, C., X. Gao, and M. Wang (2024). Robust explicit estimators using the power-weighted repeated medians. Journal of Applied Statistics, 51(8), 1590–1608. doi:10.1080/02664763.2023.2229969
See Also
weibull.pwrm for parameter estimation using the PWRM method.
Examples
attach(Wdata)
# See Section 5.1 of Park et al. (2024).
breakdown.pwrm(log(bearing), power=1)
breakdown.pwrm(log(bearing), power=2)
breakdown.pwrm(log(bearing), power=3)
Exponential Probability Plot
Description
ep.plot produces a exponential probability plot.
Usage
ep.plot(x, plot.it=TRUE, a, col.line="black", lty.line=1,
xlim=NULL, ylim=NULL, main=NULL, sub=NULL, xlab=NULL, ylab="Probability", ...)
Arguments
x |
a numeric vector of data values. Missing values are allowed. |
plot.it |
logical. Should the result be plotted? |
a |
the offset fraction to be used; typically in (0,1). See |
col.line |
the color of the straight line. |
lty.line |
the line type of the straight line. |
xlim |
the x limits of the plot. |
ylim |
the y limits of the plot. |
main |
a main title for the plot, see also |
sub |
a sub title for the plot. |
xlab |
a label for the x axis, defaults to a description of x. |
ylab |
a label for the y axis, defaults to "Probability". |
... |
graphical parameters. |
Details
The exponential probability plot is based on taking the logarithm of the exponential cumulative distribution function.
Value
A list with the following components:
x |
The sorted data |
y |
-log(1-ppoints(n,a=a)) |
Author(s)
Chanseok Park
See Also
plot, qqnorm, qqplot, wp.plot, iwp.plot.
bs.plot for the Birnbaum-Saunders probability plot in package bsgof.
Examples
x = rexp(50)
# With cosmetic lines
ep.plot(x, main="Exponential Probability Plot", col.line="red", lty.line=1, pch=3)
hline = -log(1- c(0.01,(1:9)/10,0.99))
abline( h=hline, col=gray(0.1), lty=3, lwd=0.5 )
abline( v=seq(0, 5,by=0.5), col=gray(0.1), lty=3, lwd=0.5 )
The Exponential Goodness-of-Fit Test from the Exponential Probability Plot
Description
Performs Goodness-of-fit test for the exponential distribution using the sample correlation from the exponential probability plot.
Usage
ep.test(x, a)
Arguments
x |
a numeric vector of data values. Missing values are allowed, but the number of non-missing values must be between 3 and 1000. |
a |
the offset fraction to be used; typically in (0,1). See ppoints(). |
Details
The exponential goodness-of-fit test is constructed using the sample correlation
which is calculated using the associated exponential probability plot.
The critical value is then looked up in Exponential.Plot.Quantiles.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the exponential probability plot) |
p.value |
the p-value for the test. |
sample.size |
sample size (missing observations are deleted). |
method |
a character string indicating the exponential goodness-of-fit test. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Shapiro, S. S. and M. B. Wilk (1972). An Analysis of Variance Test for the Exponential Distribution (Complete Samples). Technometrics, 14(2), 355-370.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
wp.test for performing the Weibullness test based on the Weibull probability plot.
Examples
# For Exponential GOF.
# Dataset from Section 2.5 of Shapiro and Wilk (1972).
x = c(6, 1, -4, 8, -2, 5, 0)
ep.test(x)
Critical Value for the Exponential Goodness-of-Fit test
Description
Calculates the critical value for the Weibullness test
Usage
ep.test.critical(alpha, n)
Arguments
alpha |
the significance level. |
n |
the sample size. |
Details
This function calculates the critical value for the Weibullness test
which is constructed using the sample correlation
from the associated Weibull plot.
The critical value is then looked up in Weibull.Plot.Quantiles.
There is print method for class "ep.test.critical".
Value
A list with class "ep.test.critical" containing the following components:
sample.size |
sample size (missing observations are deleted). |
alpha |
significance level. |
critical.value |
critical value. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
wp.test for performing the Weibullness test.
Examples
# Critical value with alpha (significance level) and n (sample size).
ep.test.critical(alpha=0.0982, n=7)
The p-value for the Exponential Goodness-of-Fit Test
Description
Calculates the p-value for the exponential goodness-of-fit test which is based on the sample correlation from the exponential probability plot.
Usage
ep.test.pvalue(w, n)
Arguments
w |
the ANOVA test statistic; w is in (0,1). |
n |
the sample size. |
Details
The p-value for the exponential goodness-of-fit test which is based on
the sample correlation from the exponential probability plot.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the exponential probability plot) |
p.value |
the p-value for the test. |
method |
a character string indicating the exponential probability test. |
Author(s)
Chanseok Park
References
Shapiro, S. S. and M. B. Wilk (1972). An Analysis of Variance Test for the Exponential Distribution (Complete Samples). Technometrics, 14(2), 355-370.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
wp.test for performing the Weibullness test.
Examples
# p.value with w (ANOVA statistic) and n (sample size).
ep.test.pvalue(w=0.35593, n=7)
Gumbel Probability Plot
Description
gp.plot produces a Gumbel probability plot.
Usage
gp.plot(x, plot.it=TRUE, a, col.line="black", lty.line=1,
xlim=NULL, ylim=NULL, main=NULL, sub=NULL, xlab=NULL, ylab="Probability", ...)
Arguments
x |
a numeric vector of data values. Missing values are allowed. |
plot.it |
logical. Should the result be plotted? |
a |
the offset fraction to be used; typically in (0,1). See |
col.line |
the color of the straight line. |
lty.line |
the line type of the straight line. |
xlim |
the x limits of the plot. |
ylim |
the y limits of the plot. |
main |
a main title for the plot, see also |
sub |
a sub title for the plot. |
xlab |
a label for the x axis, defaults to a description of x. |
ylab |
a label for the y axis, defaults to "Probability". |
... |
graphical parameters. |
Details
The Gumbel probability plot is based on taking the logarithm of the Gumbel cumulative distribution function twice.
Value
A list with the following components:
x |
The sorted data |
y |
-log(-log(ppoints(n,a=a))) |
Author(s)
Chanseok Park
See Also
plot, qqnorm, qqplot, wp.plot, iwp.plot, ep.plot.
bs.plot for the Birnbaum-Saunders probability plot in package bsgof.
Examples
x = c(-3.16, -3.07, -2.24, -1.8, -1.48, -0.92, -0.87, -0.41, -0.06, 1.15)
# With cosmetic lines
gp.plot(x, main="Gumbel Probability Plot", col.line="red",
xlab="Lifetimes of bearings", lty.line=1, pch=3)
hline = -log(-log(c( (1:5)/100, (1:9)/10)))
abline( h=hline, col=gray(0.1), lty=3, lwd=0.5 )
abline( v=seq(-4, 2,by=0.5), col=gray(0.1), lty=3, lwd=0.5 )
Gumbel Goodness-of-Fit Test from a Gumbel Probability Plot
Description
Performs the statistical goodness-of-fit test for the Gumbel distribution using the sample correlation from the Gumbel probability plot.
Usage
gp.test(x, a)
Arguments
x |
a numeric vector of data values. Missing values are allowed, but the number of non-missing values must be between 3 and 1000. |
a |
the offset fraction to be used; typically in (0,1). See ppoints(). |
Details
The Gumbel goodness-of-fit test is constructed using the sample correlation
which is calculated using the associated Gumbel probability plot.
The critical value is then looked up in Gumbel.Plot.Quantiles.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the Gumbel probability plot) |
p.value |
the p-value for the test. |
sample.size |
sample size (missing observations are deleted). |
method |
a character string indicating the Gumbel goodness-of-fit test. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Kinnison, R. (1989). Correlation Coefficient Goodness-of-Fit Test for the Extreme-Value Distribution. The American Statistician, 43(2), 98-100.
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
wp.test for performing the Weibullness test.
Examples
# Gumbel goodness-of-fit test.
x = c(-3.16, -3.07, -2.24, -1.8, -1.48, -0.92, -0.87, -0.41, -0.06, 1.15)
gp.test(x)
Critical Value for the Gumbel Goodness-of-Fit Test
Description
Calculates the critical value for the Gumbel goodness-of-fit test
Usage
gp.test.critical(alpha, n)
Arguments
alpha |
the significance level. |
n |
the sample size. |
Details
This function calculates the critical value for the Gumbel Goodness-of-Fit test
which is constructed using the sample correlation from
the associated Gumbel probability plot.
The critical value is then looked up in Gumbel.Plot.Quantiles.
There is print method for class "gp.test.critical".
Value
A list with class "gp.test.critical" containing the following components:
sample.size |
sample size (missing observations are deleted). |
alpha |
significance level. |
critical.value |
critical value. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Kinnison, R. (1989). Correlation Coefficient Goodness-of-Fit Test for the Extreme-Value Distribution. The American Statistician, 43(2), 98-100.
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
wp.test for performing the Weibullness test.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# Critical value with alpha (significance level) and n (sample size).
gp.test.critical(alpha=0.01, n=10)
The p-value for the Gumbel goodness-of-Fit Test
Description
Calculates the p-value for the Gumbel goodness-of-fit test which is based on the sample correlation from the Gumbel probability plot.
Usage
gp.test.pvalue(r, n)
Arguments
r |
the sample correlation coefficient from the Gumbel probability plot; r is in (0,1). |
n |
the sample size. |
Details
The p-value for the Gumbel goodness-of-fit test which is based on
the sample correlation from the Gumbel probability plot.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the Gumbel probability plot) |
p.value |
the p-value for the test. |
method |
a character string indicating the Gumbel goodness-of-fit test. |
Author(s)
Chanseok Park
References
Kinnison, R. (1989). Correlation Coefficient Goodness-of-Fit Test for the Extreme-Value Distribution. The American Statistician, 43(2), 98-100.
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
wp.test for performing the Weibullness test.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# p.value with r (sample correlation from the Gumbel probability plot) and n (sample size).
gp.test.pvalue(r=0.98504, n=10)
Estimate of location and scale parameters of the Gumbel distribution
Description
Calculates the estimates of the location and scale parameters from the Gumbel probability plot.
Usage
gumbel.gp(x, n, a)
Arguments
x |
a numeric vector of observations. |
n |
The number of observations is needed if there is right-censoring. |
a |
the offset fraction to be used; typically in (0,1). See |
Details
The Gumbel distribution has the cumulative distribution function
F(x) = \exp\Big[-\exp\Big(-\frac{x-\mu}{\sigma}\Big)\Big],
where \sigma>0.
The location (\mu) and scale (\sigma) parameters are estimated
using the intercept and slope estimates from the
Gumbel probability plot.
gumbel.gp obtains the estimates of the location and scale.
Value
An object of class "gumbel.estimate", a list with
two parameter estimates
Author(s)
Chanseok Park
See Also
gumbel.mle for the maximum likelihood parameter estimates.
weibull.wp for the parameter estimation using the Weibull plot.
Examples
x = c(-2.73, 11.69, 34.85, 7.97, -0.86, 17.46, 9.69, -7.14, 14.86, 16.85)
gumbel.gp(x)
Maximum likelihood estimates of Gumbel distribution
Description
Calculates the maximum likelihood estimates of Gumbel distribution.
Usage
gumbel.mle(x, interval, tol = .Machine$double.eps^0.5, maxiter = 1000, trace = 0)
Arguments
x |
a numeric vector of observations. |
interval |
a vector containing the end-points of the interval to be estimated for the scale parameter. |
tol |
the desired accuracy (convergence tolerance). |
maxiter |
the maximum number of iterations. |
trace |
integer number; if positive, tracing information is produced. Higher values giving more details. |
Details
The Gumbel distribution has the cumulative distribution function
F(x) = \exp\Big[-\exp\Big(-\frac{x-\mu}{\sigma}\Big)\Big],
where \sigma>0.
The location (\mu) and scale (\sigma) parameters are estimated
using the maximum likelihood.
If the algorithm does not converge in maxiter steps,
a warning is printed and the current approximation is returned
(see also uniroot).
Value
An object of class "gumbel.estimate", a list with two parameter estimates.
Author(s)
Chanseok Park
References
Gumbel, E. J. (1954). Statistical Theory of Extreme Values and Some Practical Applications (National Bureau of Standards Applied Mathematics Series 33) U.S. Government Printing Office, Washington, D.C.
Gentleman, J., Whitmore, G., Zwiers, F., and Ross, W. (1994). Extreme-value analysis of canadian wind speeds. In Gentleman, J. and Whitmore, G., editors, Case Studies in Data Analysis. Lecture Notes in Statistics, vol 94, Springer, New York, NY.
See Also
weibull.mle for the maximum likelihood parameter estimates of the Weibull distribution.
gumbel.gp for the parameter estimation using the Gumbel plot.
Examples
data = c(79.5, 68.4, 74.0, 59.2, 74.0, 64.8, 64.8, 59.2, 79.5, 62.9,
59.2, 68.2, 64.8, 88.8, 88.8, 75.8, 68.4, 68.4, 61.0, 51.8,
62.9, 64.8, 61.0, 61.0, 70.3, 68.4, 55.5, 64.8, 77.7, 57.3,
48.1, 53.6, 55.5, 62.9, 61.0, 61.0, 51.8, 48.1)
gumbel.mle(data)
The inverse Weibull distribution
Description
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Inverse Weibull distribution
with parameters shape and scale.
Usage
dinvweibull(x, shape, scale = 1, log = FALSE)
pinvweibull(q, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
qinvweibull(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
rinvweibull(n, shape, scale = 1)
Arguments
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
shape, scale |
parameters. Must be positive. |
log, log.p |
logical; if |
lower.tail |
logical; if |
Details
The probability density function of the inverse Weibull distribution with parameters shape
=\beta and scale = \theta is given by
f(x) = \frac{\beta (\theta/x)^\beta e^{-(\theta/x)^\beta}}{x}
where x > 0, \beta > 0 and \theta > 0.
The cumulative distribution function is given by
F(X)=\exp(-(\theta/x)^\beta)
Value
dinvweibull gives the density, pinvweibull gives the distribution function,
qinvweibull gives the quantile function, and rinvweibull generates random deviates.
Author(s)
Chanseok Park
Examples
x = (-1):2
names(x) = letters[1:4]
dinvweibull(x, shape=2)
exp( dinvweibull(x, shape=2, log=TRUE) )
pinvweibull (1, shape=2)
exp(pinvweibull (1, shape=2, log=TRUE))
q = c(-1,0,1,2)
qinvweibull ( pinvweibull (q, shape=2), shape=2 )
Maximum likelihood estimates of the two-parameter inverse Weibull distribution
Description
Calculates the maximum likelihood estimates of the two-parameter Weibull distribution.
Usage
invweibull.mle(x, interval, tol = .Machine$double.eps^0.5, maxiter = 1000, trace = 0)
Arguments
x |
a numeric vector of observations. |
interval |
a vector containing the end-points of the interval to be estimated for the shape parameter. |
tol |
the desired accuracy (convergence tolerance). |
maxiter |
the maximum number of iterations. |
trace |
integer number; if positive, tracing information is produced. Higher values giving more details. |
Details
The two-parameter inverse Weibull distribution has the cumulative distribution function
F(X)=\exp(-(\theta/x)^\beta)
where x>0, \beta>0 and \theta>0.
The shape (\beta) and scale (\theta) parameters are estimated
by calling weibull.mle on the reciprocally transformed data.
The maximum likelihood estimation on the the reciprocally transformed data is performed using the method by
Farnum and Booth (1997).
If interval is missing, the interval is given by the method in
Farnum and Booth (1997).
Convergence is declared either if f(x) == 0
or the change in x for one step of the algorithm is less than
tol (see also uniroot).
If the algorithm does not converge in maxiter steps,
a warning is printed and the current approximation is returned
(see also uniroot).
Value
An object of class "weibull.estimate", a list with
two parameter estimates.
Author(s)
Chanseok Park
References
Farnum, N. R. and P. Booth (1997). Uniqueness of Maximum Likelihood Estimators of the 2-Parameter Weibull Distribution. IEEE Transactions on Reliability, 46, 523-525.
Examples
attach(Wdata)
invweibull.mle(1/bearing)
Inverse Weibull Probability Plot
Description
iwp.plot produces an inverse Weibull probability plot.
Usage
iwp.plot(x, plot.it=TRUE, a, col.line="black", lty.line=1,
xlim=NULL, ylim=NULL, main=NULL, sub=NULL, xlab=NULL, ylab="Probability", ...)
Arguments
x |
a numeric vector of data values. Missing values are allowed. |
plot.it |
logical. Should the result be plotted? |
a |
the offset fraction to be used; typically in (0,1). See |
col.line |
the color of the straight line. |
lty.line |
the line type of the straight line. |
xlim |
the x limits of the plot. |
ylim |
the y limits of the plot. |
main |
a main title for the plot, see also |
sub |
a sub title for the plot. |
xlab |
a label for the x axis, defaults to a description of x. |
ylab |
a label for the y axis, defaults to "Probability". |
... |
graphical parameters. |
Details
The inverse Weibull probability plot is based on taking the logarithm of the inverse Weibull cumulative distribution function twice.
Value
A list with the following components:
x |
The sorted data |
y |
-log(-log(ppoints(n,a=a))) |
Author(s)
Chanseok Park
See Also
plot, qqnorm, qqplot, wp.plot.
Examples
x = c(0.38, 0.41, 1.2, 0.52, 0.69, 0.89, 0.67, 1.59, 0.55, 0.59)
iwp.plot( x )
# With cosmetic lines
iwp.plot(x, main="Inverse Weibull Probability Plot", col.line="red",
xlab="Lifetimes of bearings", lty.line=1, pch=3)
hline = -log(-log( c( (1:5)/100, (1:9)/10) ))
abline( h=hline, col=gray(0.1), lty=3, lwd=0.5 )
abline( v= seq(0, 2,by=0.1), col=gray(0.1), lty=3, lwd=0.5 )
Weibullness Test from a inverse Weibull Plot
Description
Performs the statistical test of inverse Weibullness (Goodness-of-fit test for the inverse Weibull distribution) using the sample correlation from the inverse Weibull plot.
Usage
iwp.test(x, a)
Arguments
x |
a numeric vector of data values. Missing values are allowed, but the number of non-missing values must be between 3 and 1000. |
a |
the offset fraction to be used; typically in (0,1). See ppoints(). |
Details
The inverse Weibullness test is constructed using the sample correlation
which is calculated using the associated inverse Weibull plot.
The critical value is then looked up in IW.Plot.Quantiles.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the inverse Weibull plot) |
p.value |
the p-value for the test. |
sample.size |
sample size (missing observations are deleted). |
method |
a character string indicating the inverse Weibullness test. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
wp.test for performing the Weibullness test.
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# For inverse Weibullness hypothesis test.
attach(Wdata)
iwp.test(1/bearing)
# Note
wp.test(bearing)
Critical Value for the inverse Weibullness Test
Description
Calculates the critical value for the inverse Weibullness test
Usage
iwp.test.critical(alpha, n)
Arguments
alpha |
the significance level. |
n |
the sample size. |
Details
This function calculates the critical value for the inverse Weibullness test
which is constructed using the sample correlation
from the associated inverse Weibull plot.
The critical value is then looked up in IW.Plot.Quantiles.
There is print method for class "iwp.test.critical".
Value
A list with class "iwp.test.critical" containing the following components:
sample.size |
sample size (missing observations are deleted). |
alpha |
significance level. |
critical.value |
critical value. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice, 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# Critical value with alpha (significance level) and n (sample size).
iwp.test.critical(alpha=0.01, n=10)
The p-value for the inverse Weibullness Test
Description
Calculates the p-value for the inverse Weibullness test which is based on the sample correlation from the inverse Weibull plot.
Usage
iwp.test.pvalue(r, n)
Arguments
r |
the sample correlation coefficient from the Weibull plot; r is in (0,1). |
n |
the sample size. |
Details
The p-value for the inverse Weibullness test which is based on
the sample correlation from the inverse Weibull plot.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the Weibull plot) |
p.value |
the p-value for the test. |
method |
a character string indicating the inverse Weibullness test. |
Author(s)
Chanseok Park
References
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice, 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# p.value with r (sample correlation from the inverse Weibull plot) and n (sample size).
iwp.test.pvalue(r=0.6, n=10)
Estimates of the intercept and slope parameters of a simple linear regression using the power-weighted repeated median (PWRM).
Description
Calculates robust estimates of the intercept and slope parameters of a simple linear regression using the PWRM method.
Usage
regr.pwrm(x, y, weight = FALSE, power = 1)
Arguments
x |
a numeric vector of covariate observations. |
y |
a numeric vector of response observations. |
weight |
logical. If |
power |
the power used in the weights of the PWRM method. The default value is 1. |
Details
regr.pwrm obtains robust estimates of the intercept and slope parameters using the PWRM method. These estimates can be used to estimate the shape and scale parameters of the Weibull distribution from a Weibull plot.
The regr.pwrm function can also be used to estimate the Weibull parameters, as shown in the examples below.
Value
An object of class weibull.estimate, which is a list containing the estimates of the intercept and slope parameters.
Author(s)
Chanseok Park
References
Park, C., X. Gao, and M. Wang (2024). Robust explicit estimators using the power-weighted repeated medians. Journal of Applied Statistics, 51(8), 1590–1608. doi:10.1080/02664763.2023.2229969
See Also
regr.rm for parameter estimation using the repeated median (RM) method.
weibull.pwrm for parameter estimation using the PWRM method.
Examples
data <- rweibull(10, shape = 2)
x <- sort(log(data))
y <- log(-log(1 - ppoints(x)))
para <- as.numeric(regr.pwrm(x, y))
c(para[2], exp(-para[1] / para[2]))
weibull.pwrm(data)
Estimates of the intercept and slope parameters of a simple linear regression using the repeated median (RM).
Description
Calculates robust estimates of the intercept and slope parameters of a simple linear regression using the RM method.
Usage
regr.rm(x, y)
Arguments
x |
a numeric vector of covariate observations. |
y |
a numeric vector of response observations. |
Details
regr.rm obtains robust estimates of the intercept and slope parameters using the RM method.
These estimates can be used to estimate the shape and scale parameters of the Weibull distribution from a Weibull plot.
The regr.rm function can also be used to estimate the Weibull parameters, as shown in the examples below.
Value
An object of class weibull.estimate, which is a list containing the estimates of the intercept and slope parameters.
Author(s)
Chanseok Park
References
Boudt, K. and D. Caliskan and C. Croux (2011). obust explicit estimators of Weibull parameters. Metrika, 73, 187-209.
See Also
regr.pwrm for parameter estimation using the power-weighted repeated median (PWRM) method.
weibull.rm for parameter estimation using the PWRM method.
Examples
data <- rweibull(10, shape = 2)
x <- sort(log(data))
y <- log(-log(1 - ppoints(x)))
para <- as.numeric(regr.rm(x, y))
c(para[2], exp(-para[1] / para[2]))
weibull.rm(data)
Maximum likelihood estimates with interval censoring
Description
Calculates the maximum likelihood estimates with interval censoring using the EM Algorithm.
Usage
weibull.ic(X, start=c(1,1), maxits=10000, eps=1E-5)
Arguments
X |
a numeric matrix (n x 2) of observations. |
start |
a starting value. |
maxits |
the maximum number of iterations. |
eps |
the desired accuracy (convergence tolerance). |
Details
The expectation-maximization(EM) algorithm is used for estimating the parameters with interval-censored data.
Value
Calculates the maximum likelihood estimates with interval-censored data
Author(s)
Chanseok Park
References
Park, C. (2023). A Note on Weibull Parameter Estimation with Interval Censoring Using the EM Algorithm. Mathematics, 11(14), 3156. doi:10.3390/math11143156
Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed.; John Wiley & Sons: New York, NY.
See Also
weibull.wp for the parameter estimation using the Weibull plot with full observations.
weibull.mle for the parameter estimation using the maximum likelihood method with full observations.
Examples
attach(Wdata)
weibull.ic(radio.chemotherapy)
# Two-parameter Weibull with full observations
weibull.ic( cbind(bearing,bearing) )
# Two-parameter Weibull with full observations (using weibull.mle)
weibull.mle(bearing, threshold=0)
Maximum likelihood estimates of three-parameter Weibull distribution
Description
Calculates the maximum likelihood estimates of three-parameter Weibull distribution.
Usage
weibull.mle(x, threshold, interval, interval.threshold, extendInt="downX",
a, tol = .Machine$double.eps^0.5, maxiter = 1000, trace = 0)
Arguments
x |
a numeric vector of observations. |
threshold |
the threshold parameter value. |
interval |
a vector containing the end-points of the interval to be estimated for the shape parameter. |
interval.threshold |
a vector containing the end-points of the interval to be estimated for the threshold parameter. |
extendInt |
character string specifying if the interval c(left,right) should be extended or directly produce an error when f() has no differing signs at the endpoints. The default, "downX", keep lowering the the left end of the interval so that f() has different signs. See |
a |
the offset fraction to be used; typically in (0,1). |
tol |
the desired accuracy (convergence tolerance). |
maxiter |
the maximum number of iterations. |
trace |
integer number; if positive, tracing information is produced. Higher values giving more details. |
Details
The three-parameter Weibull distribution has the cumulative distribution function
F(x) = 1 - \exp\Big[ - \Big( \frac{x-\theta}{\beta} \Big)^{\alpha} \Big],
where x>\theta.
The shape (\alpha) and scale (\beta) parameters are estimated
using the maximum likelihood.
The maximum likelihood estimation is performed using the method by
Farnum and Booth (1997).
If the threshold (\theta) is missing, it is estimated by
weibull.threshold.
If threshold=0, then weibull.mle calculates the maximum likelihood
estimates of the two-parameter Weibull distribution.
If interval is missing, the interval is given by the method in
Farnum and Booth (1997).
If interval.threshold is missing, the interval is initally given
by (min(x)-sd(x), min(x)). If this interval does not include
the estimate, its lower bound is extended (see also uniroot).
The choice of a follows ppoints function.
Convergence is declared either if f(x) == 0
or the change in x for one step of the algorithm is less than
tol (see also uniroot).
If the algorithm does not converge in maxiter steps,
a warning is printed and the current approximation is returned
(see also uniroot).
Value
An object of class "weibull.estimate", a list with
two parameter estimates (if threshold is given) or three-parameter estimates.
Author(s)
Chanseok Park
References
Park, C. (2018). A Note on the Existence of the Location Parameter Estimate of the Three-Parameter Weibull Model Using the Weibull Plot. Mathematical Problems in Engineering, 2018, 6056975. doi:10.1155/2018/6056975
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice, 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Farnum, N. R. and P. Booth (1997). Uniqueness of Maximum Likelihood Estimators of the 2-Parameter Weibull Distribution. IEEE Transactions on Reliability, 46, 523-525.
See Also
weibull.wp for the parameter estimation using the Weibull plot.
weibull.rm for robust parameter estimation using the repeated median method.
weibull.threshold for the estimate of the threshold parameter.
fitdistr for maximum-likelihood fitting of univariate distributions in package MASS.
Examples
# Three-parameter Weibull
data = c(355,725,884,462,1092,190,166,172,188,224,267,298,355,471,
154,101,76,811,80,249,752,305,301,386,667,212,186,127,
121,214,242,237,355,210,253,400,401,514,211,285)
weibull.mle(data)
# Two-parameter Weibull
weibull.mle(data, threshold=0)
Robust estimate of shape and scale parameters of Weibull using the power-weighted repeated median (PWRM) method
Description
Calculates the estimates of the shape and scale parameters.
Usage
weibull.pwrm(x, a, weight=FALSE, power=1)
Arguments
x |
a numeric vector of observations. |
a |
the offset fraction to be used; typically in (0,1). See |
weight |
|
power |
|
Details
weibull.pwrm obtains the robust estimates of the shape and scale parameters
using the intercept and slope estimates using the PWRM method from the Weibull plot.
Value
An object of class weibull.estimate, a list with two parameter estimates
Author(s)
Chanseok Park
References
Park, C., X. Gao, and M. Wang (2024). Robust explicit estimators using the power-weighted repeated medians. Journal of Applied Statistics, 51(8), 1590-1608. doi:10.1080/02664763.2023.2229969
See Also
weibull.mle for the parameter estimation using the maximum likelihood method.
weibull.wp for the parameter estimation using the Weibull plot.
weibull.rm for the parameter estimation using the repeated median (RM) method.
fitdistr for maximum-likelihood fitting of univariate distributions in package MASS.
Examples
attach(Wdata)
weibull.pwrm(bearing)
Robust estimate of shape and scale parameters of Weibull using the repeated median method
Description
Calculates the estimates of the shape and scale parameters.
Usage
weibull.rm(x, a)
Arguments
x |
a numeric vector of observations. |
a |
the offset fraction to be used; typically in (0,1). See |
Details
weibull.rm obtains the robust estimates of the shape and scale
parameters using the intercept and slope estimates using the repeated median method from the
Weibull plot.
Value
An object of class "weibull.estimate", a list with
two parameter estimates
Author(s)
Chanseok Park
References
Siegel, A. F. (1982). Robust Regression Using Repeated Medians. Biometrika, 69, 242-244.
See Also
weibull.mle for the parameter estimation using the maximum likelihood method.
weibull.wp for the parameter estimation using the Weibull plot.
weibull.pwrm for the parameter estimation using the power-weighted repeated median (PWRM) method.
fitdistr for maximum-likelihood fitting of univariate distributions in package MASS.
Examples
attach(Wdata)
weibull.rm(bearing)
Maximum likelihood estimates of two-parameter Weibull distribution
Description
Calculates the maximum likelihood estimates of two-parameter Weibull distribution.
Usage
weibull.seki(x)
Arguments
x |
a numeric vector of observations. |
Details
The two-parameter Weibull distribution has the cumulative distribution function
F(x) = 1 - \exp\Big[ - \Big( \frac{x}{\beta} \Big)^{\alpha} \Big]
.
The shape (\alpha) and scale (\beta) parameters are estimated the Seki method.
Value
An object of class "weibull.estimate", a list with two parameter estimates.
Author(s)
Chanseok Park
References
Seki, T. and S. Yokoyama (1993). Simple and robust estimation of the Weibull parameters. Microelectronics Reliability, 33, 45-52.
See Also
weibull.mle for the parameter estimation using the maximum likelihood method.
weibull.wp for the parameter estimation using the Weibull plot.
weibull.rm for robust parameter estimation using the repeated median (RM) method.
weibull.pwrm for robust parameter estimation using the power-weighted repeated median (PWRM) method.
fitdistr for maximum-likelihood fitting of univariate distributions in package MASS.
Examples
attach(Wdata)
weibull.seki(bearing)
Estimate of threshold parameter of three-parameter Weibull distribution
Description
Calculates the estimate of the threshold parameter.
Usage
weibull.threshold(x, a, interval.threshold, extendInt="downX")
Arguments
x |
a numeric vector of observations. |
a |
the offset fraction to be used; typically in (0,1). |
interval.threshold |
a vector containing the end-points of the interval to be estimated for the threshold parameter. |
extendInt |
character string specifying if the interval c(left,right) should be extended or directly produce an error when f() has no differing signs at the endpoints. The default, "downX", keep lowering the the left end of the interval so that f() has different signs. See |
Details
The three-parameter Weibull distribution has the cumulative distribution function
F(x) = 1 -\exp\Big[-\Big( \frac{x-\theta}{\beta}\Big)^{\alpha}\Big],
where x>\theta.
The threshold parameter (\theta) is estimated
by maximizing the correlation function from the Weibull plot.
The choice of a follows ppoints function.
If interval.threshold is missing, the interval is initially given
by (min(x)-sd(x), min(x)). If this interval does not include
the estimate, its lower bound is extended (see also uniroot).
Value
weibull.threshold returns a numeric value.
Author(s)
Chanseok Park
References
Park, C. (2018). A Note on the Existence of the Location Parameter Estimate of the Three-Parameter Weibull Model Using the Weibull Plot. Mathematical Problems in Engineering, 2018, 6056975. doi:10.1155/2018/6056975
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice, 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
See Also
weibull.mle for the maximum likelihood estimate.
weibull.wp for the parameter estimation using the Weibull plot.
Examples
attach(Wdata)
weibull.threshold(bearing)
Estimate of shape and scale parameters of Weibull from the Weibull plot
Description
Calculates the estimates of the shape and scale parameters.
Usage
weibull.wp(x, n, a)
Arguments
x |
a numeric vector of observations. |
n |
The number of observations is needed if there is right-censoring. |
a |
the offset fraction to be used; typically in (0,1). See |
Details
weibull.wp obtains the estimates of the shape and scale
parameters using the intercept and slope estimates from the
Weibull plot.
Value
An object of class "weibull.estimate", a list with
two parameter estimates
Author(s)
Chanseok Park
See Also
weibull.mle for the parameter estimation using the maximum likelihood method.
weibull.rm for robust parameter estimation using the repeated median method.
weibull.pwrm for the parameter estimation using the power-weighted repeated median
(PWRM) method.
fitdistr for maximum-likelihood fitting of univariate distributions in package MASS.
Examples
attach(Wdata)
weibull.wp(bearing)
Weighted median
Description
Calculates the weighted median of a vector x with its corresponding w.
Usage
wmedian(x, w, a=0.5)
Arguments
x |
a numeric vector of observations. |
w |
a numeric vector of weights corresponding to |
a |
the offset fraction in [0,1]. A value of zero calculates the left quantile of the weighted empirical cumulative distribution function (CDF), while a value of one calculates the right quantile of the weighted empirical CDF. |
Details
The weighted empirical CDF is defined by
F_w(t) = \sum_{j=1}^{n} w_j I(x_j \le t),
where I is the indicator function. For more details, see Section 2.1 of Park et al. (2024).
The left quantile is given by Q_L(u) = \inf\{t: F_w(t) \ge u\},
and the right quantile is given by Q_R(u) = \inf\{t: F_w(t) > u\}.
Then, wmedian calculates (1-a)Q_L(u) + aQ_R(u),
where a is the interpolation value.
Value
a numeric value.
Author(s)
Chanseok Park
References
F. Y. Edgeworth (1888). On a New Method of Reducing Observations Relating to Several Quantities. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25(154), 184-191.
Park, C., X. Gao, and M. Wang (2024). Robust explicit estimators using the power-weighted repeated medians. Journal of Applied Statistics, 51(8), 1590-1608. doi:10.1080/02664763.2023.2229969
See Also
median for the conventional median.
Examples
wmedian( c(1,2,3,4,5), c(0.15, 0.1, 0.2, 0.3, 0.25) )
wmedian( c(1,2,3,4,5) ) # is the same as median( c(1,2,3,4,5) )
Weibull Plot
Description
wp.plot produces a Weibull plot.
Usage
wp.plot(x, plot.it=TRUE, a, col.line="black", lty.line=1,
xlim=NULL, ylim=NULL, main=NULL, sub=NULL, xlab=NULL, ylab="Probability", ...)
Arguments
x |
a numeric vector of data values. Missing values are allowed. |
plot.it |
logical. Should the result be plotted? |
a |
the offset fraction to be used; typically in (0,1). See |
col.line |
the color of the straight line. |
lty.line |
the line type of the straight line. |
xlim |
the x limits of the plot. |
ylim |
the y limits of the plot. |
main |
a main title for the plot, see also |
sub |
a sub title for the plot. |
xlab |
a label for the x axis, defaults to a description of x. |
ylab |
a label for the y axis, defaults to "Probability". |
... |
graphical parameters. |
Details
The Weibull plot is based on taking the logarithm of the Weibull cumulative distribution function twice. The horizontal axis is logarithmic.
Value
A list with the following components:
x |
The sorted data |
y |
log(-log(1-ppoints(n,a=a))) |
Author(s)
Chanseok Park
See Also
plot, qqnorm, qqplot, iwp.plot, ep.plot.
bs.plot for the Birnbaum-Saunders probability plot in package bsgof.
Examples
attach(Wdata)
wp.plot(bearing)
# With cosmetic lines
wp.plot(bearing, main="Weibull Plot", col.line="red",
xlab="Lifetimes of bearings", lty.line=1, pch=3)
hline = log(-log(1- c( (1:5)/100, (1:9)/10) ))
abline( h=hline, col=gray(0.1), lty=3, lwd=0.5 )
abline( v=seq(15, 200,by=5), col=gray(0.1), lty=3, lwd=0.5 )
Weibullness Test from a Weibull Plot
Description
Performs the statistical test of Weibullness (Goodness-of-fit test for the Weibull distribution) using the sample correlation from the Weibull plot.
Usage
wp.test(x, a)
Arguments
x |
a numeric vector of data values. Missing values are allowed, but the number of non-missing values must be between 3 and 1000. |
a |
the offset fraction to be used; typically in (0,1). See ppoints(). |
Details
The Weibullness test is constructed using the sample correlation
which is calculated using the associated Weibull plot.
The critical value is then looked up in Weibull.Plot.Quantiles.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the Weibull plot) |
p.value |
the p-value for the test. |
sample.size |
sample size (missing observations are deleted). |
method |
a character string indicating the Weibullness test. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# For Weibullness hypothesis test.
attach(Wdata)
wp.test(urinary)
Critical Value for the Weibullness Test
Description
Calculates the critical value for the Weibullness test
Usage
wp.test.critical(alpha, n)
Arguments
alpha |
the significance level. |
n |
the sample size. |
Details
This function calculates the critical value for the Weibullness test
which is constructed using the sample correlation
from the associated Weibull plot.
The critical value is then looked up in Weibull.Plot.Quantiles.
There is print method for class "wp.test.critical".
Value
A list with class "wp.test.critical" containing the following components:
sample.size |
sample size (missing observations are deleted). |
alpha |
significance level. |
critical.value |
critical value. |
data.name |
a character string giving the name(s) of the data. |
Author(s)
Chanseok Park
References
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering - Theory, Applications and Practice, 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test
for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# Critical value with alpha (significance level) and n (sample size).
wp.test.critical(alpha=0.01, n=10)
The p-value for the Weibullness Test
Description
Calculates the p-value for the Weibullness test which is based on the sample correlation from the Weibull plot.
Usage
wp.test.pvalue(r, n)
Arguments
r |
the sample correlation coefficient from the Weibull plot; r is in (0,1). |
n |
the sample size. |
Details
The p-value for the Weibullness test which is based on
the sample correlation from the Weibull plot.
There is print method for class "htest".
Value
A list with class "htest" containing the following components:
statistic |
the value of the test statistic (sample correlation from the Weibull plot) |
p.value |
the p-value for the test. |
method |
a character string indicating the Weibullness test. |
Author(s)
Chanseok Park
References
Park, C. (2017). Weibullness test and parameter estimation of the three-parameter Weibull model using the sample correlation coefficient. International Journal of Industrial Engineering – Theory, Applications and Practice, 24(4), 376-391. doi:10.23055/ijietap.2017.24.4.2848
Vogel, R. M. and C. N. Kroll (1989). Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients. Journal of Water Resources Planning and Management, 115, 338-357.
See Also
ks.test for performing the Kolmogorov-Smirnov test
for the goodness of fit test of two samples.
shapiro.test for performing the Shapiro-Wilk test for normality.
Examples
# p.value with r (sample correlation from the Weibull plot) and n (sample size).
wp.test.pvalue(r=0.6, n=10)