Package {weibullness}

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.Quantiles

Format

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.Quantiles

Format

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.Quantiles

Format

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.Quantiles

Format

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

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

Details

The exponential probability plot is based on taking the logarithm of the exponential cumulative distribution function.

Value

A list with the following components:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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

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

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

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

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:

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

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:

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

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:

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

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:

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

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

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

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

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

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

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

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

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

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

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

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:

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

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:

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

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:

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

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:

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)