Package 'weibullness'

Title:	Goodness-of-Fit Test for Weibull Distribution (Weibullness)
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 (MSIT) (No. 2022R1A2C1091319, RS-2023-00242528).
Authors:	Chanseok Park [aut, cre]
Maintainer:	Chanseok Park <[email protected]>
License:	GPL-2 \| GPL-3
Version:	1.24.1
Built:	2024-11-01 11:14:52 UTC
Source:	https://github.com/appliedstat/r

Help Index

Exponential Probability Plot
The Exponential Goodness-of-Fit Test from the Exponential Probability Plot
Critical Value for the Exponential Goodness-of-Fit test
The p-value for the Exponential Goodness-of-Fit Test
Exponential quantile values
Gumbel Probability Plot
Gumbel Goodness-of-Fit Test from a Gumbel Probability Plot
Critical Value for the Gumbel Goodness-of-Fit Test
The p-value for the Gumbel goodness-of-Fit Test
Estimate of location and scale parameters of the Gumbel distribution
Gumbel quantile values
The inverse Weibull distribution
Maximum likelihood estimates of the two-parameter inverse Weibull distribution
Inverse Weibull quantile values
Inverse Weibull Probability Plot
Weibullness Test from a inverse Weibull Plot
Critical Value for the inverse Weibullness Test
The p-value for the inverse Weibullness Test
Dataset
Maximum likelihood estimates with interval censoring
Maximum likelihood estimates of three-parameter Weibull distribution
Weibull quantile values
Robust estimate of shape and scale parameters of Weibull using the repeated median method
Estimate of threshold parameter of three-parameter Weibull distribution
Estimate of shape and scale parameters of Weibull using the Weibull plot
Weibull Plot
Weibullness Test from a Weibull Plot
Critical Value for the Weibullness Test
The p-value for the Weibullness Test

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", ...) 
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 `ppoints`.
`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 `title`.
`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

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 )
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)
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.

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)
# 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)
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

Examples

# Critical value with alpha (significance level) and n (sample size).
ep.test.critical(alpha=0.0982, n=7)
# 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)
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.

Examples

# p.value with w (ANOVA statistic) and n (sample size).
ep.test.pvalue(w=0.35593, n=7)
# p.value with w (ANOVA statistic) and n (sample size).
ep.test.pvalue(w=0.35593, n=7)

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 1E08.

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.QuantilesExponential.ANOVA.Quantiles

Format

This data frame contains 998 rows and 1001 columns.

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", ...) 
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 `ppoints`.
`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 `title`.
`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

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 )
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)
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.

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)
# 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)
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.

Examples

# Critical value with alpha (significance level) and n (sample size).
gp.test.critical(alpha=0.01, n=10)
# 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)
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.

Examples

# p.value with r (sample correlation from the Gumbel probability plot) and n (sample size).
gp.test.pvalue(r=0.98504, n=10)
# 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 fromn the Gumbel probability plot.

Usage

gumbel.gp(x, n, a)
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 `ppoints`.

Details

gumbel.gp obtains the estimates of the location and scale parameters using the intercept and slope estimates from the Gumbel probability plot.

Value

An object of class "gumbel.estimate", a list with two parameter estimates

Author(s)

Chanseok Park

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)
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)

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 1E08.

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.QuantilesGumbel.Plot.Quantiles

Format

This data frame contains 998 rows and 1001 columns.

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)
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 `length(n) > 1`, the length is taken to be the number required.
`shape`, `scale`	parameters. Must be positive.
`log`, `log.p`	logical; if `TRUE`, probabilities or densities $p$ are given as $\log(p)$ .
`lower.tail`	logical; if `TRUE` (default), probabilities are $P[X \le x]$ , otherwise, $P[X > x]$ .

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 )
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$double.eps^0.25, maxiter = 1000, trace = 0)
invweibull.mle(x, interval, tol = .Machine$double.eps^0.25, 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

# 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)
invweibull.mle(data)
# 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)
invweibull.mle(data)

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 1E08.

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.QuantilesIW.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

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", ...) 
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 `ppoints`.
`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 `title`.
`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

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 )
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)
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

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.

Examples

# For inverse Weibullness hypothesis test. 
attach(Wdata)
iwp.test(urinary)
# For inverse Weibullness hypothesis test. 
attach(Wdata)
iwp.test(urinary)

Critical Value for the inverse Weibullness Test

Description

Calculates the critical value for the inverse Weibullness test

Usage

iwp.test.critical(alpha, n)
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.

Examples

# Critical value with alpha (significance level) and n (sample size).
iwp.test.critical(alpha=0.01, n=10)
# 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)
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.

Examples

# p.value with r (sample correlation from the inverse Weibull plot) and n (sample size).
iwp.test.pvalue(r=0.6, n=10)
# p.value with r (sample correlation from the inverse Weibull plot) and n (sample size).
iwp.test.pvalue(r=0.6, n=10)

Dataset

Description

bearings: It is from Lieblein and Zelen (1956). These data are deep-groove ball bearings 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.

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.

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.

Usage

Wdata
Wdata

References

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.

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.

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.

Finkelstein, D. M. (1986), A proportional hazards model for interval-censored failure time data. Biometrics, 42, 845-865.

Lindsey, J. C. and L. M. Ryan (1998). Tutorial in biostatistics: Methods for interval-censored data. Stat. Med., 17, 219-238.

Examples

# Attach datasets
attach(Wdata)
bearings
glassfiber1.5
glassfiber15
urinary
radiotherapy
radio.chemotherapy
# Attach datasets
attach(Wdata)
bearings
glassfiber1.5
glassfiber15
urinary
radiotherapy
radio.chemotherapy

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)
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.

Examples

attach(Wdata)
weibull.ic(radio.chemotherapy)

# Two-parameter Weibull with full observations
weibull.ic( cbind(bearings,bearings) )

# Two-parameter Weibull with full observations (using weibull.mle)
weibull.mle(bearings, threshold=0)
attach(Wdata)
weibull.ic(radio.chemotherapy)

# Two-parameter Weibull with full observations
weibull.ic( cbind(bearings,bearings) )

# Two-parameter Weibull with full observations (using weibull.mle)
weibull.mle(bearings, 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$double.eps^0.25, maxiter = 1000, trace = 0)
weibull.mle(x, threshold, interval, interval.threshold, extendInt="downX", 
            a, tol = .Machine$double.eps^0.25, 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 `uniroot`.
`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.

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)
# 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)

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 1E08.

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.QuantilesWeibull.Plot.Quantiles

Format

This data frame contains 998 rows and 1001 columns.

References

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)
weibull.rm(x, a)

Arguments

`x`	a numeric vector of observations.
`a`	the offset fraction to be used; typically in (0,1). See `ppoints`.

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.

Examples

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.rm(data)
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.rm(data)

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")
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 `uniroot`.

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. (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

Examples

# Data
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.threshold(data)
# Data
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.threshold(data)

Estimate of shape and scale parameters of Weibull using the Weibull plot

Description

Calculates the estimates of the shape and scale parameters.

Usage

weibull.wp(x, n, a)
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 `ppoints`.

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

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

Examples

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.wp(data)
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.wp(data)

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", ...) 
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 `ppoints`.
`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 `title`.
`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

Examples

 
attach(Wdata)

wp.plot(bearings)

# With cosmetic lines 
wp.plot(bearings, 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 )
attach(Wdata)

wp.plot(bearings)

# With cosmetic lines 
wp.plot(bearings, 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)
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

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.

Examples

# For Weibullness hypothesis test. 
attach(Wdata)
wp.test(urinary)
# 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)
wp.test.critical(alpha, n)

Arguments

`alpha`	the significance level.
`n`	the sample size.

Details

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.

Examples

# Critical value with alpha (significance level) and n (sample size).
wp.test.critical(alpha=0.01, n=10)
# 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)
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.

Examples

# p.value with r (sample correlation from the Weibull plot) and n (sample size).
wp.test.pvalue(r=0.6, n=10)
# p.value with r (sample correlation from the Weibull plot) and n (sample size).
wp.test.pvalue(r=0.6, n=10)

Package 'weibullness'

Help Index

Exponential Probability Plot

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

The Exponential Goodness-of-Fit Test from the Exponential Probability Plot

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Critical Value for the Exponential Goodness-of-Fit test

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

The p-value for the Exponential Goodness-of-Fit Test

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Exponential quantile values

Description

Usage

Format

Gumbel Probability Plot

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Gumbel Goodness-of-Fit Test from a Gumbel Probability Plot

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Critical Value for the Gumbel Goodness-of-Fit Test

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The p-value for the Gumbel goodness-of-Fit Test

Description

Usage

Arguments

Details

Value

Author(s)