PTLENKies

PTLENKies provides statistical tools for the Power Topp Leone Exponential Negative Kies Burr (PTLENKBurr) distribution, a flexible seven-parameter distribution for positive data built on a Burr XII baseline. The package lets you:

• evaluate the density, the cumulative distribution function and the quantiles;
• simulate random samples;
• estimate the seven parameters by maximum likelihood;
• plot the theoretical curves, with the mode and the median marked;
• compare a fitted density with your data.

The distribution

For $x > 0$, the cumulative distribution function is

$$F(x; m, n, p, \lambda, \beta, c, k) = \exp\left( m,n \left(1 - \frac{1}{K(x)^{p}}\right) \right) \left[, 2 - \exp\left( n \left(1 - \frac{1}{K(x)^{p}}\right) \right) \right]^{m}$$

where

$$K(x) = 1 - \exp\left( -\lambda \left( \frac{1 - (1+x^{c})^{-k}}{(1+x^{c})^{-k}} \right)^{\beta} \right)$$

and $m, n, p, \lambda, \beta, c, k > 0$. The parameters c and k are those of the Burr XII baseline.

Installation

Install the released version from CRAN:

install.packages("PTLENKies")

Or the development version from GitHub:

# install.packages("pak")
# pak::pak("Otodji/PTLENKies")
install.packages("PTLENKies", repos = c('https://otodji.r-universe.dev', 'https://cloud.r-project.org'))

Functions

Function	Purpose
P_PTLENKBurr(x, m, n, p, lambda, beta, c, k)	Probability density function
C_PTLENKBurr(x, m, n, p, lambda, beta, c, k)	Cumulative distribution function
Q_PTLENKBurr(u, m, n, p, lambda, beta, c, k)	Quantiles for a vector of probabilities u
R_PTLENKBurr(n_samples, m, n, p, lambda, beta, c, k)	Random sample of size n_samples
E_PTLENKBurr(data)	Maximum likelihood estimation of the seven parameters
Sim_PTLENKBurr(data)	Histogram of the data with the fitted density overlaid
Plot_PPTLENKBurr(m, n, p, lambda, beta, c, k)	Plot of the density, with mode and median
Plot_CPTLENKBurr(m, n, p, lambda, beta, c, k)	Plot of the cumulative distribution function, with median

The probability argument of the quantile function is named u, and the sample size of the random generator is named n_samples, because p and n are parameters of the distribution.

The package also ships the dataset GlassFibers: breaking strengths of 1.5 cm glass fibers measured at the UK National Physical Laboratory.

Examples

library(PTLENKies)

Density, distribution function and quantiles

x <- c(0.5, 1.0, 1.5)

# Density and cumulative probabilities
P_PTLENKBurr(x, m = 1, n = 1, p = 1, lambda = 0.5, beta = 1.5, c = 2, k = 1)
#> [1] 3.542693e-05 1.977328e+00 3.041824e-01
C_PTLENKBurr(x, m = 1, n = 1, p = 1, lambda = 0.5, beta = 1.5, c = 2, k = 1)
#> [1] 3.691507e-07 3.822999e-01 9.587699e-01

# Quartiles
Q_PTLENKBurr(u = c(0.25, 0.5, 0.75),
             m = 1, n = 1, p = 1, lambda = 0.5, beta = 1.5, c = 2, k = 1)
#> [1] 0.9312389 1.0601357 1.2133246

Random samples

set.seed(123)
sim_data <- R_PTLENKBurr(n_samples = 100,
                         m = 1, n = 1, p = 1, lambda = 0.5, beta = 1.5, c = 2, k = 1)
head(sim_data)
#> [1] 0.9513949 1.2446528 1.0134882 1.3464558 1.4488376 0.7777750

Plotting the density and the distribution function

Plot_PPTLENKBurr(m = 1, n = 1, p = 1, lambda = 0.5, beta = 1.5, c = 2, k = 1)

Plot_CPTLENKBurr(m = 1, n = 1, p = 1, lambda = 0.5, beta = 1.5, c = 2, k = 1)

Fitting the distribution to data

E_PTLENKBurr() returns the maximum likelihood estimates as a named vector.

data(GlassFibers)

E_PTLENKBurr(GlassFibers)
#>            m            n            p       lambda         beta            c 
#>   0.66034955 166.13384622   0.01840540  19.79164556   4.82830738   3.67202817 
#>            k 
#>   0.06630026

Sim_PTLENKBurr() draws the histogram of the data and overlays the fitted density:

Sim_PTLENKBurr(GlassFibers)

References

If you use this package, please cite the related articles:

• Atchadé, M. N., Otodji, T., & Djibril, A. (2024). https://doi.org/10.1063/5.0179458
• Atchadé, M. N., Otodji, T., Djibril, A., & N’bouké, M. (2023). https://doi.org/10.1515/phys-2023-0151

The citation of the package itself is given by:

citation("PTLENKies")

Authors

• Mintodê Nicodème Atchadé
• Théophile Otodji (maintainer) — ORCID 0009-0003-0389-9007

Bug reports

Please report bugs and suggestions on the issue tracker.

License

GPL-2