Package 'PTLENKies'

Cumulative Distribution Function (CDF) of the PTLENKBurr Distribution

Description

This function calculates the Cumulative Distribution Function (CDF) of the PTLENKBurr distribution.

Usage

C_PTLENKBurr(x, m, n, p, lambda, beta, c, k)

Arguments

x	Value or vector of positive values up to which to calculate the CDF.
m	Parameter m of the distribution (m > 0).
n	Parameter n of the distribution (n > 0).
p	Parameter p of the distribution (p > 0).
lambda	Parameter lambda of the distribution (lambda > 0).
beta	Parameter beta of the distribution (beta > 0).
c	Parameter c of the Burr XII baseline component (c > 0).
k	Parameter k of the Burr XII baseline component (k > 0).

Details

It takes parameters x, m, n, p, lambda, beta, c, and k, and returns the CDF value at x based on these parameters.

The Cumulative Distribution Function (CDF) of the PTLENKBurr distribution is defined as:

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

where

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

and $m, n, p, \lambda, \beta, c, k > 0$, $x > 0$.

Value

A numeric vector containing the values of the CDF evaluated at x.

Examples

C_PTLENKBurr(
  x = c(0.5, 1.0, 1.5),
  m = 1, n = 1, p = 1,
  lambda = 0.5, beta = 1.5, c = 2, k = 1
)

---

Maximum Likelihood Estimation (MLE) of the PTLENKBurr Distribution

Description

Estimates the parameters of the Power Transformed Lindley Extended Modified Kies Burr XII (PTLENKBurr) distribution using maximum likelihood estimation (MLE) via constrained optimization with nlminb.

Usage

E_PTLENKBurr(data, start_params = NULL, lower = 1e-05)

Arguments

data	Numeric vector of positive observations.
start_params	Optional numeric vector of length 7 containing initial parameter values for c(m, n, p, lambda, beta, c, k). Default is c(1, 1, 1, 1, 1, 1, 1).
lower	Lower bounds for parameters. Default is 1e-5 for all 7 parameters.

Details

Optimization is performed using nlminb. Infeasible iterations returning non-positive densities, NA, or NaN values are assigned a high penalty.

Value

A named numeric vector of estimated parameters: m, n, p, lambda, beta, c, and k.

Examples

data_sample <- c(0.5, 1.2, 0.8, 2.1, 0.3, 1.5, 0.9)
E_PTLENKBurr(data_sample)

---

Dataset: GlassFibers

Description

The dataset contains observations collected at the UK National Physical Laboratory regarding the breaking strengths of 1.5 cm glass fibers, previously reported in Atchade (2023).

Usage

data(GlassFibers)

Format

A numeric vector of breaking strengths.

Details

This dataset contains measurements corresponding to the breaking strengths of 1.5 cm glass fibers. The data are presented as a numeric vector.

References

Atchade, N. (2023). Statistical modeling of lifetime data and probability distributions.

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Probability Density Function (PDF) of the PTLENKBurr Distribution

Description

This function calculates the Probability Density Function (PDF) of the PTLENKBurr distribution.

Usage

P_PTLENKBurr(x, m, n, p, lambda, beta, c, k)

Arguments

x	Value or vector of positive values to evaluate the PDF at.
m	Parameter m of the distribution (m > 0).
n	Parameter n of the distribution (n > 0).
p	Parameter p of the distribution (p > 0).
lambda	Parameter lambda of the distribution (lambda > 0).
beta	Parameter beta of the distribution (beta > 0).
c	Parameter c of the Burr XII baseline component (c > 0).
k	Parameter k of the Burr XII baseline component (k > 0).

Details

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

where

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

and $m, n, p, \lambda, \beta, c, k > 0$, $x > 0$.

Value

A numeric vector containing the values of the PDF evaluated at x.

Examples

P_PTLENKBurr(
  x = c(0.5, 1.0, 1.5),
  m = 1, n = 1, p = 1,
  lambda = 0.5, beta = 1.5, c = 2, k = 1
)

---

Plot the Cumulative Distribution Function (CDF) of the PTLENKBurr Distribution

Description

Generates an enhanced plot of the Cumulative Distribution Function (CDF) of the PTLENKBurr distribution over [0, +Inf), automatically calculating and displaying the numerical median.

Usage

Plot_CPTLENKBurr(
  m,
  n,
  p,
  lambda,
  beta,
  c,
  k,
  max_x = NULL,
  show_median = TRUE,
  col_line = "deepskyblue4",
  col_median = "firebrick3",
  legend_pos = "right",
  ...
)

Arguments

m	Parameter m of the distribution (m > 0).
n	Parameter n of the distribution (n > 0).
p	Parameter p of the distribution (p > 0).
lambda	Parameter lambda of the distribution (lambda > 0).
beta	Parameter beta of the distribution (beta > 0).
c	Parameter c of the Burr XII baseline component (c > 0).
k	Parameter k of the Burr XII baseline component (k > 0).
max_x	Optional maximum x-axis value. If NULL (default), it is automatically computed using the CDF.
show_median	Logical. If TRUE (default), calculates and plots the median.
col_line	Color of the CDF curve. Default is "deepskyblue4".
col_median	Color of the median lines and point. Default is "firebrick3".
legend_pos	Position of the legend. Default is "right".
...	Additional graphical parameters passed to plot.

Details

The median is numerically calculated by finding the root of $F(x) - 0.5 = 0$ using uniroot.

Value

A plot of the CDF and an invisible list containing the evaluation grid data frame and the numerical value of the median.

Examples

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

---

Plot the Probability Density Function (PDF) of the PTLENKBurr Distribution

Description

Generates an enhanced plot of the Probability Density Function (PDF) of the PTLENKBurr distribution over [0, +Inf), automatically calculating and displaying the numerical mode and median.

Usage

Plot_PPTLENKBurr(
  m,
  n,
  p,
  lambda,
  beta,
  c,
  k,
  max_x = NULL,
  show_mode = TRUE,
  show_median = TRUE,
  col_line = "deepskyblue4",
  col_mode = "darkorange2",
  col_median = "firebrick3",
  legend_pos = "topright",
  ...
)

Arguments

m	Parameter m of the distribution (m > 0).
n	Parameter n of the distribution (n > 0).
p	Parameter p of the distribution (p > 0).
lambda	Parameter lambda of the distribution (lambda > 0).
beta	Parameter beta of the distribution (beta > 0).
c	Parameter c of the Burr XII baseline component (c > 0).
k	Parameter k of the Burr XII baseline component (k > 0).
max_x	Optional maximum x-axis value. If NULL (default), it is automatically computed using the CDF.
show_mode	Logical. If TRUE (default), calculates and plots the mode.
show_median	Logical. If TRUE (default), calculates and plots the median.
col_line	Color of the PDF curve. Default is "deepskyblue4".
col_mode	Color of the mode marker. Default is "darkorange2".
col_median	Color of the median marker. Default is "firebrick3".
legend_pos	Position of the legend. Default is "topright".
...	Additional graphical parameters passed to plot.

Details

The upper bound max_x is determined automatically by finding where $F(x) \ge 0.999$. The mode is calculated numerically by maximizing the PDF via optimize, and the median is found via uniroot on the CDF (C_PTLENKBurr).

Value

A plot of the PDF and an invisible list containing the evaluation grid data frame, calculated mode, and median.

Examples

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

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Quantile Function of the PTLENKBurr Distribution

Description

Calculates the quantile value (inverse CDF) of the PTLENKBurr distribution for a given probability u or a vector of probabilities.

Usage

Q_PTLENKBurr(u, m, n, p, lambda, beta, c, k, tol = 1e-08)

Arguments

u	Numeric vector of probabilities ($0 \le u \le 1$).
m	Parameter m of the distribution (m > 0).
n	Parameter n of the distribution (n > 0).
p	Parameter p of the distribution (p > 0).
lambda	Parameter lambda of the distribution (lambda > 0).
beta	Parameter beta of the distribution (beta > 0).
c	Parameter c of the Burr XII baseline component (c > 0).
k	Parameter k of the Burr XII baseline component (k > 0).
tol	Tolerance level for numerical root finding via uniroot. Default is 1e-8.

Details

The quantile function $Q(u)$ is obtained by numerically solving $F(x) - u = 0$ using uniroot. The probability argument is named u to avoid name collision with the distribution parameter p.

Value

A numeric vector of quantiles corresponding to the probabilities in u.

Examples

# Calculate quartiles (25%, 50%, 75%)
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
)

---

Random Generation for the PTLENKBurr Distribution

Description

Generates random samples from the PTLENKBurr distribution using inverse transform sampling (quantile method).

Usage

R_PTLENKBurr(n_samples, m, n, p, lambda, beta, c, k)

Arguments

n_samples	Number of random samples to generate (must be a positive integer).
m	Parameter m of the distribution (m > 0).
n	Parameter n of the distribution (n > 0).
p	Parameter p of the distribution (p > 0).
lambda	Parameter lambda of the distribution (lambda > 0).
beta	Parameter beta of the distribution (beta > 0).
c	Parameter c of the Burr XII baseline component (c > 0).
k	Parameter k of the Burr XII baseline component (k > 0).

Details

The parameter for the sample size is named n_samples to avoid naming conflicts with the distribution parameter n. Random variates are generated by applying the quantile function Q_PTLENKBurr to uniform random variables: $X = Q(U)$ where $U \sim \text{Uniform}(0, 1)$.

Value

A numeric vector of n_samples random values generated from the PTLENKBurr distribution.

Examples

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)

---

Histogram and Fitted Density Function (PDF) of the PTLENKBurr Distribution

Description

Generates a histogram of the provided dataset and overlays the fitted Probability Density Function (PDF) of the PTLENKBurr distribution with parameters estimated via Maximum Likelihood Estimation using E_PTLENKBurr.

Usage

Sim_PTLENKBurr(
  data,
  breaks = 15,
  col_hist = "lightblue",
  col_line = "firebrick3",
  ...
)

Arguments

data	Numeric vector of positive data values.
breaks	Number of breaks or break specification for the histogram. Default is 15.
col_hist	Fill color for the histogram bars. Default is "lightblue".
col_line	Color for the fitted PDF curve. Default is "firebrick3".
...	Additional graphical parameters passed to hist.

Details

The function uses E_PTLENKBurr to estimate distribution parameters and evaluates P_PTLENKBurr over a sequence from near zero to 1.2 times the maximum data value.

Value

An invisible named numeric vector containing the 7 estimated parameters c(m, n, p, lambda, beta, c, k).

Examples

sample_data <- c(0.5, 1.2, 0.8, 2.1, 0.3, 1.5, 0.9)
Sim_PTLENKBurr(sample_data)

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