Installation

You can install the “TLCAR” package from CRAN using the following command:

install.packages("ggplot2")
install.packages("dplyr")
install.packages("TLCAR")

Overview of TLCAR Distribution

The TLCAR distribution is particularly useful for modeling data with heavy tails, skewness, and positive values. It is a versatile distribution that can handle diverse characteristics in the data.

Probability Density Function (PDF) for TLCAR distribution

The probability density function (PDF) for the TLCAR distribution is given by the formula:

\[ f(x;\nu)=\frac{2\alpha}{\pi a} \times \left[\frac{1+\left(\frac{x^2}{\theta^2}-1\right)e^{-\frac{x^2}{2\theta^2}}+m}{1+\left(\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right) -b}{a}\right)^2}\right] \times\left[\frac{1}{2}-\frac{1}{\pi}\arctan\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right) -b}{a}\right] \times \left[ 1-\left(\frac{1}{2}-\frac{1}{\pi}\arctan\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right)-b}{a}\right)^2\right]^{\alpha-1}\] where \(\alpha\) , \(a\) , \(\theta\) , \(m\) > 0 .

Example 1

Let’s calculate the PDF for \(x=1\) , \(\alpha=1\) , \(a=1\) , \(b=0\) , \(\theta=2\) and \(m=1\)

dTLCAR(x = 1, alpha = 1, a = 1, b = 0, theta = 2, m=1)

## [1] 0.08801882

Example 2

Let’s calculate the PDF for \(x=3\) , \(\alpha=2\) , \(a=1\) , \(b=2\) , \(\theta=2\) and \(m=4\)

dTLCAR(x = 3, alpha = 2, a = 1, b = 2, theta = 2, m=4)

## [1] 0.001247268

Cumulative Distribution Function (CDF) for TLCAR distribution

The cumulative distribution function (CDF) for the TLCAR distribution is given by the formula:

\[F(x;\nu)=\left[ 1-\left(\frac{1}{2}-\frac{1}{\pi}\arctan\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right) -b}{a}\right)^2\right]^\alpha \]

where \(\alpha\) , \(a\) , \(\theta\) , \(m\) > 0 .

Example 1

Let’s calculate the CDF for \(x=1\) , \(\alpha=1\) , \(a=1\) , \(b=0\) , \(\theta=2\) and \(m=1\)

cTLCAR(x = 1, alpha = 1, a = 1, b = 0, theta = 2,m=1)

## [1] 0.9460113

Example 2

Let’s calculate the CDF for \(x=3\) , \(\alpha=2\) , \(a=1\) , \(b=2\) , \(\theta=2\) and \(m=4\)

cTLCAR(x = 3, alpha = 2, a = 1, b = 2, theta = 2,m=4)

## [1] 0.9986058

Graphical Plot of the TLCAR Distribution

This function generates a graphical plot of the probability density function (PDF) or cumulative distribution function (CDF) for the TLCAR distribution.

Example

Let’s plot the PDF for a range of values with parameters \(\alpha=1\) , \(a=1\) , \(b=0\) , \(\theta=2\) and \(m=1\)

ploTLCAR(x = seq(-4, 10, by = 0.1), alpha = 1, a = 1,b=0, theta = 2,m=1, type = "pdf")

Let’s plot the CDF for a range of values with parameters \(\alpha=1\) , \(a=1\) , \(b=0\) , \(\theta=2\) and \(m=1\)

ploTLCAR(x = seq(-4, 10, by = 0.1), alpha = 1, a = 1,b=0, theta = 2, m=1,type = "cdf")

Quantile function for TLCAR distribution

The quantile function calculates the quantile value for a given probability using the TLCAR distribution.

Example 1

Let’s calculate the 0.5 quantile (median) using parameters \(\alpha=1\) , \(a=1\) , \(b=0\) ,\(\theta=2\) and \(m=1\)

qTLCAR(p = 0.5, alpha = 1, a = 1, b= 0, theta = 2,m=1)

## [1] -0.7166792

Example 2

Let’s calculate the 0.75 quantile using parameters \(\alpha=2\) , \(a=2\) , \(b=1\) ,\(\theta=2\) and \(m=2\)

qTLCAR(p = 0.75, alpha = 2, a = 2, b= 1, theta = 2,m=2)

## [1] 0.9037692

Generate random samples from the TLCAR distribution

This function generates random samples from the TLCAR distribution using the Box-Muller algorithm.

Example 1

Let’s generate 25 random samples with parameters \(\alpha=1\) , \(a=1\) , \(b=0\) ,\(\theta=2\) and \(m=1\)

rTLCAR(n = 25, alpha = 1, a = 2, b = 0, theta = 2,m=1)

##  [1]  -0.7025289  -1.0591749   1.9324451  -0.3384789   2.0136376   0.8894991
##  [7]  -4.6530735  -0.6864125  -0.5854632   0.0678766  -0.7867331 -45.4280152
## [13]  -1.6517409  -3.9558302   0.4378498 -16.5788655  -1.2367879  -2.1140315
## [19]   1.7325325  -3.9671058  -2.8861827  -1.5670429  -3.3882363 -10.3419322
## [25]   1.4776296

Example 2

Let’s generate 50 random samples with parameters \(\alpha=2\) , \(a=2\) , \(b=1\) ,\(\theta=2\) and \(m=2\)

rTLCAR(n = 50, alpha = 2, a = 2, b = 1, theta = 2,m=2)

##  [1] -0.3367058  1.7285812  1.1875268  0.4120580  1.0581625  0.3990519
##  [7]  1.0071567  2.1849633  2.3598769  0.2319956 -0.5748939  1.0236759
## [13]  0.1246406  0.5529991 -1.2433635  0.3173053 -1.3641320 -0.7273208
## [19]  1.9033033  1.5304993  1.0602287 -0.6917317  0.6620348  1.3377633
## [25]  0.3161296  0.3535551  0.1526248  0.3199435  1.1434471 -1.5357639
## [31]  0.7347157  0.5214707  0.1937358  0.8818820 -0.8341408  0.8520853
## [37]  0.2262596  0.1394313  0.5857642  0.6400057 -1.3768247 -1.0428210
## [43]  0.8939435  0.5473589 -1.7624686 -0.6047640 -0.4882200  0.2529339
## [49] -2.8642775 -0.4241723

Estimate parameters with constraints

This function estimates the parameters of the TLCAR distribution while respecting constraints on the parameters.

Example 1

Let’s estimate parameters from a sample data vector.

data(ConductorFailureTimes)
 fTLCAR(ConductorFailureTimes)

## [1]  3.017587  3.495700 13.258842  1.696800  1.000000

Example 2

data(Tree_diameters)
fTLCAR(Tree_diameters)

## [1]  3.447945  6.540533 26.392380  1.000000  1.000000

Estimate parameters with constraints and plot histogram with estimated density

This function estimates parameters and plots the histogram of the data along with the estimated density curve.

Example 1

sTLCAR(ConductorFailureTimes)

Example 2

sTLCAR(Tree_diameters)

This concludes the overview of the “TLCAR” package and its functionalities for working with the Topp-Leone Cauchy Rayleigh distribution.

References

Atchadé, M.N., Bogninou, M.J., Djibril, A.M. et al. Topp-Leone Cauchy Family of Distributions with Applications in Industrial Engineering. J Stat Theory Appl 22, 339–365 (2023). https://doi.org/10.1007/s44199-023-00066-4