Package 'ierd'

Density of the Inverted Exponentiated Rayleigh Distribution

Description

This function computes the probability density function (PDF) of the Inverted Exponentiated Rayleigh distribution.

Usage

dierd(x, shape, scale)

Arguments

x	A numeric vector of quantiles.
shape	A strictly positive numeric value for the shape parameter ($\alpha$).
scale	A strictly positive numeric value for the scale parameter ($\beta$).

Details

The probability density function is mathematically defined as:

$f(x) = 2 \alpha \beta x^{-3} \exp(-\beta / x^2) (1 - \exp(-\beta / x^2))^{\alpha - 1}$

for $x > 0$, where $\alpha$ is the shape parameter and $\beta$ is the scale parameter.

Value

A numeric vector of density values evaluated at x.

Examples

# Compute the density at various values of x
dierd(x = c(0.5, 1, 1.5, 2), shape = 2, scale = 1)

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Cumulative Distribution Function of the Inverted Exponentiated Rayleigh Distribution

Description

This function computes the Cumulative Distribution Function (CDF) of the Inverted Exponentiated Rayleigh distribution.

Usage

pierd(t, shape, scale)

Arguments

t	A numeric vector of quantiles.
shape	A strictly positive numeric value for the shape parameter ($\alpha$).
scale	A strictly positive numeric value for the scale parameter ($\beta$).

Details

The cumulative distribution function is mathematically defined as:

$F(t) = 1 - (1 - \exp(-\beta / t^2))^\alpha$

for $t > 0$, where $\alpha$ is the shape parameter and $\beta$ is the scale parameter.

Value

A numeric vector of cumulative probabilities evaluated at t.

Examples

# Compute the cumulative probabilities at various values of t
pierd(t = c(0.5, 1, 1.5, 2), shape = 2, scale = 1)

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Plot Multiple Inverted Exponentiated Rayleigh Densities

Description

This function creates a ggplot2 visualization comparing the dierd PDF across combinations of shape and scale parameters.

Usage

plot_dierd(shape, scale, lower = 0.01, upper = 5, paired = FALSE)

Arguments

shape	A numeric vector of strictly positive shape parameters.
scale	A numeric vector of strictly positive scale parameters.
lower	A numeric value for the lower bound of the x-axis (default is 0.01).
upper	A numeric value for the upper bound of the x-axis (default is 5).
paired	Logical. If FALSE (default), creates a full grid of all possible shape and scale combinations. If TRUE, pairs the shape and scale vectors element-by-element (vectors must be the same length).

Value

A ggplot object showing the density curves.

Examples

# Full grid: 2 shapes * 2 scales = 4 curves
plot_dierd(shape = c(1, 2), scale = c(1, 2))

# Paired: 2 specific combinations = 2 curves
plot_dierd(shape = c(1, 2), scale = c(1, 2), paired = TRUE)

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Plot Multiple Inverted Exponentiated Rayleigh CDFs

Description

This function creates a ggplot2 visualization of the pierd CDF across combinations of shape and scale parameters.

Usage

plot_pierd(shape, scale, lower = 0.01, upper = 7.5, paired = FALSE)

Arguments

shape	A numeric vector of strictly positive shape parameters.
scale	A numeric vector of strictly positive scale parameters.
lower	A numeric value for the lower bound of the x-axis (default is 0.01).
upper	A numeric value for the upper bound of the x-axis (default is 7.5).
paired	Logical. If FALSE (default), creates a full grid of all possible shape and scale combinations. If TRUE, pairs the shape and scale vectors element-by-element (vectors must be the same length).

Value

A ggplot object showing the cumulative distribution curves.

Examples

# Full grid: 2 shapes * 2 scales = 4 curves
plot_pierd(shape = c(1, 2), scale = c(1, 2))

# Paired: 2 specific combinations = 2 curves
plot_pierd(shape = c(1, 2), scale = c(1, 2), paired = TRUE)

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Quantile Function of the Inverted Exponentiated Rayleigh Distribution

Description

This function computes the quantile function (inverse CDF) of the Inverted Exponentiated Rayleigh distribution.

Usage

qierd(p, shape, scale)

Arguments

p	A numeric vector of probabilities.
shape	A strictly positive numeric value for the shape parameter ($\alpha$).
scale	A strictly positive numeric value for the scale parameter ($\beta$).

Details

The quantile function is mathematically defined as:

$Q(p) = \sqrt{\frac{-\beta}{\log(1 - (1 - p)^{1/\alpha})}}$

for $0 \le p \le 1$, where $\alpha$ is the shape parameter and $\beta$ is the scale parameter.

Value

A numeric vector of quantiles evaluated at p.

Examples

# Compute the quantiles at various probability values (e.g., quartiles)
qierd(p = c(0.25, 0.5, 0.75), shape = 2, scale = 1)

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Random Numbers from the Inverted Exponentiated Rayleigh Distribution

Description

This function generates random numbers from an Inverted Exponentiated Rayleigh distribution using inverse transform sampling.

Usage

rierd(n, shape, scale)

Arguments

n	An integer specifying the number of random values to return.
shape	A strictly positive numeric value for the shape parameter.
scale	A strictly positive numeric value for the scale parameter.

Value

A numeric vector of length n containing the generated random numbers.

Examples

# Generate 10 random numbers with shape = 2 and scale = 1
rierd(n = 10, shape = 2, scale = 1)

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