expgrowth: Exponential growth distribution functions

In primarycensored: Primary Event Censored Distributions

Exponential growth distribution functions

Description

Density, distribution function, and random generation for the exponential growth distribution.

Usage

dexpgrowth(x, min = 0, max = 1, r, log = FALSE)

pexpgrowth(q, min = 0, max = 1, r, lower.tail = TRUE, log.p = FALSE)

rexpgrowth(n, min = 0, max = 1, r)

Arguments

x, q	Vector of quantiles.
min	Minimum value of the distribution range. Default is 0.
max	Maximum value of the distribution range. Default is 1.
r	Rate parameter for the exponential growth.
log, log.p	Logical; if TRUE, probabilities p are given as log(p).
lower.tail	Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
n	Number of observations. If length(n) > 1, the length is taken to be the number required.

Details

The exponential growth distribution is defined on the interval [min, max] with rate parameter (r). Its probability density function (PDF) is:

f(x) = \frac{r \cdot \exp(r \cdot (x - min))}{\exp(r \cdot max) - \exp(r \cdot min)}

The cumulative distribution function (CDF) is:

F(x) = \frac{\exp(r \cdot (x - min)) - \exp(r \cdot min)}{ \exp(r \cdot max) - \exp(r \cdot min)}

For random number generation, we use the inverse transform sampling method:

1. Generate u \sim \text{Uniform}(0,1)

2. Set F(x) = u and solve for x:

   x = min + \frac{1}{r} \cdot \log(u \cdot (\exp(r \cdot max) - \exp(r \cdot min)) + \exp(r \cdot min))

This method works because of the probability integral transform theorem, which states that if X is a continuous random variable with CDF F(x), then Y = F(X) follows a \text{Uniform}(0,1) distribution. Conversely, if U is a \text{Uniform}(0,1) random variable, then F^{-1}(U) has the same distribution as X, where F^{-1} is the inverse of the CDF.

In our case, we generate u from \text{Uniform}(0,1), then solve F(x) = u for x to get a sample from our exponential growth distribution. The formula for x is derived by algebraically solving the equation:

u = \frac{\exp(r \cdot (x - min)) - \exp(r \cdot min)}{\exp(r \cdot max) - \exp(r \cdot min)}

When r is very close to 0 (|r| < 1e-10), the distribution approximates a uniform distribution on [min, max], and we use a simpler method to generate samples directly from this uniform distribution.

Value

dexpgrowth gives the density, pexpgrowth gives the distribution function, and rexpgrowth generates random deviates.

The length of the result is determined by n for rexpgrowth, and is the maximum of the lengths of the numerical arguments for the other functions.

Examples

x <- seq(0, 1, by = 0.1)
probs <- dexpgrowth(x, r = 0.2)
cumprobs <- pexpgrowth(x, r = 0.2)
samples <- rexpgrowth(100, r = 0.2)

primarycensored documentation built on April 3, 2025, 6:24 p.m.