composite: The two- or three-composite distribution
In distributionsrd: Distribution Fitting and Evaluation

Description Usage Arguments Details Value References Examples

Density, distribution function, quantile function, raw moments and random generation for the two- or three-composite distribution.

dcomposite(x, dist, coeff, startc = c(1, 1), log = FALSE)

pcomposite(q, dist, coeff, startc = c(1, 1), log.p = FALSE, lower.tail = TRUE)

qcomposite(p, dist, coeff, startc = c(1, 1), log.p = FALSE, lower.tail = TRUE)

mcomposite(
  r = 0,
  truncation = 0,
  dist,
  coeff,
  startc = c(1, 1),
  lower.tail = TRUE
)

rcomposite(n, dist, coeff, startc = c(1, 1))

`x, q`	vector of quantiles
`dist`	character vector denoting the distribution of the first-, second- (and third) component respectively. If only two components are provided, the distribution reduces to the two-component distribution.
`coeff`	named numeric vector holding the coefficients of the first-, second- (and third) component, predeced by coeff1., coeff2. (and coeff3.), respectively. Coefficients for the last component do not have to be provided for the two-component distribution and will be disregarded.
`startc`	starting values for the lower and upper cutoff, defaults to c(1,1).
`log, log.p`	logical; if TRUE, probabilities p are given as log(p).
`lower.tail`	logical; if TRUE (default), probabilities (moments) are P[X ≤q x] (E[x^r\|X ≤q y]), otherwise, P[X > x] (E[x^r\|X > y])
`p`	vector of probabilities
`r`	rth raw moment of the Pareto distribution
`truncation`	lower truncation parameter
`n`	number of observations

These derivations are based on the two-composite distribution proposed by \insertCitebakar2015modellingdistributionsrd. Probability Distribution Function:

f(x) = \{ \begin{array}{lrl} \frac{α_1}{1 + α_1 + α_2} \frac{m_1(x)}{M_1(c_1)} & { if} & 0<x ≤q c_1 \\ \frac{1}{1 + α_1 + α_2} \frac{m_2(x)}{M_2(c_2) - M_2(c_1)} & { if} & c_1<x ≤q c_2 \\ \frac{α_2}{1 + α_1 + α_2} \frac{m_3(x)}{1-M_3(c_2)} & { if} & c_{2}<x < ∞ \\ \end{array} .

Cumulative Distribution Function:

\{ \begin{array}{lrl} \frac{α_1}{1 + α_1 + α_2} \frac{M_1(x)}{M_1(c_1)} & { if} & 0<x ≤q c_1 \\ \frac{α_1}{1 + α_1 + α_2} + \frac{1}{1 + α_1 + α_2}\frac{M_2(x) - M_2(c_1)}{M_2(c_2) - M_2(c_1)} & { if} & c_1<x ≤q c_2 \\ \frac{1+α_1}{1 + α_1 + α_2} + \frac{α_2}{1 + α_1 + α_2} \frac{M_3(x) - M_3(c_2)}{1-M_3(c_2)} & { if} & c_{2}<x < ∞ \\ \end{array} .

Quantile function

Q(p) = \{ \begin{array}{lrl} Q_1( \frac{1 + α_1 + α_2}{α_1} p M_1(c_1) ) & { if} & 0<x ≤q \frac{α_1}{1 + α_1 + α_2} \\ Q_2[((p - \frac{α_1}{1 + α_1 + α_2})(1 + α_1 + α_2)(M_2(c_2) - M_2(c_1))) + M_2(c_1)] & { if} & \frac{α_1}{1 + α_1 + α_2}<x ≤q \frac{1+α_1}{1 + α_1 + α_2} \\ Q_3[((p - \frac{1+α_1}{1 + α_1 + α_2})(\frac{1 + α_1 + α_2}{α_2})(1 - M_3(c_2))) + M_3(c_2)] & { if} & \frac{1+α_1}{1 + α_1 + α_2}<x < ∞ \\ \end{array} .

The lower y-bounded r-th raw moment of the distribution equals

μ_y^r = \{ \begin{array}{lrl} \frac{α_1}{1 + α_1 + α_2}\frac{ (μ_1)_y^r - (μ_1)_{c_1}^r}{M_1(c_1)} + \frac{1}{1 + α_1 + α_2}\frac{ (μ_2)_{c_1}^r - (μ_2)_{c_2}^r }{M_2(c_2) - M_2(c_1)} + \frac{α_2}{1 + α_1 + α_2} \frac{(μ_3)_y^r}{1-M_3(c_2)} & { if} & 0< y ≤q c_2 \\ \frac{1}{1 + α_1 + α_2} \frac{(μ_2)_y^r - (μ_2)_{c_2}^r }{M_2(c_2) - M_2(c_1)} + \frac{α_2}{1 + α_1 + α_2} \frac{(μ_3)_{c_2}^r}{1-M_3(c_2)} & { if} & c_1< y ≤q c_2\\ \frac{α_2}{1 + α_1 + α_2} \frac{(μ_3)_y^r}{1-M_3(c_2)} & { if} & c_2< y < ∞ \\ \end{array} .

dcomposite returns the density, pcomposite the distribution function, qcomposite the quantile function, mcomposite the rth moment of the distribution and rcomposite generates random deviates.

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

\insertAllCited

#' ## Three-component distribution
dist <- c("invpareto", "lnorm", "pareto")
coeff <- c(coeff2.meanlog = -0.5, coeff2.sdlog = 0.5, coeff3.k = 1.5, coeff1.k = 1.5)

# Compare density with the Double-Pareto Lognormal distribution
plot(x = seq(0, 5, length.out = 1e3), y = dcomposite(x = seq(0, 5, length.out = 1e3),
dist = dist, coeff = coeff))
lines(x = seq(0, 5, length.out = 1e3), y = ddoubleparetolognormal(x = seq(0, 5, length.out = 1e3)))

# Demonstration of log functionality for probability and quantile function
qcomposite(pcomposite(2, dist = dist, coeff = coeff, log.p = TRUE), dist = dist,
coeff = coeff, log.p = TRUE)

# The zeroth truncated moment is equivalent to the probability function
pcomposite(2, dist = dist, coeff = coeff)
mcomposite(truncation = 2, dist = dist, coeff = coeff)

# The (truncated) first moment is equivalent to the mean of a (truncated) random sample,
#for large enough samples.
coeff <- c(coeff2.meanlog = -0.5, coeff2.sdlog = 0.5, coeff3.k = 3, coeff1.k = 1.5)
x <- rcomposite(1e5, dist = dist, coeff = coeff)

mean(x)
mcomposite(r = 1, lower.tail = FALSE, dist = dist, coeff = coeff)

sum(x[x > quantile(x, 0.1)]) / length(x)
mcomposite(r = 1, truncation = quantile(x, 0.1), lower.tail = FALSE, dist = dist, coeff = coeff)

## Two-component distribution
dist <- c("lnorm", "pareto")
coeff <- coeff <- c(coeff2.k = 1.5, coeff1.meanlog = -0.5, coeff1.sdlog = 0.5)

# Compare density with the Right-Pareto Lognormal distribution
plot(x = seq(0, 5, length.out = 1e3), y = dcomposite(x = seq(0, 5, length.out = 1e3),
dist = dist, coeff = coeff))
lines(x = seq(0, 5, length.out = 1e3), y = drightparetolognormal(x = seq(0, 5, length.out = 1e3)))

# Demonstration of log functionality for probability and quantile function
qcomposite(pcomposite(2, dist = dist, coeff = coeff, log.p = TRUE), dist = dist,
coeff = coeff, log.p = TRUE)

# The zeroth truncated moment is equivalent to the probability function
pcomposite(2, dist = dist, coeff = coeff)
mcomposite(truncation = 2, dist = dist, coeff = coeff)

# The (truncated) first moment is equivalent to the mean of a (truncated) random sample,
#for large enough samples.
coeff <- c(coeff1.meanlog = -0.5, coeff1.sdlog = 0.5, coeff2.k = 3)
x <- rcomposite(1e5, dist = dist, coeff = coeff)

mean(x)
mcomposite(r = 1, lower.tail = FALSE, dist = dist, coeff = coeff)

sum(x[x > quantile(x, 0.1)]) / length(x)
mcomposite(r = 1, truncation = quantile(x, 0.1), lower.tail = FALSE, dist = dist, coeff = coeff)