DiscreteWeibull (version 1.1)

Discrete Weibull (Type 3): The type 3 discrete Weibull distribution

Description

Probability mass function, distribution function, quantile function, random generation, and hazard function for the type 3 discrete Weibull distribution with parameters $c$ and $\beta$

Usage

ddweibull3(x, c, beta)
pdweibull3(x, c, beta)
qdweibull3(p, c, beta)
rdweibull3(n, c, beta)
hdweibull3(x, c, beta)

Arguments

x

vector of values/quantiles

p

vector of probabilities

c

first parameter

beta

second parameter

n

sample size

Value

ddweibull3 gives the probability function, pdweibull3 gives the distribution function, qdweibull3 gives the quantile function, hdweibull3 gives the hazard function, and rdweibull generates random values.

Details

The type 3 discrete Weibull distribution is characterized by the following cumulative distribution function: $F(x;c,\beta)=1-\exp(-c\sum_{j=0}^{x+1}j^{\beta})$, for $x=0,1,2,\dots$, with $c>0$ and $\beta>=-1$.

Examples

# ddweibull3
x <- 0:10
c <- 0.3
beta <- 0.75
p <- ddweibull3(x, c, beta)
p
plot(x, p, type="b", ylab=expression(P(X)==x))
# pdweibull3
x <- 0:10
c <- 0.5
beta <- 0.5
p <- pdweibull3(x, c, beta)
p
cumsum(ddweibull3(x, c, beta))
plot(x, p, type="s", ylab=expression(P(X<=x)))
# qdweibull3
p <- c(1:9)/10
p
c <- 0.1
beta <- 0.5
qdweibull3(p, c, beta)
pdweibull3(10, c, beta)
pdweibull3(9, c, beta)
# rdweibull3
n <- 20
c <- 0.25
beta <- -0.25
x <- rdweibull3(n, c, beta)
x
beta <- 0
x <- rdweibull3(n, c, beta)
x
beta <- 0.25
x <- rdweibull3(n, c, beta)
x
n <- 1000
x <- rdweibull3(n, c, beta)
obs <- c(sum(x==0), tabulate(x))
obs <- obs/sum(obs)
theo <- ddweibull3(min(x):max(x), c, beta)
barplot(rbind(obs, theo), beside=TRUE, names.arg=min(x):max(x),
ylab="relative frequency/probability", col=1:2)
legend(24, 0.1, c("observed", "theoretical"), pch=15, col=1:2)
#hdweibull3
x <- 0:15
c <- 0.5
hn<-hdweibull3(x, c, beta = -0.5)
h0<-hdweibull3(x, c, beta = 0)
hp<-hdweibull3(x, c, beta = 0.5)
plot(x, hn, type="b", ylim = c(0, 1), ylab="hazard rate")
points(x, h0, type = "b", col=2)
points(x, hp, type = "b", col=3)
legend(11, 0.5, c("beta<0", "beta=0", "beta>0"), col=1:3, pch=21)