DiscreteWeibull: Discrete Weibull distribution (type I)

In extraDistr: Additional Univariate and Multivariate Distributions

Discrete Weibull distribution (type I)

Description

Density, distribution function, quantile function and random generation for the discrete Weibull (type I) distribution.

Usage

ddweibull(x, shape1, shape2, log = FALSE)

pdweibull(q, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

qdweibull(p, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rdweibull(n, shape1, shape2)

Arguments

x, q	vector of quantiles.
shape1, shape2	parameters (named q, \beta). Values of shape2 need to be positive and 0 < shape1 < 1.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x] otherwise, P[X > x].
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x) = q^{x^\beta} - q^{(x+1)^\beta}

Cumulative distribution function

F(x) = 1-q^{(x+1)^\beta}

Quantile function

F^{-1}(p) = \left \lceil{\left(\frac{\log(1-p)}{\log(q)}\right)^{1/\beta} - 1}\right \rceil

References

Nakagawa, T. and Osaki, S. (1975). The Discrete Weibull Distribution. IEEE Transactions on Reliability, R-24, 300-301.

Kulasekera, K.B. (1994). Approximate MLE's of the parameters of a discrete Weibull distribution with type I censored data. Microelectronics Reliability, 34(7), 1185-1188.

Khan, M.A., Khalique, A. and Abouammoh, A.M. (1989). On estimating parameters in a discrete Weibull distribution. IEEE Transactions on Reliability, 38(3), 348-350.

See Also

Weibull

Examples

x <- rdweibull(1e5, 0.32, 1)
xx <- seq(-2, 100, by = 1)
plot(prop.table(table(x)), type = "h")
lines(xx, ddweibull(xx, .32, 1), col = "red")

# Notice: distribution of F(X) is far from uniform:
hist(pdweibull(x, .32, 1), 50)

plot(ecdf(x))
lines(xx, pdweibull(xx, .32, 1), col = "red", lwd = 2, type = "s")

extraDistr documentation built on May 29, 2024, 9:31 a.m.