Discrete Weibull distribution (type I)
Density, distribution function, quantile function and random generation for the discrete Weibull (type I) distribution.
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)
x, q |
vector of quantiles. |
shape1, shape2 |
parameters (named q, β). Values of |
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]. |
p |
vector of probabilities. |
n |
number of observations. If |
Probability mass function
f(x) = q^x^β - q^(x+1)^β
Cumulative distribution function
F(x) = 1-q^(x+1)^β
Quantile function
F^-1(p) = ceiling((log(1-p)/log(q))^(1/β) - 1)
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.
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")
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