reliaR (version 0.2)

Loglog: The Loglog distribution

Description

Density, distribution function, quantile function and random generation for the Loglog distribution with shape parameter alpha and scale parameter lambda.

Usage

dloglog(x, alpha, lambda, log = FALSE)
ploglog(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qloglog(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rloglog(n, alpha, lambda)

Value

dloglog gives the density,

ploglog gives the distribution function,

qloglog gives the quantile function, and

rloglog generates random deviates.

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

alpha

shape parameter.

lambda

scale parameter.

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, \(P[X > x]\).

Details

The loglog(Pham) distribution has density

$$f(x) = \alpha \ln \left(\lambda\right) x^{\alpha - 1} \lambda^{x^\alpha} \exp\left\{{1 - \lambda ^{x^\alpha}}\right\};\; x > 0, \lambda > 0, \alpha > 0$$ where \(\alpha\) and \(\lambda\) are the shape and scale parameters, respectively. (Pham, 2002)

References

Pham, H.(2002). A Vtub-Shaped Hazard Rate Function with Applications to System Safety, International Journal of Reliability and Applications. ,Vol. 3, No. l, pp. 1-16.

Pham, H.(2006). System Software Reliability, Springer-Verlag.

See Also

.Random.seed about random number; sloglog for Loglog survival / hazard etc. functions;

Examples

data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.9058689 lambda.est = 1.0028228

dloglog(sys2, 0.9058689, 1.0028228, log = FALSE)
ploglog(sys2, 0.9058689, 1.0028228, lower.tail = TRUE, log.p = FALSE)
qloglog(0.25, 0.9058689, 1.0028228, lower.tail=TRUE, log.p = FALSE)
rloglog(30, 0.9058689, 1.0028228)