NeuDist (version 1.0.1)

LindleyInvWeibull: Lindley-Inverse Weibull Distribution

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Lindley-Inverse Weibull distribution.

Usage

dlindley.inv.weib(x, alpha, beta, theta, log = FALSE)
plindley.inv.weib(q, alpha, beta, theta, lower.tail = TRUE, log.p = FALSE)
qlindley.inv.weib(p, alpha, beta, theta, lower.tail = TRUE, log.p = FALSE)
rlindley.inv.weib(n, alpha, beta, theta)
hlindley.inv.weib(x, alpha, beta, theta)

Value

• dlindley.inv.weib: numeric vector of (log-)densities

• plindley.inv.weib: numeric vector of probabilities

• qlindley.inv.weib: numeric vector of quantiles

• rlindley.inv.weib: numeric vector of random variates

• hlindley.inv.weib: numeric vector of hazard values

Arguments

x, q

numeric vector of quantiles (x, q)

alpha

positive numeric parameter

beta

positive numeric parameter

theta

positive numeric parameter

log

logical; if TRUE, returns log-density

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\).

log.p

logical; if TRUE, probabilities are given as log(p)

p

numeric vector of probabilities (0 < p < 1)

n

number of observations (integer > 0)

Details

The Lindley-Inverse Weibull distribution is parameterized by the parameters \(\alpha > 0\), \(\beta > 0\), and \(\theta > 0\).

The Lindley-Inverse Weibull distribution has CDF:

$$ F(x; \alpha, \beta, \theta) = \quad 1 - {\left( {1 - {e^{ - \alpha {x^{ - \beta }}}}} \right)^\theta } \left\{ {1 - \left( {\frac{\theta }{{\theta + 1}}} \right)\ln \left( {1 - {e^{ - \alpha {x^{ - \beta }}}}} \right)} \right\} \, ; \quad x > 0. $$

where \(\alpha\), \(\beta\), and \(\theta\) are the parameters.

The functions available are listed below:

• dlindley.inv.weib() — Density function

• plindley.inv.weib() — Distribution function

• qlindley.inv.weib() — Quantile function

• rlindley.inv.weib() — Random generation

• hlindley.inv.weib() — Hazard function

References

Joshi, R. K., & Kumar, V. (2020). Lindley inverse Weibull distribution: Theory and Applications. Bull. Math. & Stat. Res., 8(3), 32--46.

Examples

x <- seq(0.1, 1, 0.1)
dlindley.inv.weib(x, 1.5, 2.0, 0.5)
plindley.inv.weib(x, 1.5, 2.0, 0.5)
qlindley.inv.weib(0.5, 2.0, 5.0, 0.1)
rlindley.inv.weib(10, 1.5, 2.0, 0.5)
hlindley.inv.weib(x, 1.5, 2.0, 0.5)

# Data
x <- waiting
# ML estimates
params = list(alpha=9.3340, beta=0.3010, theta=104.4248)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = plindley.inv.weib, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qlindley.inv.weib, fit.line=FALSE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dlindley.inv.weib, pfun=plindley.inv.weib, plot=FALSE)
print.gofic(out)