NeuDist (version 1.0.1)

InvPham: Inverse Pham Distribution

Description

Provides density, distribution, quantile, random generation, and hazard functions for the Inverse Pham distribution.

Usage

dinv.pham(x, beta, delta, log = FALSE)
pinv.pham(q, beta, delta, lower.tail = TRUE, log.p = FALSE)
qinv.pham(p, beta, delta, lower.tail = TRUE, log.p = FALSE)
rinv.pham(n, beta, delta)
hinv.pham(x, beta, delta)

Value

• dinv.pham: numeric vector of (log-)densities

• pinv.pham: numeric vector of probabilities

• qinv.pham: numeric vector of quantiles

• rinv.pham: numeric vector of random variates

• hinv.pham: numeric vector of hazard values

Arguments

x, q

numeric vector of quantiles (x, q)

beta

positive numeric parameter

delta

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 Inverse Pham distribution is parameterized by the parameters \(\beta > 0\), and \(\delta > 0\).

The Inverse Pham distribution has CDF:

$$ F(x; \beta, \delta) = \exp \left( {1 - {\delta ^{{x^{ - \beta }}}}} \right) \quad ;\;x > 0. $$

where\(\beta\) and \(\delta\) are the parameters.

The following functions are included:

• dinv.pham() — Density function

• pinv.pham() — Distribution function

• qinv.pham() — Quantile function

• rinv.pham() — Random generation

• hinv.pham() — Hazard function

References

Elbatal, M., Araibi, M.I.A., Ocloo, S.K., Almetwally, E.M., Sapkota, L.P., & Gemeay, A.M. (2025). Classical and Bayesian Methodology for a New Inverse Statistical Model. Engineering Reports, 7(8), 1--33. tools:::Rd_expr_doi("10.1002/eng2.70323")

Srivastava, A.K., & Kumar, V. (2011). Analysis of Pham (Loglog) Reliability Model Using Bayesian Approach. Computer Science Journal, 1(2), 79--100.

Pham, H. (2002). A Vtub-Shaped Hazard Rate Function With Applications to System Safety. International Journal of Reliability and Applications, 3(1), 1--16.

Examples

x <- seq(1, 10, 0.5)
dinv.pham(x, 0.5, 1.5)
pinv.pham(x, 0.5, 1.5)
qinv.pham(0.5, 0.5, 1.5)
rinv.pham(10, 0.5, 1.5)
hinv.pham(x, 0.5, 1.5)

# Data
x <- relief
# ML estimates
params = list(beta=2.8287, delta=9.6044)
#P–P (probability–probability) plot
pp.plot(x, params = params, pfun = pinv.pham, fit.line=TRUE)

#Q-Q (quantile–quantile) plot 
qq.plot(x, params = params, qfun = qinv.pham, fit.line=TRUE)

# Goodness-of-Fit(GoF) and Model Diagnostics 
out <- gofic(x, params = params,
             dfun = dinv.pham, pfun=pinv.pham, plot=FALSE)
print.gofic(out)