Beta Extended G Distribution

Description

Computes the pdf, cdf, quantile and random numbers of the beta extended distribution due to Cordeiro et al. (2012) specified by the pdf

f (x) = \frac {\displaystyle α g (x)}{\displaystyle B (a, b)} ≤ft\{ 1 - \exp ≤ft[ -α G (x) \right] \right\}^{a - 1} \exp ≤ft[ -α b G (x) \right]

for G any valid cdf, g the corresponding pdf, α > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter. Also computes the Cramer-von Misses statistic, Anderson Darling statistic, Kolmogorov Smirnov test statistic and p-value, maximum likelihood estimates, Akaike Information Criterion, Consistent Akaikes Information Criterion, Bayesian Information Criterion, Hannan-Quinn information criterion, standard errors of the maximum likelihood estimates, minimum value of the negative log-likelihood function and convergence status when the distribution is fitted to some data

Usage

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dbeg(x, spec, alpha = 1, a = 1, b = 1, log = FALSE, ...)
pbeg(x, spec, alpha = 1, a = 1, b = 1, log.p = FALSE, lower.tail = TRUE, ...)
qbeg(p, spec, alpha = 1, a = 1, b = 1, log.p = FALSE, lower.tail = TRUE, ...)
rbeg(n, spec, alpha = 1, a = 1, b = 1, ...)
mbeg(g, data, starts, method = "BFGS")

Arguments

x

scaler or vector of values at which the pdf or cdf needs to be computed

p

scaler or vector of probabilities at which the quantile needs to be computed, must be between 0 and 1 - exp(-alpha)

n

number of random numbers to be generated

alpha

the value of the scale parameter, must be positive, the default is 1

a

the value of the first shape parameter, must be positive, the default is 1

b

the value of the second shape parameter, must be positive, the default is 1

spec

a character string specifying the distribution of G and g (for example, "norm" if G and g correspond to the standard normal).

log

if TRUE then log(pdf) are returned

log.p

if TRUE then log(cdf) are returned and quantiles are computed for exp(p)

lower.tail

if FALSE then 1-cdf are returned and quantiles are computed for 1-p

...

other parameters

g

same as spec but must be one of chisquare ("chisq"), exponential ("exp"), F ("f"), gamma ("gamma"), lognormal ("lognormal"), Weibull ("weibull"), Burr XII ("burrxii"), Chen ("chen"), Frechet ("frechet"), Gompertz ("gompertz"), linear failure rate ("lfr"), log-logistic ("log-logistic"), Lomax ("lomax") and Rayleigh ("rayleigh"). Each of these distributions has one parameter (r) or two parameters (r, s), for details including the density function and parameter specifications see Nadarajah and Rocha (2014)

data

a vector of data values for which the distribution is to be fitted

starts

initial values of (alpha, a, b, r) if g has one parameter or initial values of (alpha, a, b, r, s) if g has two parameters

method

the method for optimizing the log likelihood function. It can be one of "Nelder-Mead", "BFGS", "CG", "L-BFGS-B" or "SANN". The default is "BFGS". The details of these methods can be found in the manual pages for optim

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the quantile values computed at p or an object of the same length as n, giving the random numbers generated or an object giving the values of Cramer-von Misses statistic, Anderson Darling statistic, Kolmogorov Smirnov test statistic and p-value, maximum likelihood estimates, Akaike Information Criterion, Consistent Akaikes Information Criterion, Bayesian Information Criterion, Hannan-Quinn information criterion, standard errors of the maximum likelihood estimates, minimum value of the negative log-likelihood function and convergence status.

Note

NaNs can be produced if p \not \in [0, 1 - \exp (-α)].

Author(s)

Saralees Nadarajah, Ricardo Rocha

References

S. Nadarajah and R. Rocha, Newdistns: An R Package for New Families of Distributions, Journal of Statistical Software, 69(10), 1-32, doi:10.18637/jss.v069.i10

G. M. Cordeiro, E. M. M. Ortega, G. Silva, The beta extended Weibull family, Journal of Probability and Statistical Science 10 (2012) 15-40

Examples

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x=runif(10,min=0,max=1)
dbeg(x,"exp",alpha=1,a=1,b=1)
pbeg(x,"exp",alpha=1,a=1,b=1)
qbeg(x,"exp",alpha=1,a=1,b=1)
rbeg(10,"exp",alpha=1,a=1,b=1)
mbeg("exp",rexp(100),starts=c(1,1,1,1),method="BFGS")