The DBURR12()
function defines the discrete Burr type XII distribution, a three parameter discrete distribution, for a gamlss.family
object to be used in GAMLSS fitting using the function gamlss()
.
The functions dDBURR12()
, pDBURR12()
, qDBURR12()
and rDBURR12()
define the density, distribution function, quantile function and random generation for the discrete Burr type XII DBURR12()
, distribution.
DBURR12(mu.link = "log", sigma.link = "log", nu.link = "log")
dDBURR12(x, mu = 5, sigma = 2, nu = 2, log = FALSE)
pDBURR12(q, mu = 5, sigma = 2, nu = 2, lower.tail = TRUE,
log.p = FALSE)
qDBURR12(p, mu = 5, sigma = 2, nu = 2, lower.tail = TRUE,
log.p = FALSE)
rDBURR12(n, mu = 5, sigma = 2, nu = 2)
The function DBURR12()
Returns a gamlss.family
object which can be used to fit a discrete Burr XII distribution in the gamlss()
function.
Defines the mu.link
, with "log" link as the default for the mu
parameter
Defines the sigma.link
, with "log" link as the default for the sigma
parameter
Defines the nu.link
, with "log" link as the default for the nu
parameter
vector of (non-negative integer) quantiles
vector of probabilities
vector of quantiles
vector of positive mu
vector of positive dispersion parameter sigma
vector of nu
logical; if TRUE, probabilities p are given as log(p)
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
number of random values to return
Rigby, R. A., Stasinopoulos D. M., Fernanda De Bastiani.
The probability function of the discrete Burr XII distribution is given by $$f(y|\mu,\sigma,\nu)= (1+(y/\mu)^\sigma)^\nu - (1+((y+1)/\mu)^\sigma)^\nu$$
for \(y=0,1,2,...,\infty\), \(\mu>0\) , \(\sigma>0\) and \(\mu>0\) see pp 504-505 of Rigby et al. (2019).
Note that the above parametrization is different from Para and Jan (2016).
Para, B. A. and Jan, T. R. (2016). On discrete three parameter Burr type XII and discrete Lomax distributions and their applications to model count data from medical science. Biometrics and Biostatistics International Journal, 54, part 3, pp 507-554.
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 4, pp 1-15.
Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://www.gamlss.com/.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, tools:::Rd_expr_doi("10.18637/jss.v023.i07").
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. tools:::Rd_expr_doi("10.1201/b21973")
(see also https://www.gamlss.com/).
gamlss.family
, DPO
DBURR12()#
#plot the pdf using plot
plot(function(y) dDBURR12(y, mu=10, sigma=1, nu=1), from=0, to=100, n=100+1, type="h") # pdf
# plot the cdf
plot(seq(from=0,to=100),pDBURR12(seq(from=0,to=100), mu=10, sigma=1, nu=1), type="h") # cdf
# generate random sample
tN <- table(Ni <- rDBURR12(100, mu=5, sigma=1, nu=1))
r <- barplot(tN, col='lightblue')
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