Logarithmic and zero adjusted logarithmic distributions for fitting a GAMLSS model
The function LG defines the logarithmic distribution, a one parameter distribution, for a gamlss.family object to be
used in GAMLSS fitting using the function gamlss(). The functions dLG, pLG, qLG and rLG define the
density, distribution function, quantile function
and random generation for the logarithmic , LG(), distribution.
The function ZALG defines the zero adjusted logarithmic distribution, a two parameter distribution, for a gamlss.family object to be
used in GAMLSS fitting using the function gamlss(). The functions dZALG, pZALG, qZALG and rZALG define the
density, distribution function, quantile function
and random generation for the inflated logarithmic , ZALG(), distribution.
LG(mu.link = "logit") dLG(x, mu = 0.5, log = FALSE) pLG(q, mu = 0.5, lower.tail = TRUE, log.p = FALSE) qLG(p, mu = 0.5, lower.tail = TRUE, log.p = FALSE, max.value = 10000) rLG(n, mu = 0.5) ZALG(mu.link = "logit", sigma.link = "logit") dZALG(x, mu = 0.5, sigma = 0.1, log = FALSE) pZALG(q, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) qZALG(p, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) rZALG(n, mu = 0.5, sigma = 0.1)
mu.link |
defines the |
sigma.link |
defines the |
x |
vector of (non-negative integer) |
mu |
vector of positive means |
sigma |
vector of probabilities at zero |
p |
vector of probabilities |
q |
vector of quantiles |
n |
number of random values to return |
log, log.p |
logical; if TRUE, probabilities p are given as log(p) |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |
max.value |
valued needed for the numerical calculation of the q-function |
For the definition of the distributions see Rigby and Stasinopoulos (2010) below.
The parameterization of the logarithmic distribution in the function LM is
f(y|mu) = α μ^y / y
where for y>=1 and μ>0 and
α= [log(1-μ)]^{-1}
The function LG and ZALG return a gamlss.family object which can be used to fit a
logarithmic and a zero inflated logarithmic distributions respectively in the gamlss() function.
Mikis Stasinopoulos, Bob Rigby
Johnson, Norman Lloyd; Kemp, Adrienne W; Kotz, Samuel (2005). "Chapter 7: Logarithmic and Lagrangian distributions". Univariate discrete distributions (3 ed.). John Wiley & Sons. ISBN 9780471272465.
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.
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. 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, https://www.jstatsoft.org/v23/i07.
Rigby, R. A. and Stasinopoulos D. M. (2010) The gamlss.family distributions, (distributed with this package or see https://www.gamlss.com/)
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.
LG() ZAP() # creating data and plotting them dat <- rLG(1000, mu=.3) r <- barplot(table(dat), col='lightblue') dat1 <- rZALG(1000, mu=.3, sigma=.1) r1 <- barplot(table(dat1), col='lightblue')
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