# LG: Logarithmic and zero adjusted logarithmic distributions for... In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

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.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```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) ```

## Arguments

 `mu.link` defines the `mu.link`, with `logit` link as the default for the `mu` parameter `sigma.link` defines the `sigma.link`, with `logit` link as the default for the sigma parameter which in this case is the probability at zero. `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

## Details

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}

## Value

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.

## Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby

## References

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.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://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, http://www.jstatsoft.org/v23/i07.

Rigby, R. A. and Stasinopoulos D. M. (2010) The gamlss.family distributions, (distributed with this package or see http://www.gamlss.org/)

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.

`gamlss.family`, `PO`, `ZAP`
 ```1 2 3 4 5 6 7``` ```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') ```