R: The Discrete Burr type XII distribution for fitting a GAMLSS...

DBURR12 {gamlss.dist} | R Documentation

## The Discrete Burr type XII distribution for fitting a GAMLSS model

### Description

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.

### Usage

    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)

### Arguments

`mu.link` | Defines the `mu.link`, with "log" link as the default for the `mu` parameter
`sigma.link` | Defines the `sigma.link`, with "log" link as the default for the `sigma` parameter
`nu.link` | Defines the `nu.link`, with "log" link as the default for the `nu` parameter
`x` | vector of (non-negative integer) quantiles
`p` | vector of probabilities
`q` | vector of quantiles
`mu` | vector of positive `mu`
`sigma` | vector of positive dispersion parameter `sigma`
`nu` | vector of `nu`
`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]
`n` | number of random values to return

### Details

The probability function of the discrete Burr XII distribution is given by

f(y|μ,σ,ν)= (1+(y/μ)^σ)^ν - (1+((y+1)/μ)^σ)^ν

for y=0,1,2,..., mu>0 , σ>0 and nu>0.

Note that the above parametrization is different from Para and Jan (2016).

### Value

The function `DBURR12()` Returns a `gamlss.family` object which can be used to fit a discrete Burr XII distribution in the `gamlss()` function.

### Note

The parameters of the distributioins are highly correlated so the argument of `gamlss` `method=mixed(10,100)` may have to be used.

The distribution can be under/over dispersed and also with long tails.

### Author(s)

Rigby, R. A., Stasinopoulos D. M., Fernanda De Bastiani.

### References

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.

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.

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.

### See Also

`gamlss.family`, `DPO`

### Examples

    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')

[Package gamlss.dist version 5.3-2 Index]