# GeoGG: Geometrical Generalized Gamma Distribution In aneisse/survdistr: Survival Analysis Distributions

## Description

Density, distribution function, quantile function and random generation for the GeoGG distribution with parameters `tau`, `alpha`, `k` and `pg`.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```dgeogg(x, alpha = 1, tau, k, pg, log = FALSE) pgeogg(q, alpha = 1, tau, k, pg, lower.tail = TRUE, log.p = FALSE) qgeogg(p, alpha = 1, tau, k, pg, lower.tail = TRUE, log.p = FALSE) rgeogg(n, alpha = 1, tau, k, pg, cens.prop = 0) ml.geogg(x, alpha.ini, tau.ini, k.ini, pg.ini) ```

## Arguments

 `x, q` numeric vector of quantiles. `alpha` scale parameter α from the Generalized Gama (Stacy, 1962), α > 0. `tau` shape parameter τ from the Generalized Gama (Stacy, 1962), τ > 0. `k` shape parameter k from the Generalized Gama (Stacy, 1962), k > 0. `pg` is the probability of success p from the Geometric, 0 < p < 1. `log, log.p` logical; if `TRUE`, probabilities/densities `p` are given as `log(p)`. `lower.tail` logical; if `TRUE`, probabilities are P[X ≤ x], otherwise, P[X ≥ x] `n` desired size of the random number sample. `cens.prop` proportion of censored data to be simulated. If greater than `0`, a matrix will be returned instead of a vector. The matrix will contain the random values and a censorship indicator variable.

## Details

The GeoGG distribution has density

f(x) = ((τ(1-p))/(α Γ(k)))(x/α))^(τ k-1) e^(-(t/α)^τ)(1 - p*(1-γ(k, (t/α)^τ)))^(-2)

with scale parameter α, shape parameters τ and k and p is the degeneration parameter as described by Ortega et al (2011).

If `alpha` is not specified it assumes the default value of 1.

With `pg = 0` GeoGG equals the Generalized Gamma with parameterizantion described on Stacy's (1962).

When `pg = 0` and `tau = 1` then it equals a Gama with `shape = k` and `scale = alpha`.

With `pg = 0` and `k = 1` it equals a Weibull with `shape = tau` and `scale = alpha`.

Finally, when `pg = 0` and `tau = k = 1` it becomes the Exponential with `rate = 1/alpha`.

For the cases described above, when `0 < pg < 1` then the GeoGG will result in the Geometrical Gamma, Geometrical Weibull and Geometrical Exponential, respectively, as described by Ortega et al (2011).

## Value

`dgeogg` gives the density, `pgeogg` gives the distribution function, `qgeogg` gives the quantile function, and `rgeogg` generates random values.

The length of the result is determined by `n` for `rgeogg`, for the other fucntions the length is the same as the vector passed to the first argument.

Only the first element of the logical arguments are used.

## Author(s)

Anderson Neisse <[email protected]>

## Source

The source code of all distributions in this package can also be found on the survdistr Github repository.

## References

STACY, E. W., et al. A generalization of the gamma distribution. The Annals of mathematical statistics, 1962, 33.3: 1187-1192.

ORTEGA, E. M. M.; CORDEIRO, G. M.; PASCOA, M. A. R. The generalized gamma geometric distribution. Journal of Statistical Theory and Applications, 2011, 10.3: 433-454.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ```#Equivalency to the Generalized Gamma on it's original parameterization all.equal(dgeogg(5, alpha = 2, tau = 3, k = 0.5, pg = 0), dgengamma.orig(5, shape = 3, scale = 2, k = 0.5)) #Equivalency to the Gamma all.equal(dgeogg(5, alpha = 2, tau = 1, k = 0.5, pg = 0), dgamma(5, shape = 0.5, scale = 2)) #Equivalency to the Weibull all.equal(dgeogg(5, alpha = 2, tau = 3, k = 1, pg = 0), dweibull(5, shape = 3, scale = 2)) #Equivalency to the Exponential all.equal(dgeogg(5, alpha = 2, tau = 1, k = 1, pg = 0), dexp(5, rate = 1/2)) # Generating values and comparing with the function x <- rgeogg(10000, alpha = 2, tau = 3, k = 0.5, pg = 0.35) hist(x, probability = T, breaks = 100) curve(dgeogg(x, alpha = 2, tau = 3, k = 0.5, pg = 0.35), from = 0, to = 4, add = T) ```