# dbivgeocure: Joint Probability Mass Function for the Basu-Dhar Bivariate... In BivGeo: Basu-Dhar Bivariate Geometric Distribution

## Description

This function computes the joint probability mass function of the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values in presence of cure fraction.

## Usage

 `1` ```dbivgeocure(x, y, theta, phi11, log = FALSE) ```

## Arguments

 `x` matrix or vector containing the data. If x is a matrix then it is considered as x the first column and y the second column (y argument need be setted to NULL). Additional columns and y are ignored. `y` vector containing the data of y. It is used only if x is also a vector. Vectors x and y should be of equal length. `theta` vector (of length 3) containing values of the parameters θ_1, θ_2 and θ_{3} of the Basu-Dhar bivariate Geometric distribution. The parameters are restricted to 0 < θ_i < 1, i = 1,2 and 0 < θ_{3} ≤ 1. `phi11` real number containing the value of the cure fraction incidence parameter φ_{11} restricted to 0 < φ_{11} < 1 and φ_{11} + φ_{10} + φ_{01} + φ_{00}= 1 where φ_{10}, φ_{01} and φ_{00} are the complementary cure fraction incidence parameters for the joint cdf and sf functions. `log` logical argument for calculating the log probability or the probability function. The default value is FALSE.

## Details

The joint probability mass function for a random vector (X, Y) following a Basu-Dhar bivariate geometric distribution in presence of cure fraction could be written as:

P(X = x, Y = y) = φ_{11}(θ_{1}^{x - 1} θ_{2}^{y - 1} θ_{3}^{z_1} - θ_{1}^{x} θ_{2}^{y - 1} θ_{3}^{z_2} - θ_{1}^{x - 1} θ_{2}^{y} θ_{2}^{z_3} + θ_{1}^{x} θ_{2}^{y} θ_{3}^{z_4})

where x,y > 0 are positive integers and z_1 = \max(x - 1, y - 1),z_2 = \max(x, y - 1), z_3 = \max(x - 1, y), z_4 = \max(x, y).

## Value

`dbivgeocure` gives the values of the probability mass function in presence of cure fraction.

Invalid arguments will return an error message.

## Author(s)

Ricardo P. Oliveira rpuziol.oliveira@gmail.com

Jorge Alberto Achcar achcar@fmrp.usp.br

## Source

`dbivgeocure` is calculated directly from the definition.

## References

Basu, A. P., & Dhar, S. K. (1995). Bivariate geometric distribution. Journal of Applied Statistical Science, 2, 1, 33-44.

Achcar, J. A., Davarzani, N., & Souza, R. M. (2016). Basu<e2><80><93>Dhar bivariate geometric distribution in the presence of covariates and censored data: a Bayesian approach. Journal of Applied Statistics, 43, 9, 1636-1648.

de Oliveira, R. P., & Achcar, J. A. (2018). Basu-Dhar's bivariate geometric distribution in presence of censored data and covariates: some computational aspects. Electronic Journal of Applied Statistical Analysis, 11, 1, 108-136.

de Oliveira, R. P., Achcar, J. A., Peralta, D., & Mazucheli, J. (2018). Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach. Journal of Applied Statistics, 1-19.

`Geometric` for the univariate geometric distribution.
 ```1 2 3 4 5 6 7 8 9``` ```# If log = FALSE: dbivgeocure(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), phi11 = 0.4, log = FALSE) # [1] 0.064512 # If log = TRUE: dbivgeocure(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), phi11 = 0.4, log = TRUE) # [1] -2.740904 ```