# pbivgeocure: Joint Cumulative Function for the Basu-Dhar Bivariate... In BivGeo: Basu-Dhar Bivariate Geometric Distribution

## Description

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

## Usage

 `1` ```pbivgeocure(x, y, theta, phi, lower.tail = TRUE) ```

## 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. `phi` vector (of length 4) containing values of the cure fraction incidence parameters φ_{11}, φ_{10}, φ_{01} and φ_{00}. The parameters are restricted to φ_{11} + φ_{10} + φ_{01} + φ_{00}= 1. `lower.tail` logical; If TRUE (default), probabilities are P(X ≤ x, Y ≤ y) otherwise P(X > x, Y > y).

## Details

The joint cumulative 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) = 1 - (φ_{11} + φ_{10}) (θ_1 θ_3)^x - (φ_{01} + φ_{00}) - (φ_{11} + φ_{01}) (θ_2 θ_3)^y - (φ_{10} + φ_{00})

+ φ_{11} (θ_{1}^{x} θ_{2}^{y}θ_{3}^{\max(x,y)}) + φ_{10} (θ_1 θ_{3})^x + φ_{01} (θ_2 θ_{3})^y + φ_{00}

and the joint survival function is given by:

P(X > x, Y > y) = φ_{11} (θ_{1}^{x} θ_{2}^{y}θ_{3}^{\max(x,y)}) + φ_{10} (θ_1 θ_{3})^x + φ_{01} (θ_2 θ_{3})^y + φ_{00}

## Value

`pbivgeocure` gives the values of the cumulative 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

`pbivgeocure` 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.

## See Also

`Geometric` for the univariate geometric distribution.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```# If lower.tail = TRUE: pbivgeocure(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), phi = c(0.2, 0.3, 0.3, 0.2), lower.tail = TRUE) #  0.159456 matr <- matrix(c(1,2,3,5), ncol = 2) pbivgeocure(x=matr,y=NULL,theta=c(0.2, 0.4, 0.7),phi=c(0.2, 0.3, 0.3, 0.2),lower.tail = TRUE) #  0.1684877 0.1957496 # If lower.tail = FALSE: pbivgeocure(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), phi = c(0.2, 0.3, 0.3, 0.2), lower.tail = FALSE) #  0.268656 matr <- matrix(c(1,2,3,5), ncol = 2) pbivgeocure(x=matr,y=NULL,theta=c(0.2, 0.4, 0.7),phi=c(0.2, 0.3, 0.3, 0.2),lower.tail = FALSE) #  0.2494637 0.2064101 ```

BivGeo documentation built on May 2, 2019, 6:12 a.m.