dbivgeocure function

Joint Probability Mass Function for the Basu-Dhar Bivariate Geometric Distribution in Presence of Cure Fraction

Joint Probability Mass Function for the Basu-Dhar Bivariate Geometric Distribution in Presence of Cure Fraction

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

Source

dbivgeocure is calculated directly from the definition.

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 θ1,θ2\theta_1, \theta_2 and θ3\theta_{3} of the Basu-Dhar bivariate Geometric distribution. The parameters are restricted to 0<θi<1,i=1,20 < \theta_i < 1, i = 1,2 and 0<θ3≤10 < \theta_{3} \le 1.
  • phi11: real number containing the value of the cure fraction incidence parameter ϕ11\phi_{11} restricted to 0<ϕ11<10 < \phi_{11} < 1 and ϕ11+ϕ10+ϕ01+ϕ00=1\phi_{11} + \phi_{10} + \phi_{01} + \phi_{00}= 1 where ϕ10,ϕ01\phi_{10}, \phi_{01} and ϕ00\phi_{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.

Returns

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

Invalid arguments will return an error message.

Details

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

P(X=x,Y=y)=ϕ11(θ1x−1θ2y−1θ3z1−θ1xθ2y−1θ3z2−θ1x−1θ2yθ2z3+θ1xθ2yθ3z4) P(X = x, Y = y) = \phi_{11}(\theta_{1}^{x - 1} \theta_{2}^{y - 1} \theta_{3}^{z_1} - \theta_{1}^{x} \theta_{2}^{y - 1} \theta_{3}^{z_2} - \theta_{1}^{x - 1} \theta_{2}^{y} \theta_{2}^{z_3} + \theta_{1}^{x} \theta_{2}^{y} \theta_{3}^{z_4})

where x,y>0x,y > 0 are positive integers and z1=max⁡(x−1,y−1),z2=max⁡(x,y−1),z3=max⁡(x−1,y),z4=max⁡(x,y)z_1 = \max(x - 1, y - 1),z_2 = \max(x, y - 1), z_3 = \max(x - 1, y), z_4 = \max(x, y).

Examples

# 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

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–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.

Author(s)

Ricardo P. Oliveira rpuziol.oliveira@gmail.com

Jorge Alberto Achcar achcar@fmrp.usp.br

BivGeo packageRead PDF manual