pbivgeocure {BivGeo}

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

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

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 \theta_1, \theta_2 and \theta_{3} of the Basu-Dhar bivariate Geometric distribution. The parameters are restricted to 0 < \theta_i < 1, i = 1,2 and 0 < \theta_{3} \le 1.

phi

vector (of length 4) containing values of the cure fraction incidence parameters \phi_{11}, \phi_{10}, \phi_{01} and \phi_{00}. The parameters are restricted to \phi_{11} + \phi_{10} + \phi_{01} + \phi_{00}= 1.

lower.tail

logical; If TRUE (default), probabilities are P(X \le x, Y \le 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 \le x, Y \le y) = 1 - (\phi_{11} + \phi_{10}) (\theta_1 \theta_3)^x - (\phi_{01} + \phi_{00}) - (\phi_{11} + \phi_{01}) (\theta_2 \theta_3)^y - (\phi_{10} + \phi_{00})

+ \phi_{11} (\theta_{1}^{x} \theta_{2}^{y}\theta_{3}^{\max(x,y)}) + \phi_{10} (\theta_1 \theta_{3})^x + \phi_{01} (\theta_2 \theta_{3})^y + \phi_{00}

and the joint survival function is given by:

P(X > x, Y > y) = \phi_{11} (\theta_{1}^{x} \theta_{2}^{y}\theta_{3}^{\max(x,y)}) + \phi_{10} (\theta_1 \theta_{3})^x + \phi_{01} (\theta_2 \theta_{3})^y + \phi_{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 [e-mail]

Jorge Alberto Achcar [e-mail]

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–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

# 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)
# [1] 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)
# [1] 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)
# [1] 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)
# [1] 0.2494637 0.2064101