pbivgeo function

Joint Cumulative Function for the Basu-Dhar Bivariate Geometric Distribution

Joint Cumulative Function for the Basu-Dhar Bivariate Geometric Distribution

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

Source

pbivgeo is calculated directly from the definition.

pbivgeo(x, y, theta, 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 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.
  • lower.tail: logical; If TRUE (default), probabilities are P(X≤x,Y≤y)P(X \le x, Y \le y) otherwise P(X>x,Y>y)P(X > x, Y > y).

Returns

pbivgeo gives the values of the cumulative function.

Invalid arguments will return an error message.

Details

The joint cumulative function for a random vector (XX, YY) following a Basu-Dhar bivariate geometric distribution could be written as:

P(X≤x,Y≤y)=1−(θ1θ3)x−(θ2θ3)y+θ1xθ2yθ3max⁡(x,y) P(X \le x, Y \le y) = 1 - (\theta_{1}\theta_3)^{x} - (\theta_{2}\theta_3)^{y} + \theta_{1}^{x}\theta_{2}^{y} \theta_{3}^{\max(x,y)}

and the joint survival function is given by:

P(X>x,Y>y)=θ1xθ2yθ3max⁡(x,y) P(X > x, Y > y) = \theta_{1}^{x}\theta_{2}^{y} \theta_{3}^{\max(x,y)}

Examples

# If x and y are integer numbers:

pbivgeo(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), lower.tail = TRUE)
# [1] 0.79728

# If x is a matrix:

matr    <-       matrix(c(1,2,3,5), ncol = 2)
pbivgeo(x = matr, y = NULL, theta = c(0.2,0.4,0.7), lower.tail = TRUE)
# [1] 0.8424384 0.9787478

# If lower.tail = FALSE:

pbivgeo(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), lower.tail = FALSE)
# [1] 0.01568

matr    <-       matrix(c(1,2,3,5), ncol = 2)
pbivgeo(x = matr, y = NULL, theta = c(0.9,0.4,0.7), lower.tail = FALSE)
# [1] 0.01975680 0.00139404

References

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

Li, J., & Dhar, S. K. (2013). Modeling with bivariate geometric distributions. Communications in Statistics-Theory and Methods, 42, 2 , 252-266.

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.

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