dbivgeo function

Joint Probability Mass Function for the Basu-Dhar Bivariate Geometric Distribution

Joint Probability Mass Function for the Basu-Dhar Bivariate Geometric Distribution

This function computes the joint probability mass function of the Basu-Dhar bivariate geometric distribution for arbitrary parameter values.

Source

dbivgeo1 and dbivgeo2 are calculated directly from the definitions.

dbivgeo1(x, y = NULL, theta, log = FALSE)
dbivgeo2(x, y = NULL, theta, 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.
  • log: logical argument for calculating the log probability or the probability function. The default value is FALSE.

Returns

dbivgeo1 gives the values of the probability mass function using the first form of the joint pmf.

dbivgeo2 gives the values of the probability mass function using the second form of the joint pmf.

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 could be written in two forms. The first form is described by:

P(X=x,Y=y)=θ1x−1θ2y−1θ3z1−θ1xθ2y−1θ3z2−θ1x−1θ2yθ2z3+θ1xθ2yθ3z4 P(X = x, Y = y) = \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). The second form is given by the conditions:

If X < Y, then

P(X=x,Y=y)=θ1x−1(θ2θ3)y−1(1−θ2θ3)(1−θ1) P(X = x, Y = y) = \theta_1^{x - 1} (\theta_2 \theta_{3})^{y - 1}(1 - \theta_{2} \theta_{3}) (1 - \theta_1)

If X = Y, then

P(X=x,Y=y)=(θ1θ2θ3)x−1(1−θ1θ3−θ2θ3+θ1θ2θ3) P(X = x, Y = y) = (\theta_1 \theta_2 \theta_{3})^{x - 1}(1 - \theta_1 \theta_{3} - \theta_2 \theta_{3} + \theta_1 \theta_2 \theta_{3})

If X > Y, then

P(X=x,Y=y)=θ2y−1(θ1θ3)x−1(1−θ1θ3)(1−θ2) P(X = x, Y = y) = \theta_2^{y - 1} (\theta_1 \theta_{3})^{x - 1}(1 - \theta_{1} \theta_{3}) (1 - \theta_2)

Examples

# If x and y are integer numbers:

dbivgeo1(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), log = FALSE)
# [1] 0.16128
dbivgeo2(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), log = FALSE)
# [1] 0.16128

# If x is a matrix:

matr    <-       matrix(c(1,2,3,5), ncol = 2)

dbivgeo1(x = matr, y = NULL, theta = c(0.2,0.4,0.7), log = FALSE)
# [1] 0.0451584000 0.0007080837
dbivgeo2(x = matr, y = NULL, theta = c(0.2,0.4,0.7), log = FALSE)
# [1] 0.0451584000 0.0007080837

# If log = TRUE:

dbivgeo1(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), log = TRUE)
# [1] -1.824613
dbivgeo2(x = 1, y = 2, theta = c(0.2, 0.4, 0.7), log = TRUE)
# [1] -1.824613

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