| Type: | Package |
| Title: | Basu-Dhar Bivariate Geometric Distribution |
| Version: | 2.1.1 |
| Maintainer: | Ricardo Puziol de Oliveira <rpuziol.oliveira@gmail.com> |
| Description: | Provides functions to compute the joint probability mass function (pmf), cumulative distribution function (cdf), and survival function (sf) of the Basu-Dhar bivariate geometric distribution. Additional functionalities include the calculation of the correlation coefficient, covariance, and cross-factorial moments, as well as the generation of random variates. The package also implements parameter estimation based on the method of moments. |
| Depends: | R (≥ 3.0.2) |
| Imports: | stats |
| URL: | https://doi.org/10.1285/i20705948v11n1p108 |
| RoxygenNote: | 7.3.2 |
| Encoding: | UTF-8 |
| NeedsCompilation: | no |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| Packaged: | 2025-05-13 15:55:03 UTC; rpuzi |
| Author: | Ricardo Puziol de Oliveira [aut, cre], Jorge Alberto Achcar [aut] |
| Repository: | CRAN |
| Date/Publication: | 2025-05-16 17:10:02 UTC |
Cross-factorial Moment for the Basu-Dhar Bivariate Geometric Distribution
Description
This function computes the cross-factorial moment for the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values.
Usage
cfbivgeo(theta)
Arguments
theta |
vector (of length 3) containing values of the parameters |
Details
The cross-factorial moment between X and Y, assuming the Basu-Dhar bivariate geometric distribution, is given by,
E[XY] = \frac{1 - \theta_1 \theta_2 \theta_{3}^2}{(1 - \theta_1\theta_3)(1 - \theta_2\theta_3)(1 - \theta_1 \theta_2 \theta_{3})}
Note that the cross-factorial moment is always positive.
Value
cfbivgeo computes the cross-factorial moment for the Basu-Dhar bivariate geometric distribution for arbitrary parameter values.
Invalid arguments will return an error message.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
cfbivgeo 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.
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.
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.
Examples
cfbivgeo(theta = c(0.5, 0.5, 0.7))
# [1] 2.517483
cfbivgeo(theta = c(0.2, 0.5, 0.7))
# [1] 1.829303
cfbivgeo(theta = c(0.8, 0.9, 0.1))
# [1] 1.277864
cfbivgeo(theta = c(0.9, 0.9, 0.9))
# [1] 35.15246
Correlation Coefficient for the Basu-Dhar Bivariate Geometric Distribution
Description
This function computes the correlation coefficient analogous of the Pearson correlation coefficient for the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values.
Usage
corbivgeo(theta)
Arguments
theta |
vector (of length 3) containing values of the parameters |
Details
The correlation coefficient between X and Y, assuming the Basu-Dhar bivariate geometric distribution, is given by,
\rho = \frac{(1 - \theta_{3})(\theta_1 \theta_2)^{1/2}}{1 - \theta_1 \theta_2 \theta_{3}}
Note that the correlation coefficient is always positive which implies that the Basu-Dhar bivariate geometric distribution is useful for bivariate lifetimes with positive correlation.
Value
corbivgeo computes the correlation coefficient analogous to the Pearson correlation coefficient for the Basu-Dhar bivariate geometric distribution for arbitrary parameter values.
Invalid arguments will return an error message.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
corbivgeo 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.
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.
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.
Examples
corbivgeo(theta = c(0.5, 0.5, 0.7))
# [1] 0.1818182
corbivgeo(theta = c(0.2, 0.5, 0.7))
# [1] 0.102009
corbivgeo(theta = c(0.8, 0.9, 0.1))
# [1] 0.822926
corbivgeo(theta = c(0.9, 0.9, 0.9))
# [1] 0.3321033
Covariance for the Basu-Dhar Bivariate Geometric Distribution
Description
This function computes the covariance for the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values.
Usage
covbivgeo(theta)
Arguments
theta |
vector (of length 3) containing values of the parameters |
Details
The covariance between X and Y, assuming the Basu-Dhar bivariate geometric distribution, is given by,
Cov(X,Y) = \frac{\theta_1 \theta_2 \theta_{3}(1 - \theta_3)}{(1 - \theta_1\theta_3)(1 - \theta_2\theta_3)(1 - \theta_1 \theta_2 \theta_{3})}
Note that the covariance is always positive.
Value
covbivgeo computes the covariance for the Basu-Dhar bivariate geometric distribution for arbitrary parameter values.
Invalid arguments will return an error message.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
covbivgeo 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.
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.
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.
Examples
covbivgeo(theta = c(0.5, 0.5, 0.7))
# [1] 0.1506186
covbivgeo(theta = c(0.2, 0.5, 0.7))
# [1] 0.04039471
covbivgeo(theta = c(0.8, 0.9, 0.1))
# [1] 0.0834061
covbivgeo(theta = c(0.9, 0.9, 0.9))
# [1] 7.451626
Joint Probability Mass Function for the Basu-Dhar Bivariate Geometric Distribution
Description
This function computes the joint probability mass function of the Basu-Dhar bivariate geometric distribution for arbitrary parameter values.
Usage
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 |
log |
logical argument for calculating the log probability or the probability function. The default value is FALSE. |
Details
The joint probability mass function for a random vector (X, Y) following a Basu-Dhar bivariate geometric distribution could be written in two forms. The first form is described by:
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 > 0 are positive integers and 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) = \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) = (\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) = \theta_2^{y - 1} (\theta_1 \theta_{3})^{x - 1}(1 - \theta_{1} \theta_{3}) (1 - \theta_2)
Value
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.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
dbivgeo1 and dbivgeo2 are calculated directly from the definitions.
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.
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
Moments Estimator for the Basu-Dhar Bivariate Geometric Distribution
Description
This function computes the estimators based on the method of the moments for each parameter of the Basu-Dhar bivariate geometric distribution.
Usage
mombivgeo(x, y)
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. |
Details
The moments estimators of \theta_1, \theta_2, \theta_3 of the Basu-Dhar bivariate geometric distribution are given by:
\hat \theta_1 = \frac{\bar{Y}(1 - \bar{W})}{\bar{W}(1 - \bar{Y})}
\hat \theta_2 = \frac{\bar{X}(\bar{W} - 1)}{\bar{W}(\bar{X} - 1)}
\hat \theta_3 = \frac{\bar{X}(\bar{X} - 1)(\bar{Y} - 1)}{(\bar{W} - 1)\bar{X} \bar{Y}}
Value
mombivgeo gives the values of the moments estimator.
Invalid arguments will return an error message.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
mombivgeo 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.
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.
Examples
# Generate the data set:
set.seed(123)
x1 <- rbivgeo1(n = 1000, theta = c(0.5, 0.5, 0.7))
set.seed(123)
x2 <- rbivgeo2(n = 1000, theta = c(0.5, 0.5, 0.7))
# Compute de moment estimator by:
mombivgeo(x = x1, y = NULL) # For data set x1
# [,1]
# theta1 0.5053127
# theta2 0.5151873
# theta3 0.6640406
mombivgeo(x = x2, y = NULL) # For data set x2
# [,1]
# theta1 0.4922327
# theta2 0.5001577
# theta3 0.6993893
Joint Cumulative Function for the Basu-Dhar Bivariate Geometric Distribution
Description
This function computes the joint cumulative function of the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values.
Usage
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 |
lower.tail |
logical; If TRUE (default), probabilities are |
Details
The joint cumulative function for a random vector (X, Y) following a Basu-Dhar bivariate geometric distribution could be written as:
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) = \theta_{1}^{x}\theta_{2}^{y} \theta_{3}^{\max(x,y)}
Value
pbivgeo gives the values of the cumulative function.
Invalid arguments will return an error message.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
pbivgeo 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.
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.
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
Generates Random Deviates from the Basu-Dhar Bivariate Geometric Distribution
Description
This function generates random values from the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values.
Usage
rbivgeo1(n, theta)
rbivgeo2(n, theta)
Arguments
n |
number of observations. If length(n) |
theta |
vector (of length 3) containing values of the parameters |
Details
The conditional distribution of X given Y is given by:
If X < Y, then
P(X = x | Y = y) = \theta_1^{x - 1}(1 - \theta_1)
If X = Y, then
P(X = x | Y = y) = \frac{\theta_1^{x - 1}(1 - \theta_1 \theta_{3} - \theta_2 \theta_{3} + \theta_1 \theta_2 \theta_{3})}{1 - \theta_2 \theta_{3}}
If X > Y, then
P(X = x | Y = y) = \frac{\theta_1^{x - 1} \theta_{3}^{x - y}(1 - \theta_{1} \theta_{3}) (1 - \theta_2)}{1 - \theta_2 \theta_{3}}
Value
rbivgeo1 and rbivgeo2 generate random deviates from the Bash-Dhar bivariate geometric distribution. The length of the result is determined by n, and is the maximum of the lengths of the numerical arguments for the other functions.
Invalid arguments will return an error message.
Author(s)
Ricardo P. Oliveira rpuziol.oliveira@gmail.com
Jorge Alberto Achcar achcar@fmrp.usp.br
Source
rbivgeo1 generates random deviates using the inverse transformation method. Returns a matrix that the first column corresponds to X generated random values and the second column corresponds to Y generated random values.
rbivgeo2 generates random deviates using the shock model. Returns a matrix that the first column corresponds to X generated random values and the second column corresponds to Y generated random values. See Marshall and Olkin (1967) for more details.
References
Marshall, A. W., & Olkin, I. (1967). A multivariate exponential distribution. Journal of the American Statistical Association, 62, 317, 30-44.
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.
Examples
rbivgeo1(n = 10, theta = c(0.5, 0.5, 0.7))
# [,1] [,2]
# [1,] 2 1
# [2,] 3 1
# [3,] 1 1
# [4,] 1 1
# [5,] 2 2
# [6,] 1 3
# [7,] 2 2
# [8,] 1 1
# [9,] 1 1
# [10,] 2 2
rbivgeo2(n = 10, theta = c(0.5, 0.5, 0.7))
# [,1] [,2]
# [1,] 1 1
# [2,] 2 1
# [3,] 2 1
# [4,] 4 1
# [5,] 1 1
# [6,] 2 2
# [7,] 3 2
# [8,] 3 1
# [9,] 3 2
# [10,] 1 1