TPmsm (version 1.2.12)

corrTP: Correlation between two gap times

Description

Provides the correlation between the bivariate times for some copula distributions.

Usage

corrTP(dist, corr, dist.par)

Arguments

dist

The distribution. Possible bivariate distributions are “exponential” and “weibull”.

corr

Correlation parameter. Possible values for the bivariate exponential distribution are between -1 and 1 (0 for independency). Any value between 0 (not included) and 1 (1 for independency) is accepted for the bivariate Weibull distribution.

dist.par

Vector of parameters for the allowed distributions. Two (scale) parameters for the bivariate exponential distribution and four (2 location parameters and 2 scale parameters) for the bivariate Weibull distribution. See details below.

Author

Artur Araújo, Javier Roca-Pardiñas and Luís Meira-Machado

Details

The bivariate exponential distribution, also known as Farlie-Gumbel-Morgenstern distribution is given by $$F(x,y)=F_1(x)F_2(y)[1+\alpha(1-F_1(x))(1-F_2(y))]$$ for \(x\ge0\) and \(y\ge0\). Where the marginal distribution functions \(F_1\) and \(F_2\) are exponential with scale parameters \(\theta_1\) and \(\theta_2\) and correlation parameter \(\alpha\), \(-1 \le \alpha \le 1\).

The bivariate Weibull distribution with two-parameter marginal distributions. It's survival function is given by $$S(x,y)=P(X>x,Y>y)=e^{-[(\frac{x}{\theta_1})^\frac{\beta_1}{\delta}+(\frac{y}{\theta_2})^\frac{\beta_2}{\delta}]^\delta}$$

Where \(0 < \delta \le 1\) and each marginal distribution has shape parameter \(\beta_i\) and a scale parameter \(\theta_i\), \(i = 1, 2\).

References

Araújo A, Meira-Machado L, Roca-Pardiñas J (2014). TPmsm: Estimation of the Transition Probabilities in 3-State Models. Journal of Statistical Software, 62(4), 1-29. tools:::Rd_expr_doi("10.18637/jss.v062.i04")

Johnson N., Kotz S. (1972). Distributions in statistics: continuous multivariate distributions, John Wiley and Sons.

Lu J., Bhattacharya G. (1990). Some new constructions of bivariate weibull models. Annals of Institute of Statistical Mathematics, 42(3), 543-559. tools:::Rd_expr_doi("10.1007/BF00049307")

See Also

dgpTP.

Examples

# Example for the bivariate Weibull distribution
corrTP( dist = "weibull", corr = 0.5, dist.par = c(2, 7, 2, 7) );

# Example for the bivariate Exponential distribution
corrTP( dist = "exponential", corr = 1, dist.par = c(1, 1) );