# The GTDL distribution Density function, survival function, failure function and random generation for the GTDL distribution. ## Source [d-p-q-r]GTDL are calculated directly from the definitions. ```r dGTDL(t, param, log = FALSE) hGTDL(t, param) sGTDL(t, param) rGTDL(n, param) ``` ## Arguments - `t`: vector of integer positive quantile. - `param`: parameters (alpha and gamma are scalars, lambda non-negative). - `log`: logical; if TRUE, probabilities p are given as log(p). - `n`: number of observations. ## Returns `dGTDL` gives the density function, `hGTDL` gives the failure function, `sGTDL` gives the survival function and `rGTDL` generates random samples. Invalid arguments will return an error message. ## Details * Density function $$ f(t\mid \boldsymbol{\theta})=\lambda\left(\frac{\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)\times\left(\frac{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)^{-\lambda/\alpha} $$ * Survival function $$ S(t \mid \boldsymbol{\theta})=\left(\frac{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)^{-\lambda/\alpha} $$ * Failure function $$ h(t\mid\boldsymbol{\theta})=\lambda\left(\frac{\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right) $$ ## Examples ```r library(GTDL) t <- seq(0,20,by = 0.1) lambda <- 1.00 alpha <- -0.05 gamma <- -1.00 param <- c(lambda,alpha,gamma) y1 <- hGTDL(t,param) y2 <- sGTDL(t,param) y3 <- dGTDL(t,param,log = FALSE) tt <- as.matrix(cbind(t,t,t)) yy <- as.matrix(cbind(y1,y2,y3)) matplot(tt,yy,type="l",xlab="time",ylab="",lty = 1:3,col=1:3,lwd=2) y1 <- hGTDL(t,c(1,0.5,-1.0)) y2 <- hGTDL(t,c(1,0.25,-1.0)) y3 <- hGTDL(t,c(1,-0.25,1.0)) y4 <- hGTDL(t,c(1,-0.50,1.0)) y5 <- hGTDL(t,c(1,-0.06,-1.6)) tt <- as.matrix(cbind(t,t,t,t,t)) yy <- as.matrix(cbind(y1,y2,y3,y4,y5)) matplot(tt,yy,type="l",xlab="time",ylab="Hazard function",lty = 1:3,col=1:3,lwd=2) ``` ## References * Mackenzie, G. (1996). Regression Models for Survival Data: The Generalized Time-Dependent Logistic Family. Journal of the Royal Statistical Society. Series D (The Statistician). 45. 21-34.

fGTDL function