Often the dependence in multivariate survival data is modeled through an individual level effect called the frailty. Due to its mathematical simplicity the gamma distribution is often used as the frailty distribution. However, it is well known that the gamma distribution for frailty has many drawbacks. For example, it weakens the effect of covariates. To overcome such drawbacks more heavy tailed distributions are needed to model the appropriate frailty distribution, e.g. the positive stable distribution. In this paper we develop a class of log-skew-$t$ distributions for the frailty. This class includes the log-normal distribution along with many other heavy tailed distributions, e.g. log-Cauchy or log-$t$ as special cases. The distributions often have heavier tails than the gamma and even the positive stable distributions.
Conditional on the frailty, the survival times are assumed to be independent with proportional hazard structure. The modeling process is then completed by assuming an appropriate baseline hazard function. There are many prior processes for modelling the baseline hazard. An attractive choice here is a correlated prior process which offers a great deal of flexibility. We consider such a process which jumps according to a time-homogeneous Poisson process. We develop Bayesian methods to obtain posterior inference using a variable dimensional Markov chain Monte Carlo method. We illustrate and compare our methods using two practical examples.