Infection is one of the most common complications after hematopoietic cell transplantation. post-transplant infection data will inevitably lead to incorrect inferential results because the time from transplant to the first infection has a different biological meaning than the gap times between consecutive recurrent infections. Some unbiased yet inefficient methods include univariate survival analysis methods based on data from the first infection or bivariate serial event data methods based on the first and second infections. In this paper we propose a non-parametric estimator of the joint distribution of time from transplant to the first infection and the gap times between consecutive infections. The proposed estimator takes into account the potentially different distributions of the two types of gap times and better uses the recurrent infection data. Asymptotic properties of the proposed estimators are established. = 1 … = 1 2 … the gap times between the following infections. Let = = 0 1 … denote the collection of all gap times since transplant for subject and the recurrent gap times after the first infection by be the censoring time from transplant which has a survival function > = sup{: denote the number of completely observed infectious episodes for subject infections are observed without censoring while the infection is censored at time (i.e. subjects are assumed to be independent and identically distributed (i.i.d.). As in existing recurrent gap time methods such as the ones considered by Wang and Chang (1999) and many others we assume there exists a subject-specific latent variable or vector (i.e. frailty) characterizing the within-subject association among the gap times of the same subject whose distribution is left unspecified. Then we make the following assumptions: Figure 1 Illustration of time from Rabbit Polyclonal to Tip60 (phospho-Ser90). transplant to first infection and gap times between recurrent infections Assumption 1: Given KU 0060648 are independent and moreover are correlated. Also under Assumption 1 the gap times of subject are exchangeable and hence the gap time pairs are also exchangeable. Note that both the distribution of and the dependency between and the gap times are left unspecified under Assumption 1. Also note that under Assumption 1 the correlation between the first gap time and a subsequent gap time is allowed to be different than that between two subsequent gap times is independent of KU 0060648 and is subject to independent censoring by are subject to dependent censoring by and the gap times between following consecutive infections represents the time from transplant to the second infection and is the pair of the first two gap times. As discussed in Huang and Louis (1998) and Huang and Wang (2005) the equality + and = (is subject to the independent censoring by with a survival function denoted KU 0060648 by and KU 0060648 can be thought of as the time from the transplant to the artificial second infection KU 0060648 time with the true second gap time being replaced by are identically (but not independently) distributed. For ease of discussion we let denote the number of completely observed gap time pairs when ≥ 2 otherwise and and and ≥ 2 the variables in are observed quantities. Let + ≤ | | > 0 unless the support for is large enough the conditional distribution | : 1 ? = 0.5 : 1 ? with the variance estimate ≤ is any number smaller than In this region the proposed estimator can obviously be identified. We assume that and with and and and involved in both converge weakly to the same limit as their counter parts in the Huang-Louis estimator. The mapping Φ is compactly differentiable at each point of &.