Many systems experience recurrent events. Recurrence data are collected to analyze quantities of interest, such as the mean cumulative number of events. Methods of analysis are available for recurrence data with left and/or right censoring. Due to practical constraints, however, recurrence data are sometimes recorded only in windows, with gaps in between. Nelson (2003, page 75) gives one example, and Chapter 2 describes two other applications that window-observation recurrence data arise. With the need for analytical methods with window-observation recurrence data, our research achieves the following three objectives:

(1). Extend the existing statistical methods, both nonparametric and parametric, to analyze window-observation recurrence data, and our focus is to estimate the mean cumulative function (MCF).

(2). Study and compare CI procedures for the MCF estimators with window-observation recurrence data, and make recommendations when the amount of observed information in the data is small.

(3). Establish the asymptotic (i.e., large-sample) properties for the MCF estimators with window-observation recurrence data.

Our research shows that the existing statistical methods, both nonparametric and parametric, can be extended to estimate the MCF with window-observation recurrence data. Chapter 2 provides the details on four MCF estimators, including the NP estimator, the nonhomogeneous Poisson process (NHPP) estimator, the local hybrid estimator, and the NHPP hybrid estimator. The NP estimator and the NHPP estimator for analysis with window-observation recurrence data are straight-forward extensions. The NP estimator requires minimum assumptions, but will be inconsistent if the size of the risk set is not positive over the entire period of interest. There is no such difficulty when using a parametric model for the recurrence data, yet the assumption on the recurrence rate form needs careful diagnoses and checking. When risk-set-size-zero (RSSZ) intervals exist, the two hybrid estimators are alternatives to the NP estimator, which generates downwardly biased estimates. Chapter 2 also presents the summary results from a simulation study that can be used as references to select the MCF estimators to use.

The four MCF estimators described in Chapter 2 are relatively easy to calculate. Besides point estimates, however, confidence intervals are useful in many applications. When the amount of observed information is large, for example, the number of units is large, the number of observed recurrences is large, and there is no or very small amount of time with RSSZ, various CI procedures generate similar results for each of the MCF estimators. However, when the number of units is small, and/or the number of observed recurrences is small, and/or there is relatively large amount of time with RSSZ or risk-set-size-one (RSSONE), the choice of which CI procedure to use makes a difference. Our research carries out an extensive simulation study on five CI procedures for each of the four MCF estimators described in Chapter 2, and the details of the simulation studies and the summary results are in Chapter 3. Chapter 3 also makes suggestions on the CI procedures to use based on the number of units, the number of recurrences observed, and the amount of time with RSSZ or RSSONE.

Chapter 4 establishes the asymptotic properties for the NP and NHPP MCF estimators, and outlines the assumptions and conditions that are needed for the MCF estimators to be consistent and asymptotically normal.