A diverse variety of processes -including recurrent disease episodes, neuron firing, and communication patterns among humans -can be described using inter-event time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this paper, we illustrate how this length bias is born and how it can be corrected. To do this, we model event sequences using stationary renewal processes, and we formulate simple heuristics for determining the severity of the bias. To illustrate our results, we focus on the example of empirical communication networks, which are temporal networks that are constructed from communication events. The IET distributions of such systems guide efforts to build models of human behavior, and the variance of IETs is very important for estimating the spreading rate of information in networks of temporal interactions. It is thus crucial to have good estimates for the statistics of IET tails. We analyze several well-known data sets from the literature, and we find that the vast majority of event sequences of these data sets are susceptible to finite window-size effects. We demonstrate that the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions -and this is especially true for IETs with heavy-tails, which are often of interest. The newfound wealth of large data sets in the modern era of "Big Data" necessitates statistical analyses of such data. This has been especially prevalent in the study of human behavior, and the digital footprints left behind by electronic activities have provided a deluge of data. One of the most important problems in the study of human dynamics, which benefits directly from such data, is to quantify temporal activity patterns in human behavior. For example, this problem has been approached via the characterization of time sequences of human activities [1] [2] [3] [4] [5] [6] [7] and the analysis of "temporal networks" [8] (i.e., networks that change in time). Inter-event times (IETs) give the times between each pair of events (e.g., sending an e-mail, making a phone call, or doing any other activity), and the way they are distributed has received intense scrutiny because they can be used to characterize temporal processes. IET distributions have a long historyfor example, they are important in fields such as medicine and engineering [9-11] -and they have become particularly important for the study of large temporal data sets. Recently, they have been used to guide the modeling of human-activity dynamics [2-4, 6, 12, 13], and they are also important for temporal networks, where they can have an large effect on processes that act on top of such networks [8, 14] . For example, spreading processes are sensitive to the bursty nature of the time sequences of contacts, which tend to be characterized by heavy-tailed inter-contact time distributions [15] [16] [17] [18] [19] . Electronic records often have a huge number of data points. Such data often includes many subjects, but it may or may not also include a similar wealth of longitudinal points. For example, there exist data sets with thousands or even millions of people but with observation periods that only last a few months [1, 5, 17] . Moreover, even when the observation period is long, a given individual might rarely be active during that time. This is the case, for example, in recent studies of e-mail communication [1-3, 6, 20], mobile phone calling [17, 18] , website usage [4] , and donations to charities [21] . Data sets that describe human activity in which the observation windows are comparable in scale to the IETs are vulnerable to finite-size biases, because the time interval between events is more likely to be truncated artificially by the boundaries of a time window when it is longer, then. This can arise due to short observation windows and/or sparse records of activity. This effect will bias the tails of the observed IET distributions, which is a very serious issue, as the properties of distribution tails are often among the most important empirical features that one needs to consider [8] and models of human dynamics have been validated or refuted based on their predictions of the shape of the IET distributions [2, 3, 6, 7, 12, 13] . Furthermore, the variance of the IET distributions can have a large effect on the speed of spreading processes [15] [16] [17] [18] , and it has been used to classify the processes that produce them [20] . Several approaches have been used to deal with the bias introduced by a finite temporal-window size. In particular, it is common to disregard all of the boundary effects and use the observed IETs [4, 5, 20] . Such biases are sometimes acknowledged: for example, the exponential tail of the IET distribution is sometimes construed as arXiv:1412.8388v1 [physics.soc-ph]
This emphasis impacts the observation bias and the accuracy and significance of analyses, as dynamic network processes (e.g., the formation and dissolution of ties) can begin or end during inter-event periods REF .
Estimating inter-event time distributions from finite observation periods in communication networks
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