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The Bootstrap (Efron 1979) assumes that the data are IID.

Obviously, if we have time series data then we probably cannot make that assumption unless we have a special case that we a time series of IID noise. Under what other, non-obvious conditions is this assumption invalid?

How about the following scenario?

I have a table of events, which were triggered by different individuals at different points in time. At certain times, it is more likely that an event will be triggered. Some events are more common that others. However if individual A triggers an event, this has no influence on whether or not individual B will trigger an event. Some types of events are more popular among most individuals. It is attributes of the events that drive their popularity. Some events only occur at certain times.

Events are assigned "minute buckets", so that their actual time is not used. These minute buckets are of varying sizes.

I am trying to answer the following question: "what attributes of these events attract more users to trigger them?"

An individual may trigger multiple events. There could be two cases here:

  1. Multiple events could be so compelling that they will both be triggered
  2. An individual could have persistence to their actions This is probably due to two kinds of individuals.

The data described above would not be called "panel data", right?

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  • $\begingroup$ This is a confusing description. Can you please be more specific? $\endgroup$
    – StasK
    Commented Jun 3, 2014 at 0:38
  • $\begingroup$ @StasK, which part is confusing? Then I will focus on re-writing that part. $\endgroup$
    – power
    Commented Jun 3, 2014 at 0:41
  • $\begingroup$ Events and individuals. Are they going shopping together? Are they going to the movies? Are they checking into foursquare? Also, events usually do not form time series. An event is best characterized by a random time $t$ and possibly a label, may be even a multivariate label (I bought blue jeans at GAP size 34/34 on a Thursday night -- count how many variables there are in this statement). To me, time series is a sequence of random variables $y_0, y_1, \ldots$ where the subindex is time measured in some regular intervals. Random events will not afford the regularity. $\endgroup$
    – StasK
    Commented Jun 3, 2014 at 2:19
  • $\begingroup$ I am concerned about your title. Is your question really "when is it so" (a question you in effect answer by example; if you want a complete list your question is too broad) or is it more like "how does one deal with the fact that sometimes an assumption of IID is too strict"? $\endgroup$
    – Glen_b
    Commented Jun 3, 2014 at 4:00
  • $\begingroup$ @StasK I have added some more info. However, I am not comfortable giving the exact application because it is not academic. $\endgroup$
    – power
    Commented Jun 3, 2014 at 8:27

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"Not identically distributed" is generally is a conditional property, even though sometimes the conditioning variable is not observed, or at least not very obvious. For example, in a heteroskedastic fan-shaped plot with $E[y|x]=0, V[y|x]=x$, you can derive the marginal distribution of $y$ by integrating over $x$; and typically that would mean that if $y|x \sim N(0,x)$, then for most distributions of $x$, $y$ will be heavy tailed. If the goal of inference is the mean of $y$, then you can honestly bootstrap from that full marginal distribution; it will be the same as bootstrapping the pairs $(x,y)$, although arguably the latter will allow you to produce all the relevant analyses.

Violation of independence is far more serious. Time series is one example (and block bootstrap is an answer to that). Cluster sampling is another (and cluster bootstrap is an answer to that). If an individual triggers multiple events (whatever that means in your application), then these events may or may not be independent.

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