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I have some summary measures on a time series process for a large number of time intervals, all of the same length. The summary measures are the initial value (i), which I will take to be zero without loss of generality, the final value (f), the maximum value (x), and the minimum value (v). If i is not zero, assume henceforth that it is subtracted from all four variables. I believe the internal process in each interval is best modeled as a random walk (with an unknown, but large, number of steps).

My problem is to find, from the data given, the probability that the max occurs prior in time to the min.

At first glance the data would not seem to be sufficient for that determination, but I believe that it is. However, I have not yet been able to find an analytic solution. That is my question: do the four numbers supplied suffice to determine the probability that the max comes first, and if so, how.

Let me present a simplified model to motivate my intuition that this data is sufficient. Say I have a one dimensional random walk consisting of a series of n coin tosses, with heads = 1 and tails = -1. Now suppose I reach the final value and find that it is equal to n. In this case, I we know the min comes before the max, because to reach a value of n in n turns, every outcome must be heads. So the min must be equal to the initial value and the max to the final value. On the other hand, suppose my final value is zero. Because the constraint this imposes is entirely symmetric, it can not favor either either order of precedence over the other. I conclude that the probability that the max is first (second) is monotonically increasing (decreasing) in the final vale observed, and is equal to 1 at n, one half at zero, and -1 at minus n.

In my actual problem the number of turns is large but not fixed, and the innovations (first difference of the series) are i.i.d. from an unknown continuous distribution, probably non-normal. I think, but am by no means sure, that the essentially qualitative results above still hold. But I would like a quantitative relationship between P(max before min) and my observed i, f, x, and v. Id accept a solution for either my large-but-finite n or a continuous approximation.

This question may be related.

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