Abstract
The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as It,It+1,It+2 = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold. The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model. Note that if {1 0 1} is known to not occur, for example, starting at position t = 1, then this information impacts the probability that {1 0 1} occurs starting at position t = 2 or t = 3, because the trials to obtain {1 0 1} are overlapping and thus not independent, so the Poisson distribution assumptions are not met. Nevertheless, the results shown in four examples demonstrate that Poisson-based approximation (that is strictly correct only for independent trials) can be remarkably accurate, and the SC method provides a bound on the total variation distance between the true and approximate PDF.
Highlights
Introduction and BackgroundSuppose there is interest in the probability that a pattern such as {1 0 1} or {1 1 1} occurs in a sequence of N = 10 independent Bernoulli trails
The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as It, It+1, It+2 = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold
The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model
Summary
Suppose there is interest in the probability that a pattern such as {1 0 1} or {1 1 1} occurs in a sequence of N = 10 independent Bernoulli trails. The main interest in this paper is the case with a small Bernoulli success probability, p = P (Ii = 1), consisting, for example, of whether a residual from a fitted time series model exceeds a threshold. A pattern such as {1 0 1} or {1 1 1} could indicate a depar-. Henderson ture from the fitted model, perhaps indicating that a signal of interest is present. This paper considers scanning for {1 x 1} with x = 0 or 1, with p = P (Ii =1)
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