On Thresholds of Moving Averages with Given On-Target Significant Levels

Abstract

Let X ₁, X ₂,… be a sequence of independent and identically distributed random variables, and let Y _n , n = K, K + 1, K + 2,… be the corresponding backward moving average of order K. At epoch n ≥ K, the process Y _n will be off target by the input X _n if it exceeds a threshold. By introducing a two-state Markov chain, we define a level of significance (1 − a)% to be the percentage of times that the moving average process stays on target. We establish a technique to evaluate, or estimate, a threshold, to guarantee that {Y _n } will stay (1 − a)% of times on target, for a given (1 − a)%. It is proved that if the distribution of the inputs is exponential or normal, then the threshold will be a linear function in the mean of the distribution of inputs μ _X . The slope and intercept of the line, in each case, are specified. It is also observed that for the gamma inputs, the threshold is merely linear in the reciprocal of the scale parameter. These linear relationships can be easily applied to estimate the desired thresholds by samples from the inputs.

Keywords:

• Moving average
• On-target significant level
• Threshold

Mathematics Subject Classification:

• Primary 60J20, 60G10, 62F10
• Secondary 62P12, 62P30

Acknowledgment

The authors are thankful to the referees for providing constructive comments.