Abstract
In this paper we study approximations for the boundary crossing probabilities of moving sums of i.i.d. normal random variables. We approximate a discrete time problem with a continuous time problem allowing us to apply established theory for stationary Gaussian processes. By then subsequently correcting approximations for discrete time, we show that the developed approximations are very accurate even for a small window length. Also, they have high accuracy when the original r.v. are not exactly normal and when the weights in the moving window are not all equal. We then provide accurate and simple approximations for ARL, the average run length until crossing the boundary.
Highlights
Statement of the ProblemLet ε1, ε2, . . . be a sequence of i.i.d. normal random variables (r.v.) with mean θ and variance σ 2 > 0
Developing accurate approximations for the boundary crossing probability (BCP) PS(M, H, L) for generic parameters H, M and L is very important in various areas of statistics, predominantly in applications related to change-point detection; see, for example, papers (Bauer and Hackl 1980; Chu et al 1995; Glaz et al 2012; Moskvina and Zhigljavsky 2003; Xia et al 2009) and especially books (Glaz et al 2001; Glaz et al 2009)
The main option in the ‘GenzBretz’ package requires the use of Monte-Carlo simulations so that for reliable estimation of highdimensional integrals one needs to make a lot of averaging; see Sections 6.1 and 7 for more discussion on these issues
Summary
Let ε1, ε2, . . . be a sequence of i.i.d. normal random variables (r.v.) with mean θ and variance σ 2 > 0. See for example Glaz and Johnson (1988), Glaz et al (2012), Wang and Glaz (2014), and Wang et al (2014) (the methodology was extended to the case when εj are integer-valued r.v., see Glaz and Naus (1991)) We will call these approximations ‘Glaz approximations’ by the name of the main author of these papers; they will be formally written down in Sections 2.2 and 7.
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