00259 >3777e2c66e87259144d3f5270c7d7a

Statistical Process Monitoring with Integrated Moving Average Noise 261

The explicit eąuation for g_k ( |v) in terms of <j>(') is provided in the proof of Theorem 1 in the Appendix. Upper taił areas of the distribution of U are found by integrating the density of U given in Theorem 1. Notę that g_k (|v) in Theorem 3, and hence the entire distribution of U, depends on the

IMA parameter X.. Of course, it also depends on the sample size n. No large sample limiting form is known.

In Figurę 8 solid lines show the upper tails of the cumulative distribution functions (CDFs) of U for X = 0.5 and te{2,5,10,20,40}. Dashed lines show approximate CDFs derived in the next subsection. FORTRAN subroutines to calculate these distributions are available from the author. The complexity of the computation is 0(tm³) when m-point ąuadrature is used to approximate integrals. Only quadratures with evenly spaced points can be used because of the recursive naturę of F_k and F_k. To get good precision in Figurę 8, we used the trapazoidal rule with m = 161 ąuadrature points.

Approximate nuli distribution Worsley (1983) derived an excellent approximation for the upper taił of the nuli distribution of U for the case y=0. His approach uses an improved form of Bonferroni's ineąuality and it extends immediately to other values of y. The approximation is

P,{U>u}z £Pi{|Z,|> »}-£Pr{|Z,.,| > u and \Z_t\> u}

The terms in the first summation are trivial to compute because Z_k ~ N(0, l). The terms in the second summation come from rectangular

regions under the bivariate normal distribution of (Z_k__t,Z_k). (Notę that

{Z_k__u Z_kj is given in the proof of Theorem 1 in the Appendix.) Good

numerical functions to evaluate bivariate normal CDFs are widely available. Thus, both computational time and complexity for the approximation are minimal.

The dashed lines in Figurę 8 demonstrate that this approximation is excellent for large u. It is certainly adeąuate for gaging how much evidence there is that a shift occurred in a given set of data.