Abstract
Exact expressions axe derived for the Average Run Length (ARL) of CUSUM schemes when observations follow an Erlang distribution. Additionally, exact expressions for the quasi-stationary distribution are found. Based on the results for the ARL as function of the headstart and the quasi-stationary density function, the so-called average delay can be determined. Both functions are obtained by solving integral equations. Eventually, by using the right eigenfunction of the CUSUM transition kernel — the left one yields the quasi-stationary density function — a geometric approximation of the run length distribution is constructed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brook, D. and Evans, D. A. (1972). An approach to the probability function of CUSUM-runlength, Biometrika, 59, 539–549.
Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1953). Sequential decision problems for processes with continuous time parameter. Testing hypotheses, Annals of Mathematical Statistics, 242, 254–264.
Gold, M. (1989). The geometric approximation to the CUSUM run length distribution, Biometrika, 76, 725–733.
Harris, T. (1963). The Theory of Branching Processes, Berlin: Springer-Verlag.
Khan, R. A. (1978). Wald’s approximations to the ARL in CUSUM-procedures, Journal of Statistical Planning and Inference, 2, 63–77.
Knoth, S. (1995). Quasistationäre CUSUM-Verfahren bei Erlangverteilung. Ph.D. thesis, TU Chemnitz-Zwickau, Germany.
Knoth, S.(1997a). Exact average run lengths of CUSUM schemes for Erlang distributions Technical Report, 73, Europe-University Viadrina, Frankfurt (Oder), Germany.
Knoth, S. (1997b). Quasi-stationarity of CUSUM schemes for Erlang distributions, Technical Report, 82, Europe-University Viadrina, Frankfurt (Oder), Germany.
Kohlruss, D. (1994). Exact formulas for the OC and the ASN functions of the SPRT for Erlang distributions, Sequential Analysis, 13, 53–62.
Lorden, G. (1971). Procedures for reacting to a change in distribution, Annals of Mathematical Statistics, 42, 1897–1908.
Lorden, G. and Eisenberger, I. (1973). Detection of failure rate increases, Technometrics, 15, 167–175.
Lucas, J. M. (1985). Counted data CUSUM’s, Technometrics, 27, 129–144.
Madsen, R. W. and Conn, P. S. (1973). Ergodic behaviour for nonnegative kernels, Annals of Probability, 1, 995–1013.
Mevorach, Y. and Pollak, M. (1991). A small sample size comparison of the CUSUM and Shiryaev-Roberts approaches to change point detection, The American Journal of Mathematics and Management Science, 11, 277–298.
Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions, The Annals of Statistics, 14, 1379–1387.
Page, E. S. (1954). Continuous inspection schemes, Biometrika, 41, 100–115.
Pollak, M. (1985). Optimal detection of a change in distribution, Technometrics, 13, 206–227.
Pollak, M. and Siegmund, D. (1985). A diffusion process and its applications to detection a change in the drift of Brownian motion, Biometrika, 72, 267–280.
Pollak, M. and Siegmund, D. (1986). Approximations to the ARL of CUSUM-tests, Technical Report, Department of Statistics, Stanford University.
Raghavachari, M. (1965). Operating characteristic function and expected sample size of a Sequential Probability Ratio Test for the simple exponential distribution, Bulletin of the Calcutta Statistics Association, 14, 65–73.
Regula, G. (1975). Optimal CUSUM procedure to detect a change in distribution for the gamma family, Ph.D. thesis, Case Western University, Cleveland, OH.
Riordan, G. (1982). Kombinatornye Toždestva, Moskva: Nauka.
Ritov, Y. (1990). Decision theoretic optimality of the CUSUM procedure, The Annals of Statistics, 18, 1464–1469.
Roberts, S. W. (1966). A comparison of some control chart procedures, Technometrics, 8, 411–430.
Tartakovskij, A. G. and Ivanova, I. A. (1992). Sravnenie nekotorych posle-dovatel’nych pravil obnaruženija razladki, Probi. Pcredači Inf., 28, 21–29.
Vardeman, S. and Ray, D. (1985). Average run lengths for CUSUM schemes when observations are exponentially distributed, Technometrics, 27, 145–150.
Waldmann, K.-H. (1986). Bounds for the distribution of the run length of one-sided and two-sided CUSUM quality control schemes, Technometrics, 28, 61–67.
Wang, A.-L. (1990). Exponential CUSUM scheme for detecting a decrease in mean, Sankhyā, 52, 105–114.
Woodall, W. H. (1983). The distribution of the run length of one-sided CUSUM procedures for continuous random variables, Technometrics, 24, 295–301.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Birkhäuser Boston
About this chapter
Cite this chapter
Knoth, S. (1998). CUSUM Schemes and Erlang Distributions. In: Kahle, W., von Collani, E., Franz, J., Jensen, U. (eds) Advances in Stochastic Models for Reliability, Quality and Safety. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2234-7_22
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2234-7_22
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7466-7
Online ISBN: 978-1-4612-2234-7
eBook Packages: Springer Book Archive