Skip to main content
Springer Nature Link
Log in

Monitoring process variability using auxiliary information

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

In this study a Shewhart type control chart namely V r chart is proposed for improved monitoring of process variability (targeting large shifts) of a quality characteristic of interest Y. The proposed control chart is based on regression type estimator of variance using a single auxiliary variable X. It is assumed that (Y, X) follow a bivariate normal distribution. The design structure of V r chart is developed and its comparison is made with the well-known Shewhart control chart namely S 2 chart used for the same purpose. Using power curves as a performance measure it is observed that V r chart outperforms the S 2 chart for detecting moderate to large shifts, which is main target of Shewhart type control charts, in process variability under certain conditions on ρ yx . These efficiency conditions on ρ yx are also obtained for V r chart in this study.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Acosta-Mejia CA Jr, Pignatiello JJ, Rao BV (1999) A comparison of control charting procedures for monitoring process dispersion. IIE Trans 31:569–579

    Google Scholar 

  • Adhvaryu A (1975) Combination of some estimators using supplementary information. J Indian Soc Agric Stat 27:37–50

    Google Scholar 

  • Ahmed MS (1998) A note on regression type estimators using multiple auxiliary information. Aust New Zealand J Stat 40:373–376

    Article  MATH  Google Scholar 

  • Alwan LC (2000) Statistical process analysis. McGraw-Hill, New York

    Google Scholar 

  • Battaglia GJ (1993) Regression-based statistical process control. AMP J Technol 3:38–45

    Google Scholar 

  • Chand L (1975) Some ratio type estimators based on two or more auxiliary variables. Iowa State University, Ames, Iowa (USA), Unpublished Ph.D. Thesis

    Google Scholar 

  • Chang TC, Gan FF (2004) Shewhart charts for monitoring the variance components. J Qual Technol 36:293–308

    Google Scholar 

  • Chiu WK (1974) The economic design of cusum charts for controlling normal means. Appl Stat 23:420–433

    Article  Google Scholar 

  • Chiu WK, Wetherill GB (1974) A simplified scheme for the economic design of \(\bar{X}\) chart. J Qual Technol 6:63–69

    Google Scholar 

  • Ding Y, Zhou S, Chen Y (2005) A comparison of process variation estimators for in-process dimensional measurements and control. J Dyn Syst Meas Control 127:69–79

    Article  Google Scholar 

  • Duncan AJ (1951) Operating characteristics of R charts. Ind Qual Control 7:40–41

    Google Scholar 

  • Farnum NR (1994) Statistical quality control and improvement. Duxbury, Belmont

    Google Scholar 

  • Ferrell EB (1953) Control charts using midranges and medians. Ind Qual Control 9:30–34

    Google Scholar 

  • Fuller WA (1980) Comment on ratio estimator and estimators of its variance. J Am Stat Assoc 76:78–80

    Article  Google Scholar 

  • Garcia MR, Cebrian AA (1996) Repeated substitution method: the ratio estimator for the population variance. Metrika 43:101–105

    Article  MATH  MathSciNet  Google Scholar 

  • Goel AL, Wu SM (1971) Determination of ARL and a contour nomogram for cusum charts to control normal means. Technometrics 13:221–230

    Article  MATH  Google Scholar 

  • Gonzalez IM, Viles E (2000) Semi-economic design of \(\bar{X}\)-control charts assuming gamma distribution. Econ Qual Control 15:109–118

    MATH  MathSciNet  Google Scholar 

  • Gonzalez IM, Viles E (2001) Design of R control chart assuming a gamma distribution. Econ Qual Control 16(2):199–204

    Article  MATH  MathSciNet  Google Scholar 

  • Hillier FS (1969) \(\bar{X}\) and R chart control limits based on a small number of subgroups. J Qual Technol 1:17–26

    Google Scholar 

  • Isaki CT (1983) Variance estimation using auxiliary information. J Am Stat Assoc 78:117–123

    Article  MATH  MathSciNet  Google Scholar 

  • Kiregyera B (1984) Regression type estimators using two auxiliary variables and the model of double sampling form finite populations. Metrika 31:215–226

    Article  MATH  MathSciNet  Google Scholar 

  • Khoo MB, Quah SH (2004) Alternatives to the multivariate control chart for process dispersion. Qual Eng 16:423–435

    Article  Google Scholar 

  • Kuk AYC, Mak TK (1989) Median estimation in the presence of auxiliary information. J R Stat Soc B 51:261–269

    MATH  MathSciNet  Google Scholar 

  • Langenverg P, Iglewicz B (1986) Trimed mean \(\bar{X}\) and R charts. J Qual Technol 18:152–161

    Google Scholar 

  • Magnus JR (2002) Estimation of the mean of a univariate normal distribution with known variance. Econ J 5:225–236

    MATH  MathSciNet  Google Scholar 

  • Montgomery DC (1996) Introduction to statistical quality control, 3rd edn. Wiley, New York

    Google Scholar 

  • Mukerjee R, Rao TJ, Vijayan K (1987) Regression type estimator using multiple auxiliary information. Aust J Stat 29:244–254

    Article  MathSciNet  Google Scholar 

  • Naik VD, Gupta PC (1991) A general class of estimators for estimating population mean using auxiliary information. Metrika 38:11–17

    Article  MATH  MathSciNet  Google Scholar 

  • Nelson LS (1984) The Shewhart control chart—tests for special causes. J Qual Technol 21:287–289

    Google Scholar 

  • Nelson PR (1985) Power curves for the analysis of means. Technometrics 27:65–73

    Article  Google Scholar 

  • Olkin I (1958) Multivariate ratio estimation for finite population. Biometrika 45:154–165

    MATH  MathSciNet  Google Scholar 

  • Pappanastos EA, Adams BM (1996) Alternative design of Hodges–Lehmann control chart. J Qual Technol 28(2):213–223

    Google Scholar 

  • Pearson ES (1932) The percentage limits for the distribution of range in samples from a normal population. Biometrika 18:404–417

    Google Scholar 

  • Prasad B (1989) Some improved ratio type estimators of population mean and ratio in finite population sample surveys. Commun Stat Theory Methods 18:379–392

    Article  MATH  Google Scholar 

  • Quesenberry CP (1997) SPC methods for quality improvement. Wiley, New York

    Google Scholar 

  • Raj D (1965) On a method of using multi-auxiliary information in sample surveys. J Am Stat Assoc 60:271–277

    Google Scholar 

  • Ramalhoto MF, Morais M (1999) Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals. J Appl Stat 26(1):129–160

    Article  MATH  Google Scholar 

  • Rao PSRA, Mudholkar GS (1967) Generalized multivariate estimator for the mean of finite population. J Am Stat Assoc 62:1009–1012

    Article  MathSciNet  Google Scholar 

  • Reddy VN (1993) On ratio and product methods of estimation. Sankhya B 35:307–316

    Google Scholar 

  • Rocke DM (1989) Robust control charts. Technometrics 31:173–184

    Article  MathSciNet  Google Scholar 

  • Ross SM (1990) A course in simulations. Macmillan, New York

    Google Scholar 

  • Royall RM, Cumberland WG (1981) An empirical study of the ratio estimator and estimators of its variance. J Am Stat Assoc 76:66–77

    Article  MATH  MathSciNet  Google Scholar 

  • Sahoo J, Sahoo LN (1999) A comparative study of some regression type estimators in double sampling procedures. Aligarh J Stat 19:67–76

    Google Scholar 

  • Scheffe H (1949) Operating characteristics of average and range charts. Ind Qual Control 5:13–18

    MathSciNet  Google Scholar 

  • Schilling EG, Nelson PR (1976) The effect of non-normality on the control limits of \(\bar{X}\) Charts. J Qual Technol 8(4):183–188

    Google Scholar 

  • Shewhart W (1931) Economic control of quality manufactured product. D. Van Nostrand, New York (reprinted by the American Society for Quality Control in 1980, Milwauker, WI)

  • Shewhart W (1939) Statistical method from the viewpoint of Quality Control Graduate School Department of Agriculture, Washington, D.C. (reprinted by Dover publication in 1986, Mineola, NY)

  • Shu L, Tsung F, Tsui KL (2004) Run-length performance of regression control charts with estimated parameters. J Qual Technol 36:280–292

    Google Scholar 

  • Singh GN (2001) On the use of transformed auxiliary variable in the estimation of population mean in two phase sampling. Stat Transistion 5:405–416

    Google Scholar 

  • Singh HP, Gangele RK (1989–1995) An improved regression estimator with known coeficent of variation in two-phase sampling with two auxiliary variables. Vikram 25(1–3):27–40

    Google Scholar 

  • Singh HP, Upadhyaya LN, Chandra P (2004) A general family of estimators for estimating population mean using two auxiliary variables in two phase sampling. Stat Transition 6:1055–1077

    Google Scholar 

  • Srivastava SK (1965) An estimate of the mean of a finite population using several auxiliary variables. J Indian Stat Assoc 3:2–3

    Google Scholar 

  • Srivastava SK (1966) On ratio and linera regression methods of estimation with several auxiliary variables. J Indian Stat Assoc 4:66–72

    Google Scholar 

  • Tatum LG (1997) Robust estimation of the process standard deviation for control charts. Technometrics 39:127–141

    Article  MATH  MathSciNet  Google Scholar 

  • Tikkiwal BD (1960) On theory of classical regression and double sampling estimation. J R Stat Soc B 22:131–138

    MATH  MathSciNet  Google Scholar 

  • Tuprah K, Ncube M (1987) A comparison of dispersion quality control charts. Sequential Anal 6:155–163

    Article  Google Scholar 

  • Upadhyaya LN, Singh HP (2003) A note on the estimation of mean using auxiliary information. Stat Transition 6:571–575

    Google Scholar 

  • Wheeler DJ (1995) Advanced topics in statistical process control-the power of Shewhart’s charts. SPC Press, Knoxville

    Google Scholar 

  • White EM, Schroeder R (1987) A simultaneous control chart. J Qual Technol 19:1–10

    Google Scholar 

  • Woodall WH, Spitzner DJ, Montgomery DC, Gupta S (2004) Using control charts to monitor process and product quality profiles. J Qual Technol 36:309–320

    Google Scholar 

  • Yang CH, Hiller FS (1970) Mean and variance control chart limits based on a small number of subgroup. J Qual Technol 2:9–16

    Google Scholar 

  • Yeh A, Lin D, Zho H, Venkataramani C (2003) A multivariate exponentially weighted moving average control chart for monitoring process variability. J Appl Stat 30:507–536

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan

    Muhammad Riaz

Authors
  1. Muhammad Riaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Riaz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Riaz, M. Monitoring process variability using auxiliary information. Computational Statistics 23, 253–276 (2008). https://doi.org/10.1007/s00180-007-0084-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-007-0084-6

Keywords

Search

Navigation