UNIT NADARAJAH AND HAGHIGHI DISTRIBUTION: PROPERTIES AND APPLICATIONS IN QUALITY CONTROL

Document Type : Article

Authors

1 Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan

2 Department of Statistics, St. Anthony’s College, Shillong, India

Abstract

This article introduces unit Nadarajah and Haghighi distribution to deal with the inflation of ones. Besides deriving statistical properties of the proposed distribution, several estimation methods are discussed. In particular, maximum likelihood estimation (MLE), least squares estimation (LSE), weighted least squared estimation (WLSE), maximum product spacing (MPS), minimum spacing absolute distance estimation (MSADE), minimum spacing absolute log-distance estimation (MSALDE), Cram'er-Von-Mises (CVM), Anderson-Darling method (AD) and right-tail Anderson-Darling method (RAD) are considered. Using real data sets, it is shown that the new distribution outperforms some well-known existing distribution. Furthermore, the application of the proposed distribution in quality control is also discussed. A control chart using unit Nadarajah and Haghighi distribution is constructed and its performance is evaluated using the average run length.

Keywords


  1. References:

    1. Mazucheli, J., Menezes, A. F. B., Fernandes, L. B., de Oliveira, R. P., and Ghitany, M. E., “The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates”, Journal of Applied Statistics, 47(6), pp. 954–974 (2020).
    2. Mazucheli, J., Menezes, A.F.B., and Dey, S., “Improved maximum-likelihood estimators for the parameters of the unit-gamma distribution”, Communications in Statistics-Theory and Methods, 47(15), pp. 3767–3778 (2018).
    3.  Menezes, A. F. B., Mazucheli, J., and Dey, S., “The unit-logistic distribution: Different
      methods of estimation”, Pesquisa Operacional, 38(3), pp. 555–578 (2018).
      4. Mazucheli, J., Menezes, A.F.B., and Chakraborty, S., “On the one parameter unit- Lindley
      distribution and its associated regression model for proportion data”, Journal of Applied
      Statistics, 46(4), pp. 700–714 (2019).
      5. Mazucheli, J., Menezes, A.F.B., and Dey, S., “Unit-Gompertz distribution with applications”,
      Statistica, 79(1), pp. 25–43 (2019).
      6. Sangsanit, Y. and Bodhisuwan, W., “The Topp-Leone generator of distributions: properties
      and inferences.”, Songklanakarin Journal of Science & Technology, 38(5) ((2016)).
      7. Condino, F. and Domma, F., “A new distribution function with bounded support: the
      reflected generalized Topp-Leone power series distribution”, Metron, 75(1), pp. 51–68 (2017).
      8. Nadarajah, S. and Haghighi, F., “An extension of the exponential distribution”, Statistics,
      45(6), pp. 543–558 (2011).
      9. Marshall, A.W. and Olkin, I., “Life Distributions: Structure of Nonparametric, Semiparametric,
      and Parametric Families”. Springer Series in Statistics, Springer New York (2007).
      10. Aban, I. B., Meerschaert, M. M., and Panorska, A. K., “Parameter estimation for the
      truncated pareto distribution”, Journal of the American Statistical Association, 101(473),
      pp. 270–277 (2006).
      11. Zhang, T. and Xie, M., “On the upper truncated Weibull distribution and its reliability
      implications”, Reliability Engineering & System Safety, 96(1), pp. 194 – 200 (2011).
      12. Papke, L.E. and Wooldridge, J.M., “Econometric methods for fractional response variables
      with an application to 401(k) plan participation rates”, Journal of Applied Econometrics,
      11(6), pp. 619–632 (1996).
      13. Fletcher, S.G. and Ponnambalam, K., “Estimation of reservoir yield and storage distribution
      using moments analysis”, Journal of Hydrology, 182(1), pp. 259 – 275 (1996).
      14. Seifi, A., Ponnambalam, K., and Vlach, J., “Maximization of manufacturing yield of systems
      with arbitrary distributions of component values”, Annals of Operations Research, 99,
      p. 373–383 (2000).
      15. Gangi, A., K., Ponnambalam., D., Khalili., and Karamouz, M., “Grain yield reliability analysis
      with crop water demand uncertainty”, Stochastic Environmental Research and Risk Assessment,
      20(4), pp. 259 – 277 (2006).
      16. Cook, D. O., Kieschnick, R., and McCullough, B. D., “Regression analysis of proportions in
      finance with self selection”, Journal of Empirical Finance, 15(5), pp. 860 – 867 (2008).
      17. Genc, A.I., “Estimation of p(x > y) with Topp–Leone distribution”, Journal of Statistical
      Computation and Simulation, 83(2), pp. 326–339 (2013).
      UNH 15
      18. Ali, S., Dey, S., Tahir, M.H., and Mansoor, M., “Two-parameter logistic-exponential distribution:
      Some new properties and estimation methods”, American Journal of Mathematical
      and Management Sciences, 39(3), pp. 270–298 (2020).
      19. Ali, S., Dey, S., Tahir, M. H., and Mansoor, M., “A comparison of different methods of estimation
      for the flexible Weibull distribution”, Communications Faculty of Sciences University
      of Ankara Series A1 Mathematics and Statistics, 69(1), pp. 794 – 814 (2020).
      20. Cheng, R. C. H. and Amin, N. A. K., “Maximum product of spacings estimation with
      application to the lognormal distribution”, tech. rep., (1979).
      21. Cheng, R. C. H. and Amin, N. A. K., “Estimating parameters in continuous univariate distributions
      with a shifted origin”, Journal of the Royal Statistical Society. Series B (Methodological),
      45(3), pp. 394–403 (1983).
      22. Torabi, H., “A general method for estimating and hypotheses testing using spacings”, Journal
      of Statistical Theory and Practice, 8(2), pp. 163–168 (2008).
      23. MacDonald, P. D. M., “Comment on ”an estimation procedure for mixtures of distributions”
      by Choi and Bulgren”, Journal of the Royal Statistical Society. Series B (Methodological),
      33(2), pp. 326–329 (1971).
      24. Shamsuzzaman, M., Xie, X., Goh, N.T., and Zhang, H., “Integrated control chart system for
      time-between-events monitoring in a multistage manufacturing system”, The International
      Journal of Advanced Manufacturing Technology, 40(3-4), pp. 373–381 (2009).
      25. Zhang, C.W., Xie, M., Liu, J.Y., and Goh, T.N., “A control chart for the gamma distribution
      as a model of time between events”, International Journal of Production Research, 45(23),
      pp. 5649–5666 (2007).
      26. Ali, S., Pievatolo, A., and G¨ob, R., “An overview of control charts for high quality processes”,
      Quality and Reliability Engineering International, 32(7), pp. 2171–2189 (2016).
      27. Linda, L. H., Fernandes, F.H., and Bourguignon, M., “Control charts to monitor rates and
      proportions”, Quality and Reliability Engineering International, 35(1), pp. 74–83 (2019).
      28. Cruz, F. R. B., Quinino, R. C., and Ho., Linda L., “Control charts for traffic intensity monitoring
      of Markovian multiserver queues”, Quality and Reliability Engineering International,
      36(1), pp. 354–364 (2020).
      29. Lima-Filho, L. M. de A., Pereira, T.L., de Souza, T. C., and Bayer, F.M., “Inflated beta control
      chart for monitoring double bounded processes”, Computers & Industrial Engineering,
      136, pp. 265 – 276 (2019).
      30. Lima-Filho, L.M. de A. and Bayer, F. M., “Kumaraswamy control chart for monitoring double
      bounded environmental data”, Communications in Statistics - Simulation and Computation,
      50(9), pp. 2513–2528 (2021).
      31. Chukhrova, N. and Johannssen, A., “Improved control charts for fraction non-conforming
      based on hypergeometric distribution”, Computers & Industrial Engineering, 128, pp. 795 –
      806 (2019).
      32. Lemonte, A.J., “Improved point estimation for the Kumaraswamy distribution”, Journal of
      Statistical Computation and Simulation, 81(12), pp. 1971–1982 (2011).
      33. Bourguignon, M., Ghosh, I., and Cordeiro, G.M., “General results for the transmuted family
      of distributions and new models”, Journal of Probability and Statistics, 2016, pp. 1–12
      (2016).