Skip to main content
SpringerLink
Log in

Stationary GE-Process and its Application in Analyzing Gold Price Data

  • Published:
Sankhya B Aims and scope Submit manuscript

Abstract

In this paper we introduce a new discrete time and continuous state space stationary process {Xn;n = 1,2,…}, such that Xn follows a two-parameter generalized exponential (GE) distribution. Joint distribution functions, characterization and some dependency properties of this new process have been investigated. The GE-process has three unknown parameters, two shape parameters and one scale parameter, and due to this reason it is more flexible than the existing exponential process. In presence of the scale parameter, if the two shape parameters are equal, then the maximum likelihood estimators of the unknown parameters can be obtained by solving one non-linear equation and if the two shape parameters are arbitrary, then the maximum likelihood estimators can be obtained by solving a two dimensional optimization problem. Two synthetic data sets, and one real gold-price data set have been analyzed to see the performance of the proposed model in practice. Finally some generalizations have been indicated.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7

Similar content being viewed by others

References

  • Al-Hussaini, E.K. and Ahsanullah, M. (2015). Exponentiated Distributions. Atlantis Press, France.

    Book  MATH  Google Scholar 

  • Arnold, B.C. (1993). Logistic process involving Markovian minimization. Communications in Statistics - Theory and Methods 22, 1699–1707.

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold, B.C. (2001). Pareto process, Handbook of Statistics, 19. Elsevier, Amsterdam, Rao, C. R. and Shanbag, D. N. (eds.), p. 1–33.

    Google Scholar 

  • Arnold, B.C. and Hallet, T.J. (1989). A characterization of the Pareto process among stationary processes of the form \(X_{n}, = c \min \limits (X_{n-1}, Y_{n})\). Stat. Probab. Lett. 8, 377–380.

    Article  Google Scholar 

  • Bemis, B., Bain, L.J. and Higgins, J.J. (1972). Estimation and hypothesis testing for the parameters of a bivariate exponential distribution. J. Amer. Stat. Assoc. 67, 927–929.

    Article  MATH  Google Scholar 

  • Granger, C.W.J. and Andersen, A.P. (1978). An introduction to bilinear time series models. Angewandte Statistik und Okonometrie, Heft 8. Vandenhoeck & Ruprecht, Gottingen 94.

  • Gupta, R.D. and Kundu, D. (1999). Generalized exponential distribution. Austral. Zealand J. Stat. 41, 173–188.

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta, R.D. and Kundu, D. (2007). Generalized exponential distribution: Existing methods and recent developments. J. Stat. Plann. Infer. 137, 3537–3547.

    Article  MATH  Google Scholar 

  • Halmos, P.R. (1950). Measure Theory,. D. Van Nostrand Company Inc, New York.

    Book  MATH  Google Scholar 

  • Jayakumar, K. and Girish Babu, M. (2015). Some generalizations of Weibull distribution and related processes. J. Stat. Theor. Appl. 14, 425–434.

    Article  MathSciNet  Google Scholar 

  • Jose, K.K., Ristić, M.M. and Joseph, A. (2011). Marshall-Olkin bivariate Weibull distributions and processes. Stat. Papers 52, 789–798.

    Article  MathSciNet  MATH  Google Scholar 

  • Kundu, D. and Gupta, R.D. (2009). Bivariate generalized exponential distribution. J. Multivar. Anal. 100, 581–593.

    Article  MathSciNet  MATH  Google Scholar 

  • Kundu, D. and Gupta, R.D. (2010). A class of bivariate models with proportional reversed hazard marginals. Sankhya, Ser. B 72, 236–253.

    Article  MathSciNet  MATH  Google Scholar 

  • Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliab. 42, 299–302.

    Article  MATH  Google Scholar 

  • Nadarajah, S. (2011). The exponentiated exponential distribution; A survey. Adv. Stat. Anal. 95, 219–251.

    Article  MathSciNet  MATH  Google Scholar 

  • Nelsen, R.B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.

    MATH  Google Scholar 

  • Nelson (1976). The Use of Box-Cox Transformation in Econometric Time Series: An Empirical Study, Ph.D. Thesis Economics Department. The University of California, San Diego.

    Google Scholar 

  • Pillai, R.N. (1991). Semi-pareto processes. J. Appl. Probab. 28, 461–465.

    Article  MathSciNet  MATH  Google Scholar 

  • Sim, C.-H. (1986). Simulation of Weibull and gamma autoregressive stationary process. Commun. Stat. Simul. Comput. 15, 1141–1146.

    Article  MathSciNet  MATH  Google Scholar 

  • Tavares, L.V. (1980). An exponential Markovian stationary process. J. Appl. Probab. 17, 1117–1120.

    Article  MathSciNet  MATH  Google Scholar 

  • Weiss, G. (1975). Time reversibility of linear stochastic process. J. Appl. Probab. 12, 143–171.

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the unknown reviewers for making constructive suggestions which have helped to improve the earlier version of the manuscript significantly.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin, 208016, India

    Debasis Kundu

Authors
  1. Debasis Kundu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debasis Kundu.

Ethics declarations

Conflict of Interests

It is to declare that the I do not have any conflict of interest, and did not receive any financial support for this project.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kundu, D. Stationary GE-Process and its Application in Analyzing Gold Price Data. Sankhya B 84, 575–595 (2022). https://doi.org/10.1007/s13571-021-00272-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13571-021-00272-z

Keywords

AMS (2000) subject classification

Search

Navigation