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A method constructing density functions: The case of a generalized Rayleigh variable

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Abstract

In this paper we propose a new generalized Rayleigh distribution different from that introduced in Apl. Mat. 47 (1976), pp. 395–412. The construction makes use of the so-called “conservability approach” (see Kybernetika 25 (1989), pp. 209–215) namely, if X is a positive continuous random variable with a finite mean-value E(X), then a new density is set to be f 1(x) = xf(x)/E(X), where f(x) is the probability density function of X. The new generalized Rayleigh variable is obtained using a generalized form of the exponential distribution introduced by Isaic-Maniu and the present author as f(x).

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References

  1. J. C. Ahuja, S. W. Nash: The generalized Gompertz-Verhulst family of distributions. Sankhya, Ser. A 39 (1967), 141–156.

    MathSciNet  Google Scholar 

  2. J. C. Ahuja: On certain properties of the generalized Gompertz distribution. Sankhya, Ser. B 31 (1969), 541–544.

    Google Scholar 

  3. F. Babus, A. Kobi, T. Tiplica, I. C. Bacivarov: Current troubles of control charts application under non-Gaussian distributions. Proceedings of the 10th International Conference “Quality and Dependability”, September 27–29, 2006, Sinaia, Romania. MEDIAREX 21, Publ. House, Bucharest, 2006, pp. 322–328.

    Google Scholar 

  4. R. E. Barlow, F. Proschan: Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York, 1975.

    MATH  Google Scholar 

  5. N. Bârsan-Pipu, Al. Isaic-Maniu, V. Gh. Vodă: Defectarea. Modele statistice cu aplicaţii (The Failure. Statistical Models with Application). Editura Economică, Bucureşti, 1999. (In Romanian.)

    Google Scholar 

  6. W. R. Blischke, D. N. P. Murthy: Reliability. Modeling, Prediction and Optimization. John Wiley & Sons, New York, 2000.

    MATH  Google Scholar 

  7. S. Bruscantini: Origin, features and use of the pseudo-normal distribution. Statistica 28 (1968), 102–123.

    Google Scholar 

  8. R. Caulcutt: Achieving Quality Improvement. A Practical Guide. Chapman and Hall, London, 1995.

    MATH  Google Scholar 

  9. K. Ciechanowicz: Uogĺniony rozklad Gamma i rozklad potęgówy jako rozklady trvalości elementów (Generalized Gamma distribution and power distribution as models for component durability). Archiwum Electrotehniki 21 (1972), 489–512. (In Polish.)

    MathSciNet  Google Scholar 

  10. L. Curelaru, V. Gh. Vodă: Some notes on Rayleigh distribution. Revista Colombiana de Matemáticas 9 (1975), 9–22.

    MATH  Google Scholar 

  11. R. DÁddario: Un metodo per la rappresentazione analitica delle distribuzioni statistiche (A method for analytical representation of statistical distributions). Ann. dell’Istituto di Statistica dell’Università di Bari 16 (1939), 36–45. (In Italian.)

    Google Scholar 

  12. R. D’Addario: Sulle repartizioni la cui media superiormente o inferiormente “incompleta” cresce linearmente col crescere della variabile distributiva (On distributions for which their “incomplete” mean-value is linearly increasing). Giornale degli Economisti ed Annali di Economia-Roma 11–12 (1969), 20–28. (In Italian.)

    Google Scholar 

  13. W. E. Deming: Statistical Adjustment of Data. Dover Publications Inc., New York, 1964.

    Google Scholar 

  14. W. E. Deming: On the probability as a basis for action. Am. Stat. 29 (1975), 146–152.

    Article  Google Scholar 

  15. A. Dobó: Reliability of ageing components “Quality and Reliability”. Special Edition. Economisti ed Anali di Economia, Roma, 1976, pp. 53–56.

  16. Al. C. Dorin, Al Isaic-Maniu, V. Gh. Vodă: Probleme statistice ale fiabilităţii. Aplicaţii în domeniul sculelor aşchietoare (Statistical Problems of Reliability. Cutting-tool Applications). Editura Economică, Bucureşti, 1994. (In Romanian.)

    Google Scholar 

  17. J. W. Drane, T. Postelnicu, V. Gh. Vodă: New inferences on the Rayleigh distribution. Bull. Mathématique de la Societé des Sciences Mathématiques de Roumanie, Nouvelle Série 83 (1991), 235–244.

    Google Scholar 

  18. B. Drimlová: Přejímací plány pro Weibullovo rozdelení (Acceptance sampling plans for Weibull distribution). Research Report No. SVÚSS-73-01012 Běchovice (ČSSR). Běchovice, June 1973. (In Czech.)

    Google Scholar 

  19. Sz. Firkowicz: O potęgówym rozkladzie trwalości (On the power distribution). Archiwum Electrotehniki 18 (1969), 29–40. (In Polish.)

    Google Scholar 

  20. A. C. Giorski: Beware of the Weibull euphoria. IEEE Trans. Reliability R17 (1968), 202–203.

    Article  Google Scholar 

  21. G. Guerrieri: Sopra un nouvo metodo concernente la determinazione dei parametri della distribuzione lognormale e delle distribuzioni pearsoniane del III e del V tipo (On a new method for estimating the parameters of lognormal distribution and of Pearsonian distribution of III a V type). Ann. dell’Istituto di Statistica dell’Università di Bari 34 (1969–1970), 55–110. (In Italian.)

    Google Scholar 

  22. R. D. Gupta, D. Kundu: Generalized exponential distribution: statistical inferences. J. Statist. Theory Appl. 1 (2002), 101–118.

    MathSciNet  Google Scholar 

  23. J. J. Hunter: An analytical technique for urban casualty estimation from multiple nuclear weapons. Operations Research 15 (1967), 1096–1108.

    Article  MathSciNet  Google Scholar 

  24. M. Iosifescu, C. Moineagu, Vl. Trebici, E. Ursianu: Mică Enciclopedie de Statistică (A Little Statistical Encyclopedia). Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1985. (In Romanian.)

    Google Scholar 

  25. Al. Isaic-Maniu: Metoda Weibull. Aplicaţii (Weibull Method. Applications). Editura Academiei R. S. România, Bucureşti, 1983. (In Romanian.)

    Google Scholar 

  26. Al. Isaic-Maniu, V. Gh. Vodă: O nouă generalizare a repartiţiei putere (A new generalization of power distribution. Stud. Cerc. Calc. Econ. Cib. Econ. 29 (1995), no. 1, 19–25. (In Romanian.)

    Google Scholar 

  27. Al. Isaic-Maniu, V. Gh. Vodă: O nouă generalizare a repartiţiei exponenţiale (A new generalization of exponential distribution). Stud. Cerc. Calc. Econ. Cib. Econ. 30 (1996), no. 4, 9–17. (In Romanian.)

    Google Scholar 

  28. Al. Isaic-Maniu, V. Gh. Vodă: Aspecte privind repartiţia Rayleigh (Some aspects regarding Rayleigh distribution). Stud. Cerc. Calc. Econ. Cib. Econ. 32 (1998), no. 1, 5–13. (In Romanian.)

    Google Scholar 

  29. Al. Isaic-Maniu, V. Gh. Vodă: Rayleigh distribution revisited. Econ. Comp. Econ. Cyb. Stud. Res. 34 (2000), 27–32.

    Google Scholar 

  30. M. Jílek: Statistické toleranàní meze (Statistical Tolerance Limits). SNTL — Teoretická knižnice inženýra, Praha, 1988. (In Czech.)

    Google Scholar 

  31. N. L. Johnson, S. Kotz, N. Balakrishnan: Continuous Univariable Distributions. Vol. 1, 2nd Edition. John Wiley & Sons, New York, 1994.

    Google Scholar 

  32. D. Kundu, M. Z. Raqab: Generalized Rayleigh distribution: different methods of estimations. http://home.iitk.ac.in/~kundu/ (2004).

  33. U. Hjorth: A reliability distribution with increasing, decreasing, constant and bath-tub shaped failure rates. Technometrics 22 (1980), 99–112.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. S. H. Khan: A generalized exponential distribution. Biometrical J. 29 (1987), 121–127.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. K. Patel, C. B. Read: Handbook of the Normal Distribution (2nd edition, revised and expanded); Statistics: Textbooks and Monographs, Vol. 150. Marcel Dekker Inc., New York, 1996.

    MATH  Google Scholar 

  36. E. S. Pereverzev: Random Processes in Parametric Models of Reliability. Academy of Ukraine, Institute of Technical Mechanics, Naukova Dumka Publ. House, Kiev, 1987. (In Russian.)

    Google Scholar 

  37. A. Pollard, C. Rivoire: Fiabilité et statistique previsionelle. Méthode de Weibull. Eyrolles, Paris, 1971. (In French.)

    Google Scholar 

  38. M. Z. Raqab, D. Kundu: Burr type X distribution: revisited, 2003. http://home.iitk.ac.in/~kundu/.

  39. J. V. Ravenis, II: A potentially universal probability density function for scientists and engineers. Proceedings of the International Conference on Quality Control, Tokyo, Sept. 1969. pp. 523–526.

  40. R. Rodriguez: Systems of frequency curves. In: Encyclopedia of Statistical Sciences, Vol. 3 (S. Kotz, N. L. Johnson, eds.). John Wiley & Sons, New York, 1983, pp. 212–225.

    Google Scholar 

  41. I. M. Ryshyk, I. S. Gradstein: Tables of Series, Product and Integrals. 4th edition (Yu. V. Geronimus, M. Yu. Tseylin, eds.). Acadmic Press, New York, 1965.

    Google Scholar 

  42. M. A. Savageau: A suprasystem of probability distributions. Biometrical J. 24 (1982), 209–215.

    Article  MathSciNet  Google Scholar 

  43. S. Spătaru, A. Galupa: Generalizarea unei repartiţii cu aplicaţii in teoria siguranţei (Generalization of a distribution with applications in reliability theory). Stud. Cerc. Econ. Cib. Econ. 32 (1988), no. 1, 77–82. (In Romanian.)

    Google Scholar 

  44. M. El-Moniem Soleha, Iman A. Sewilam: Generalized Rayleigh distribution revisited. InterStat. http://interstat.statjournals.net/YEAR/2007/abstracts/0702006.pdf.

  45. E. W. Stacy: A generalization of the Gamma distribution. Ann. Math. Stat. 28 (1962), 1187–1192.

    Article  MathSciNet  Google Scholar 

  46. M. T. Subbotin: On the law of frequency of error. Moscow Recueil mathématique 31 (1923), 296–301.

    Google Scholar 

  47. T. Taguchi: On a generalization of Gaussian distribution, Part. A. Ann. Inst. Statist. Math. (Tokyo) 30 (1978), 221–242.

    Google Scholar 

  48. V. Gh. Vodă: Inferential procedures on a generalized Rayleigh variate (I). Apl. Mat. 21 (1976), 395–412.

    MATH  MathSciNet  Google Scholar 

  49. V. Gh. Vodă: Inferential procedures on a generalized Rayleigh variate (II). Apl. Mat. 21 (1976), 413–419.

    MathSciNet  Google Scholar 

  50. V. Gh. Vodă: Some inferences on the generalized Gompertz distribution. Rev. Roum. Math. Pures. Appl. 25 (1980), 1267–1278.

    MATH  Google Scholar 

  51. V. Gh. Vodă: New models in durability tool-testing: pseudo Weibull distribution. Kybernetika 25 (1989), 209–215.

    MATH  MathSciNet  Google Scholar 

  52. V. Gh. Vodă: A new generalization of Rayleigh distribution. Reliability: Theory & Applications 2 (2007), no. 2, 47–56. http://www.gnedenko-forum.org/Journal/files/RTA 2 2007.pdf.

    Google Scholar 

  53. E.O. Voit: The S-distribution: a tool for approximation and classification of univariable, unimodal probability distributions. Biometrical J. 34 (1992), 855–878.

    Article  MATH  Google Scholar 

  54. Sh. Yu, E.O. Voit: A simple, flexible failure model. Biometrical J. 37 (1995), 595–609.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. “Ch. Mihoc-C. Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Casa Academiei Romane, Calea 13 Septembrie nr. 13, 050711, Bucharest, Romania

    Viorel Gh. Vodă

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  1. Viorel Gh. Vodă
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The editors learnt with great sadness that Professor Viorel Vodă passed away on May 8, 2009. The galleys of this paper were therefore not proofread by the author, and the responsibility for any typesetting inaccuracies lies solely with the editors.

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Vodă, V.G. A method constructing density functions: The case of a generalized Rayleigh variable. Appl Math 54, 417–431 (2009). https://doi.org/10.1007/s10492-009-0027-3

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