| Title:
|
A method constructing density functions: the case of a generalized Rayleigh variable (English) |
| Author:
|
Vodă, Viorel Gh. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
54 |
| Issue:
|
5 |
| Year:
|
2009 |
| Pages:
|
417-431 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we propose a new generalized Rayleigh distribution different from that introduced in Apl. Mat. {\it 47} (1976), pp.\,395--412. The construction makes use of the so-called ``conservability approach'' (see Kybernetika {\it 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 $f_1(x) = x f(x)/E(X)$, 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 $f(x)$. (English) |
| Keyword:
|
generalized Rayleigh variable (GRV) |
| Keyword:
|
generalized exponential (GE) |
| Keyword:
|
generating differential equation (GDE) |
| Keyword:
|
conservability |
| Keyword:
|
probability density function (p.d.f.) |
| Keyword:
|
pseudo-Weibull variable |
| MSC:
|
60E05 |
| MSC:
|
60E10 |
| MSC:
|
60K40 |
| MSC:
|
62P30 |
| idZBL:
|
Zbl 1212.60013 |
| idMR:
|
MR2545409 |
| DOI:
|
10.1007/s10492-009-0027-3 |
| . |
| Date available:
|
2010-07-20T13:19:55Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140376 |
| . |
| Reference:
|
[1] Ahuja, J. C., Nash, S. W.: The generalized Gompertz-Verhulst family of distributions.Sankhya, Ser. A 39 (1967), 141-156. Zbl 0173.20401, MR 0234502 |
| Reference:
|
[2] Ahuja, J. C.: On certain properties of the generalized Gompertz distribution.Sankhya, Ser. B 31 (1969), 541-544. |
| Reference:
|
[3] Babus, F., Kobi, A., Tiplica, T., Bacivarov, I. C.: 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), 322-328. |
| Reference:
|
[4] Barlow, R. E., Proschan, F.: Statistical Theory of Reliability and Life Testing.Holt, Rinehart and Winston New York (1975). Zbl 0379.62080, MR 0438625 |
| Reference:
|
[5] Bârsan-Pipu, N., Isaic-Maniu, Al., Vodǎ, V. Gh.: Defectarea. Modele statistice cu aplicaţii (The Failure. Statistical Models with Application).Editura Economicǎ Bucureşti (1999), Romanian. |
| Reference:
|
[6] Blischke, W. R., Murthy, D. N. P.: Reliability. Modeling, Prediction and Optimization.John Wiley & Sons New York (2000). Zbl 0945.62102 |
| Reference:
|
[7] Bruscantini, S.: Origin, features and use of the pseudo-normal distribution.Statistica 28 (1968), 102-123. |
| Reference:
|
[8] Caulcutt, R.: Achieving Quality Improvement. A Practical Guide.Chapman and Hall London (1995). Zbl 0862.62075 |
| Reference:
|
[9] Ciechanowicz, K.: Uogólniony 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 Polish. MR 0334446 |
| Reference:
|
[10] Curelaru, L., Vodǎ, V. Gh.: Some notes on Rayleigh distribution.Revista Colombiana de Matemáticas 9 (1975), 9-22. MR 0400510 |
| Reference:
|
[11] D'Addario, R.: 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 Italian. |
| Reference:
|
[12] D'Addario, R.: 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 Italian. |
| Reference:
|
[13] Deming, W. E.: Statistical Adjustment of Data.Dover Publications Inc. New York (1964). |
| Reference:
|
[14] Deming, W. E.: On the probability as a basis for action.Am. Stat. 29 (1975), 146-152. |
| Reference:
|
[15] Dobó, A.: Reliability of ageing components ``Quality and Reliability''. Special Edition.Economisti ed Anali di Economia Roma (1976), 53-56. |
| Reference:
|
[16] Dorin, Al. C., Al Isaic-Maniu, Vodǎ, V. Gh.: Probleme statistice ale fiabilitǎţii. Aplicaţii în domeniul sculelor aşchietoare (Statistical Problems of Reliability. Cutting-tool Applications).Editura Economicǎ Bucureşti (1994), Romanian. |
| Reference:
|
[17] Drane, J. W., Postelnicu, T., Vodǎ, V. Gh.: New inferences on the Rayleigh distribution.Bull. Mathématique de la Societé des Sciences Mathématiques de Roumanie, Nouvelle Série 83 (1991), 235-244. MR 1307689 |
| Reference:
|
[18] Drimlová, B.: Přejímací plány pro Weibullovo rozdělení (Acceptance sampling plans for Weibull distribution).Research Report No. SVÚSS-73-01012 Běchovice (ČSSR) Běchovice (June 1973) Czech. |
| Reference:
|
[19] Firkowicz, Sz.: O potęgówym rozkladzie trwalości (On the power distribution).Archiwum Electrotehniki 18 (1969), 29-40 Polish. |
| Reference:
|
[20] Giorski, A. C.: Beware of the Weibull euphoria.IEEE Trans. Reliability R17 (1968), 202-203. 10.1109/TR.1968.5216949 |
| Reference:
|
[21] Guerrieri, G.: 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 Italian. |
| Reference:
|
[22] Gupta, R. D., Kundu, D.: Generalized exponential distribution: statistical inferences.J. Statist. Theory Appl. 1 (2002), 101-118. MR 1952704 |
| Reference:
|
[23] Hunter, J. J.: An analytical technique for urban casualty estimation from multiple nuclear weapons.Operations Research 15 (1967), 1096-1108. 10.1287/opre.15.6.1096 |
| Reference:
|
[24] Iosifescu, M., Moineagu, C., Trebici, Vl., Ursianu, E.: Micǎ Enciclopedie de Statisticǎ (A Little Statistical Encyclopedia).Editura Ştiinţificǎ şi Enciclopedicǎ Bucureşti (1985), Romanian. |
| Reference:
|
[25] Isaic-Maniu, Al.: Metoda Weibull. Aplicaţii (Weibull Method. Applications).Editura Academiei R. S. România Bucureşti (1983), Romanian. |
| Reference:
|
[26] Isaic-Maniu, Al., Vodǎ, V. Gh.: O nouǎ generalizare a repartiţiei putere (A new generalization of power distribution.Stud. Cerc. Calc. Econ. Cib. Econ. 29 1 (1995), 19-25 Romanian. |
| Reference:
|
[27] Isaic-Maniu, Al., Vodǎ, V. Gh.: O nouǎ generalizare a repartiţiei exponenţiale (A new generalization of exponential distribution).Stud. Cerc. Calc. Econ. Cib. Econ. 30 4 (1996), 9-17 Romanian. MR 1454922 |
| Reference:
|
[28] Isaic-Maniu, Al., Vodǎ, V. Gh.: Aspecte privind repartiţia Rayleigh (Some aspects regarding Rayleigh distribution).Stud. Cerc. Calc. Econ. Cib. Econ. 32 1 (1998), 5-13 Romanian. MR 1319940 |
| Reference:
|
[29] Isaic-Maniu, Al., Vodǎ, V. Gh.: Rayleigh distribution revisited.Econ. Comp. Econ. Cyb. Stud. Res. 34 (2000), 27-32. |
| Reference:
|
[30] Jílek, M.: Statistické toleranční meze (Statistical Tolerance Limits).SNTL -- Teoretická knižnice inženýra Praha (1988), Czech. |
| Reference:
|
[31] Johnson, N. L., Kotz, S., Balakrishnan, N.: Continuous Univariable Distributions. Vol. 1, 2nd Edition.John Wiley & Sons New York (1994). MR 1299979 |
| Reference:
|
[32] Kundu, D., Raqab, M. Z.: Generalized Rayleigh distribution: different methods of estimations.http://home.iitk.ac.in/ {kundu}/ (2004). MR 2129172 |
| Reference:
|
[33] Hjorth, U.: A reliability distribution with increasing, decreasing, constant and bath-tub shaped failure rates.Technometrics 22 (1980), 99-112. MR 0559684, 10.2307/1268388 |
| Reference:
|
[34] Khan, M. S. H.: A generalized exponential distribution.Biometrical J. 29 (1987), 121-127. Zbl 0606.62015, MR 0883115, 10.1002/bimj.4710290121 |
| Reference:
|
[35] Patel, J. K., Read, C. B.: Handbook of the Normal Distribution (2nd edition, revised and expanded); Statistics: Textbooks and Monographs, Vol. 150.Marcel Dekker Inc. New York (1996). MR 0664762 |
| Reference:
|
[36] Pereverzev, E. S.: Random Processes in Parametric Models of Reliability.Academy of Ukraine, Institute of Technical Mechanics, Naukova Dumka Publ. House Kiev (1987), Russian. MR 0913737 |
| Reference:
|
[37] Pollard, A., Rivoire, C.: Fiabilité et statistique previsionelle. Méthode de Weibull.Eyrolles Paris (1971), French. |
| Reference:
|
[38] Raqab, M. Z., Kundu, D.: Burr type $X$ distribution: revisited, 2003. http://home.iitk.ac.in/kundu/.. |
| Reference:
|
[39] Ravenis, J. V., II: A potentially universal probability density function for scientists and engineers.Proceedings of the International Conference on Quality Control, Tokyo, Sept. 1969 523-526. |
| Reference:
|
[40] Rodriguez, R.: Systems of frequency curves.In: Encyclopedia of Statistical Sciences, Vol. 3 S. Kotz, N. L. Johnson John Wiley & Sons New York (1983), 212-225. MR 0719029 |
| Reference:
|
[41] Ryshyk, I. M., Gradstein, I. S.: Tables of Series, Product and Integrals. 4th edition.Yu. V. Geronimus, M. Yu. Tseylin Acadmic Press New York (1965). |
| Reference:
|
[42] Savageau, M. A.: A suprasystem of probability distributions.Biometrical J. 24 (1982), 209-215. Zbl 0493.62026, MR 0687870, 10.1002/bimj.4710240402 |
| Reference:
|
[43] Spǎtaru, S., Galupa, A.: Generalizarea unei repartiţii cu aplicaţii in teoria siguranţei (Generalization of a distribution with applications in reliability theory).Stud. Cerc. Econ. Cib. Econ. 32 1 (1988), 77-82 Romanian. |
| Reference:
|
[44] M. El-Moniem Soleha, Sewilam, Iman A.: Generalized Rayleigh distribution revisited. InterStat.http://interstat.statjournals.net/YEAR/2007/abstracts/0702006.pdf. |
| Reference:
|
[45] Stacy, E. W.: A generalization of the Gamma distribution.Ann. Math. Stat. 28 (1962), 1187-1192. Zbl 0121.36802, MR 0143277, 10.1214/aoms/1177704481 |
| Reference:
|
[46] Subbotin, M. T.: On the law of frequency of error.Moscow Recueil mathématique 31 (1923), 296-301. |
| Reference:
|
[47] Taguchi, T.: On a generalization of Gaussian distribution, Part. A.Ann. Inst. Statist. Math. (Tokyo) 30 (1978), 221-242. MR 0514493 |
| Reference:
|
[48] Vodǎ, V. Gh.: Inferential procedures on a generalized Rayleigh variate (I).Apl. Mat. 21 (1976), 395-412. MR 0418319 |
| Reference:
|
[49] Vodǎ, V. Gh.: Inferential procedures on a generalized Rayleigh variate (II).Apl. Mat. 21 (1976), 413-419. MR 0418320 |
| Reference:
|
[50] Vodǎ, V. Gh.: Some inferences on the generalized Gompertz distribution.Rev. Roum. Math. Pures. Appl. 25 (1980), 1267-1278. MR 0598031 |
| Reference:
|
[51] Vodǎ, V. Gh.: New models in durability tool-testing: pseudo Weibull distribution.Kybernetika 25 (1989), 209-215. MR 1010183 |
| Reference:
|
[52] Vodǎ, V. Gh.: A new generalization of Rayleigh distribution.Reliability: Theory & Applications 2 2 (2007), 47-56 http://www.gnedenko-forum.org/Journal/files/RTA_2_2007.pdf. |
| Reference:
|
[53] Voit, E. O.: The $S$-distribution: a tool for approximation and classification of univariable, unimodal probability distributions.Biometrical J. 34 (1992), 855-878. 10.1002/bimj.4710340713 |
| Reference:
|
[54] Yu, Sh., Voit, E. O.: A simple, flexible failure model.Biometrical J. 37 (1995), 595-609. Zbl 0837.62097, 10.1002/bimj.4710370509 |
| . |
