Published

2017-01-16

Statistical properties and different methods of estimation of transmuted Rayleigh distribution

Propiedades estadísticas y diferentes métodos de estimación de la distribución de Rayleigh transmutada

DOI:

https://doi.org/10.15446/rce.v40n1.56153

Keywords:

Transmuted Rayleigh distribution, Hazard Rate Function, Conditional Moments, Order Statistics, Parameter Estimation (en)
Momentos distributivos, Estadísticas de orden, La estimación de parámetros, Riesgo de tipo de función, Rayleigh transmutada. (es)

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Authors

  • Sanku Dey Anthony´s College
  • Enayetur Raheem University of Northern Colorado
  • Saikat Mukherjee NIT Meghalaya
This article addresses the various properties and different methods of estimation of the unknown parameters of the Transmuted Rayleigh (TR) distribution from the frequentist point of view. Although, our main focus is on estimation from frequentist point of view,  yet, various mathematical and statistical properties of the TR distribution (such as quantiles, moments, moment generating function, conditional moments,  hazard rate, mean residual lifetime, mean past lifetime,  mean deviation about mean and median, the stochastic ordering,  various entropies, stress-strength parameter  and order statistics) are derived.  We briefly describe different frequentist methods of estimation approaches, namely, maximum likelihood estimators, moments estimators, L-moment estimators, percentile based estimators, least squares estimators, method of maximum product of spacings,  method of Cram\'er-von-Mises, methods of Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Finally, the potentiality of the model is analyzed by means of two real data sets which is further illustrated by obtaining bias and standard error of the estimates and the bootstrap percentile confidence intervals using bootstrap resampling.

Este artículo se aborda las varias propiedades y diferentes métodos para la estimación de los desconocidos parámetros de Transmuted Rayleigh (TR) distribución desde el punto de vista de un frequentist. Aunque la tema principal de este artículo es estimación, varias propiedades matemáticas y estadísticas de TR distribución (como cuantiles, momentos, una función que genera momentos, momentos condicionales, la tasa de peligro, la media vida residual, media vida pasada, la desviación media por media y mediana, organización stochastic, entropías varias, parámetros de tensión-fuerza y estadisticas de orden) están derivadas. Describimos brevemente los diferentes métodos de estimación, como máxima probabilidad, método de momentos, estimacién basada por percentil, mínimos cuadrados, método de máximos productos de espacios, el método de Cramér-von-Mises, los métodos de Anderson-Darling y right-tail Anderson-Darling, y compararlos con extensos estudios de simulaciones. Por último, la potencialidad del modelo está estudiando con dos conjuntos de datos reales. El margen de error, el promedio de error de las estimaciones y el percentage bootstrap de los confianza intervalos estan derivido por bootstrap remuestro.

https://doi.org/10.15446/rce.v40n1.56153

Statistical Properties and Different Methods of Estimation of Transmuted Rayleigh Distribution

Propiedades estadísticas y diferentes métodosde estimación de la distribución de Rayleigh transmutada

SANKU DEY1, ENAYETUR RAHEEM2, SAIKAT MUKHERJEE3

1St. Anthony's College, Department or Statistics, Shillong, India. PhD. Email: sanku_dey2k2003@yahoo.co.in
2University of Northern Colorado, Applied Statistics and Research Methods, Greeley, USA. PhD. Email: enayetur.raheem@unco.edu
3NIT Meghalaya, Shillong, India. PhD. Email: saikat.mukherjee@nitm.ac.in

Abstract

This article addresses various properties and different methods of estimation of the unknown Transmuted Rayleigh (TR) distribution parameters from a frequentist point of view. Although our main focus is on estimation, various mathematical and statistical properties of the TR distribution (such as quantiles, moments, moment generating function, conditional moments, hazard rate, mean residual lifetime, mean past lifetime, mean deviation about mean and median, the stochastic ordering, various entropies, stress-strength parameter, and order statistics) are derived. We briefly describe different methods of estimation such as maximum likelihood, method of moments, percentile based estimation, least squares, method of maximum product of spacings, method of Cramér-von-Mises, methods of Anderson-Darling and right-tail Anderson-Darling, and compare them using extensive simulations studies. Finally, the potentiality of the model is studied using two real data sets. Bias, standard error of the estimates, and bootstrap percentile confidence intervals are obtained by bootstrap resampling.

Key words: Distributional Moments, Order Statistics, Parameter Estimation; Rate Risk Function, Rayleigh Distribution, Transmuted Rayleigh Distribution.

Resumen

Este artículo se aborda las varias propiedades y diferentes métodos para la estimación de los desconocidos parámetros de Transmuted Rayleigh (TR) distribución desde el punto de vista de un frequentist. Aunque la tema principal de este artículo es estimación, varias propiedades matemáticas y estadísticas de TR distribución (como cuantiles, momentos, una función que genera momentos, momentos condicionales, la tasa de peligro, la media vida residual, media vida pasada, la desviación media por media y mediana, organización stochastic, entropías varias, parámetros de tensión-fuerza y estadisticas de orden) están derivadas. Describimos brevemente los diferentes métodos de estimación, como máxima probabilidad, método de momentos, estimacién basada por percentil, mínimos cuadrados, método de máximos productos de espacios, el método de Cramér-von-Mises, los métodos de Anderson-Darling y right-tail Anderson-Darling, y compararlos con extensos estudios de simulaciones. Por último, la potencialidad del modelo está estudiando con dos conjuntos de datos reales. El margen de error, el promedio de error de las estimaciones y el percentage bootstrap de los confianza intervalos estan derivido por bootstrap remuestro.

Palabras clave: momentos distributivos, estadísticas de orden, la estimación de parámetros, riesgo de tipo de función, rayleigh transmutada.

Texto completo disponible en PDF

References

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2. Alkasasbeh, M. R. & Raqab, M. Z. (2009a), 'Estimation of the generalized logistic distribution parameters: Comparative study', Statistical Methodology 6(3), 262-279.

3. Alkasasbeh, M. R. & Raqab, M. Z. (2009b), 'Estimation of the generalized logistic distribution parameters: Comparative study', Statistical Methodology 6(3), 262-279.

4. Anderson, T. W. & Darling, D. A. (1952), 'Asymptotic theory of certain ''goodness-of-fit'' criteria based on stochastic processes', Annals of Mathematical Statistics 23, 193-212.

5. Anderson, T. W. & Darling, D. A. (1954), 'A test of goodness-of-fit', Journal of the American Statistical Association 49, 765-769.

6. Casella, G. & Berger, R. L. (1990), Statistical Inference, Brooks/Cole Publishing Company, California.

7. Cheng, R. C. H. & Amin, N. A. K. (1979), Maximum product of spacings estimation with applications to the lognormal distribution, Department of Mathematics, University of Wales.

8. Cheng, R. C. H. & Amin, N. A. K. (1983), 'Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin', Journal of the Royal Statistical Society: Series B 45(3), 394-403.

9. Cordeiro, G. M., Ortega, E. M. M. & Popovi\'c, Bo\vzidar V (2014), 'The gammalinear failure rate distribution: theory and applications', Journal of Statistical Computation and Simulation 84(11), 2408-2426.

10. Delignette-Muller, M. & Dutang, C. (2014), 'Fitdistrplus: An R Package for Fitting Distributions', Journal of Statistical Software 64(4), 1-34.

11. Dey, S. (2009), 'Comparison of Bayes estimators of the parameter and reliability function for Rayleigh distribution under differentloss functions', Malaysian Journal of Mathematical Sciences 3(247-264).

12. Dey, S. & Das, M. K. (2007), 'A note on prediction interval for a Rayleigh distribution: Bayesian approach', American Journal of Mathematical and Management Science 12, 43-48.

13. Dey, S., Dey, T. & Kundu, D. (2014), 'Two-parameter Rayleigh distribution: Different methods of estimation', American Journal of Math- ematical and Management Sciences 33, 55-74.

14. Gupta, R. D. & Kundu, D. (2001), 'Generalized exponential distribution: different method of estimations', Journal of Statistical Computation and Simulation 69(4), 315-337.

15. Gupta, R. D. & Kundu, D. (2007), 'Generalized exponential distribution: Existing results and some recent developments', Journal of Statistical Planning and Inference 137(11), 3537-3547.

16. Hosking, J. R. M. (1990), 'L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics', Journal of the Royal Statistical Society. Series B (Methodological) 52(1), 105-124.

17. Hosking, J. R. M. (1994), 'The four-parameter kappa distribution', Journal of Research and Development - IBM 38(3), 251-258.

18. Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (2015), 'Slashed rayleigh distribution', Revista Colombiana de Estadística 38(1), 31-44.

19. Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995), Continuous univariate distributions, Wiley, New York.

20. Kao, J. (1958), 'Computer methods for estimating Weibull parameters in reliability studies', Trans. IRE Reliability Quality Control 13, 15-22.

21. Kao, J. (1959), 'A graphical estimation of mixed Weibull parameters in life testing electron tube', Technometrics 1, 389-407.

22. Koutras, V. (2011), Two-level software rejuvenation model with in- creasing failure rate degradation, 'Dependable Computer Systems', Springer-Verlag, New York, p. 101-115.

23. Kundu, D. & Howlader, H. (2010), 'Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data', Computational Statistics and Data Analysis 54(6), 1547-1558.

24. Kundu, D. & Raqab, M. Z. (2005), 'Generalized Rayleigh distribution: Different methods of estimations', Computational Statistics and Data Analysis 49(1), 187-200.

25. Lai, M.-T. (2013), 'Optimum number of minimal repairs for a system under increasing failure rate shock model with cumulative repair-cost limit', International Journal of Reliability and Safety 7, 95-107.

26. Macdonald, P. D. M. (1971), 'Comment on ''An estimation procedure for mixtures of distributions'' by Choi and Bulgren', Journal of the Royal Statistical Society B 33, 326-329.

27. Maeda, K. & Nishikawa, M. (2006), 'Duration of party control in parliamentary and presidential governments: A study of 65 democracies, 1950 to 1998', Comparative Political Studies 39, 352-374.

28. Mahmoud, M. & Mandouh, R. (2013), 'The Transmuted Marshall-Olkin Fréchet Distribution: Properties and Applications', Journal of Applied Sciences Research 9(10), 5553-5561.

29. Mann, N. R., Schafer, R. E. & Singpurwalla, N. D. (1974), Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York.

30. Mazucheli, J., Louzada, F. & Ghitany, M. (2013), 'Comparison of estimation methods for the parameters of the weighted lindley distribution', Applied Mathematics and Computation 220, 463-471.

31. Merovci, F. (2013a), 'Transmuted lindley distribution', International Journal of Open Problems in Computer Science & Mathematics 6(2), 63-72.

32. Merovci, F. (2013b), 'Transmuted generalized rayleigh distribution', Journal of Statistics Applications and Probability 2(3), 1-12.

33. Merovci, F. (2013b), 'Transmuted Rayleigh distribution', Austrian Journal of Statistics 41(1), 21-31.

34. Merovci, F., Elbatal, I. & Ahmed, A. (2014), 'The Transmuted Generalized Inverse Weibull Distribution', Austrian Journal of Statistics 43(2), 119-131.

35. Merovci, F. & Puka, L. (2014), 'Transmuted Pareto distribution', ProbStat Forum 7, 1-11.

36. Nadarajah, S., Cordeiro, G. M. & Ortega, E. M. (2013), 'The exponentiated weibull distribution: a survey', Statistical Papers 54(3), 839-877.

37. Pettitt, A. (1976), 'A two-sample Anderson-Darling rank statistic', Biometrika 63(1), 161.

38. Ranneby, B. (1984), 'The Maximum Spacing Method. An Estimation Method Related to the Maximum Likelihood Method', Scandinavian Journal of Statistics 11(2), pp. 93-112.

39. Rayleigh, L. (1880), 'On the resultant of a large number of vibrations of the same pitch and of arbitrary phase', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10(60), 73-78.

40. Renyi, A. (1961), On measures of entropy and information, 'Fourth Berkeley symposium on mathematical statistics and probability', Vol. 1, p. 547-561.

41. Saidane, S., Babai, M., Aguir, M. & Korbaa, O. (2010), Spare parts inventory systems under an increasing failure rate demand interval distribution, '41st International Conference on Computers and Industrial Engineering', p. 768-773.

42. Salinas, H. S., Iriarte, Y. A. & Bolfarine, H. (2015), 'Slashed exponentiated rayleigh distribution', Revista Colombiana de Estadística 38(2), 453-466.

43. Shaked, M. & Shanthikumar, J. G. (1994), Stochastic orders and their applications, Academic Press, Boston, MA.

44. Shannon, C. E. (1948), 'Prediction and Entropy of Printed English', Bell System Technical Journal 30(1), 50-64.

45. Shaw, W. T. & Buckley, I. R. (2007), The Alchemy of Probability Distributions: Beyond Gram Charlier & Cornish Fisher Expansions, and Skew-Normal or Kurtotic-Normal Distributions, Financial Mathematics Group, King's College, London, U.K..

46. Singh, S. K., Singh, U. & Sharma, V. K. (2013), 'Bayesian analysis for Type-II hybrid censored sample from inverse Weibull distribution', International Journal of System Assurance Engineering and Management 4(3), 241-248.

47. Song, K. S. (2001), 'Rényi information, loglikelihood and an intrinsic distribution measure', Journal of Statistical Planning and Inference 93(1), 51-69.

48. Stephens, M. A. (2013), 'EDF Statistics for Goodness of Fit and Some Comparisons', Journal of the American Statistical Association 69(347), 730-737.

49. Swain, J. J., Venkatraman, S. & Wilson, J. R. (1988), 'Least-squares estimation of distribution functions in johnson's translation system', Journal of Statistical Computation and Simulation 29(4), 271-297.

50. Teimouri, M., Hoseini, S. M. & Nadarajah, S. (2013), 'Comparison of estimation methods for the Weibull distribution', Statistics 47(1), 93-109.

51. Tian, Y., Tian, M. & Zhu, Q. (2014), 'Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution', Communications in Statistics-Simulation and Computation 43(10), 2661-2677.

52. Tsarouhas, P. & Arvanitoyannis, I. (2007), 'Reliability and maintainability analysis of bread production line', Critical Reviews in Food Science and Nutrition 50, 327-343.

53. Woosley, R. & Cossman, J. (2007), 'Drug development and the FDAs critical path initiative', Public policy 81, 129-133.

[Recibido en marzo de 2016. Aceptado en noviembre de 2016]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv40n1a08,
    AUTHOR  = {Dey, Sanku and Raheem, Enayetur and Mukherjee, Saikat},
    TITLE   = {{Statistical Properties and Different Methods of Estimation of Transmuted Rayleigh Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2017},
    volume  = {40},
    number  = {1},
    pages   = {165-203}
}

References

Akram, M. & Hayat, A. (2014), ‘Comparison of estimators of the Weibull Distribution’, Journal of Statistical Theory and Practice 8(2), 238–259.

Alkasasbeh, M. R. & Raqab, M. Z. (2009a), ‘Estimation of the generalized logistic distribution parameters: Comparative study’, Statistical Methodology 6(3), 262–279.

Alkasasbeh, M. R. & Raqab, M. Z. (2009b), ‘Estimation of the generalized logistic distribution parameters: Comparative study’, Statistical Methodology 6(3), 262–279.

Anderson, T. W. & Darling, D. A. (1952), ‘Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes’.

Anderson, T. W. & Darling, D. A. (1954), ‘A test of goodness-of-fit’, Journal of the American Statistical Association 49, 765–769.

Casella, G. & Berger, R. L. (1990), Statistical Inference, Brooks/Cole Publishing Company, California.

Cheng, R. C. H. & Amin, N. A. K. (1979), Maximum product of spacings estimation with applications to the lognormal distribution, Technical report, Department of Mathematics, University of Wales.

Cheng, R. C. H. & Amin, N. A. K. (1983), ‘Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin’, Journal of the Royal Statistical Society: Series B 45(3), 394–403.

Cordeiro, G. M., Ortega, E. M. M. & Popovic, B. V. (2014), ‘The gammalinear failure rate distribution: theory and applications’, Journal of Statistical Computation and Simulation 84(11), 2408–2426.

Delignette-Muller, M. & Dutang, C. (2014), ‘fitdistrplus: An R Package for Fitting Distributions’, Journal of Statistical Software 64(4), 1–34.

Dey, S. (2009), ‘Comparison of Bayes estimators of the parameter and reliability function for Rayleigh distribution under differentloss functions’, Malaysian Journal of Mathematical Sciences 3(247-264).

Dey, S. & Das, M. K. (2007), ‘A note on prediction interval for a Rayleigh distribution: Bayesian approach’, American Journal of Mathematical and Management Science 12, 43–48.

Dey, S., Dey, T. & Kundu, D. (2014), ‘Two-parameter Rayleigh distribution: Different methods of estimation’, American Journal of Math- ematical and Management Sciences 33, 55–74.

Gupta, R. D. & Kundu, D. (2001), ‘Generalized exponential distribution: different method of estimations’, Journal of Statistical Computation and Simulation 69(4), 315–337.

Gupta, R. D. & Kundu, D. (2007), ‘Generalized exponential distribution: Existing results and some recent developments’, Journal of Statistical Planning and Inference 137(11), 3537–3547.

Hosking, J. R. M. (1990), ‘L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics’, Journal of the Royal Statistical Society. Series B (Methodological) 52(1), 105–124.

Hosking, J. R. M. (1994), ‘The four-parameter kappa distribution’, Journal of Research and Development - IBM 38(3), 251–258.

Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (2015), ‘Slashed rayleigh distribution’, Revista Colombiana de Estadística 38(1), 31–44.

Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995), Continuous univariate distributions, Wiley, New York.

Kao, J. (1958), ‘Computer methods for estimatingWeibull parameters in reliability studies’, Trans. IRE Reliability Quality Control 13, 15–22.

Kao, J. (1959), ‘A graphical estimation of mixed Weibull parameters in life testing electron tube’, Technometrics 1, 389–4–7.

Koutras, V. (2011), Two-level software rejuvenation model with in- creasing failure rate degradation, in ‘Dependable Computer Systems’, Springer-Verlag, New York, pp. 101–115.

Kundu, D. & Howlader, H. (2010), ‘Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data’, Computational Statistics and Data Analysis 54(6), 1547–1558.

Kundu, D. & Raqab, M. Z. (2005), ‘Generalized Rayleigh distribution: Different methods of estimations’, Computational Statistics and Data Analysis 49(1), 187–200.

Lai, M.-T. (2013), ‘Optimum number of minimal repairs for a system under increasing failure rate shock model with cumulative repair-cost limit’, International Journal of Reliability and Safety 7, 95–107.

Macdonald, P. D. M. (1971), ‘Comment on “An estimation procedure for mixtures of distributions” by Choi and Bulgren’, Journal of the Royal Statistical Society B 33, 326–329.

Maeda, K. & Nishikawa, M. (2006), ‘Duration of party control in parliamentary and presidential governments: A study of 65 democracies, 1950 to 1998’, Comparative Political Studies 39(352-374).

Mahmoud, M. & Mandouh, R. (2013), ‘R.M.(2013)On the Transmuted Fréchet Distribution’, Journal of Applied Sciences Research 9(10), 5553–5561.

Mann, N. R., Singpurwalla, N. D. & Schafer, R. E. (1974), Methods for statistical analysis of reliability and life data, Wiley.

Mazucheli, J., Louzada, F. & Ghitany, M. (2013), ‘Comparison of estimation methods for the parameters of the weighted lindley distribution’, Applied Mathematics and Computation 220, 463–471.

Merovci, F. (2013a), ‘Transmuted generalized rayleigh distribution’, Journal of Statistics Applications and Probability 2(3), 1–12.

Merovci, F. (2013b), ‘Transmuted lindley distribution’, International Journal of Open Problems in Computer Science & Mathematics 6(2), 63–72.

Merovci, F. (2013c), ‘Transmuted Rayleigh distribution’, Austrian Journal of Statistics 41(1), 21–31.

Merovci, F., Elbatal, I. & Ahmed, A. (2014), ‘The Transmuted Generalized Inverse Weibull Distribution’, Austrian Journal of Statistics 43(2), 119–131.

Merovci, F. & Puka, L. (2014), ‘Transmuted Pareto distribution’, ProbStat Forum 7, 1–11.

Nadarajah, S., Cordeiro, G. M. & Ortega, E. M. (2013), ‘The exponentiated weibull distribution: a survey’, Statistical Papers 54(3), 839–877.

Pettitt, A. (1976), ‘A two-sample Anderson-Darling rank statistic’, Biometrika 63(1), 161.

Ranneby, B. (1984), ‘The Maximum Spacing Method. An Estimation Method Related to the Maximum Likelihood Method’, Scandinavian Journal of Statistics

(2), pp. 93–112.

Rayleigh, J. W. S. (1880), ‘On the resultant of a large number of vibrations of the some pitch and of arbitrary phase’, Philosophical Magazine 5th Series(10), 73–78.

Rényi, A. (1961), ‘On measures of entropy and information’, Entropy 547(c), 547–561. http://www.maths.gla.ac.uk/ tl/Renyi.pdf

Saidane, S., Babai, M., Aguir, M. & Korbaa, O. (2010), Spare parts inventory systems under an increasing failure rate demand interval distribution, in ‘41st International Conference on Computers and Industrial Engineering’, pp. 768–773.

Salinas, H. S., Iriarte, Y. A. & Bolfarine, H. (2015), ‘Slashed exponentiated rayleigh distribution’, Revista Colombiana de Estadística 38(2), 453–466.

Shaked, M. & Shanthikumar, J. G. (1994), Stochastic orders and their applications, Academic Press, Boston, MA.

Shannon, C. E. (1948), ‘Prediction and Entropy of Printed English’, Bell System Technical Journal 30(1), 50–64.

Shaw, W. T. & Buckley, I. R. (2007), The Alchemy of Probability Distributions: Beyond Gram Charlier & Cornish Fisher Expansions, and Skew-Normal or Kurtotic-Normal Distributions, Technical report, Financial Mathematics

Group, King’s College, London, U.K.

Singh, S. K., Singh, U. & Sharma, V. K. (2013), ‘Bayesian analysis for Type-II hybrid censored sample from inverse Weibull distribution’, International Journal of System Assurance Engineering and Management 4(3), 241–248.

Song, K. S. (2001), ‘Rényi information, loglikelihood and an intrinsic distribution measure’, Journal of Statistical Planning and Inference 93(1-2), 51–69.

Stephens, M. A. (2013), ‘EDF Statistics for Goodness of Fit and Some Comparisons’, Journal of the American Statistical Association 69(347), 730–737.

Swain, J. J., Venkatraman, S. & Wilson, J. R. (1988), ‘Least-squares estimation of distribution functions in johnson’s translation system’, Journal of Statistical Computation and Simulation 29(4), 271–297.

Teimouri, M., Hoseini, S. M. & Nadarajah, S. (2013), ‘Comparison of estimation methods for the Weibull distribution’, Statistics 47(1), 93–109.

Tian, Y., Tian, M. & Zhu, O. (2014), ‘Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution’, Communications

in Statistics-Simulation and Computation 43(10), 2661–2677.

Tsarouhas, P. & Arvanitoyannis, I. (2007), ‘Reliability and maintainability analysis of bread production line’, Critical Reviews in Food Science and Nutrition 50, 327–343.

Woosley, R. & Cossman, J. (2007), ‘Drug development and the FDAs critical path initiative’, Public policy 81, 129–133.

How to Cite

APA

Dey, S., Raheem, E. and Mukherjee, S. (2017). Statistical properties and different methods of estimation of transmuted Rayleigh distribution. Revista Colombiana de Estadística, 40(1), 165–203. https://doi.org/10.15446/rce.v40n1.56153

ACM

[1]
Dey, S., Raheem, E. and Mukherjee, S. 2017. Statistical properties and different methods of estimation of transmuted Rayleigh distribution. Revista Colombiana de Estadística. 40, 1 (Jan. 2017), 165–203. DOI:https://doi.org/10.15446/rce.v40n1.56153.

ACS

(1)
Dey, S.; Raheem, E.; Mukherjee, S. Statistical properties and different methods of estimation of transmuted Rayleigh distribution. Rev. colomb. estad. 2017, 40, 165-203.

ABNT

DEY, S.; RAHEEM, E.; MUKHERJEE, S. Statistical properties and different methods of estimation of transmuted Rayleigh distribution. Revista Colombiana de Estadística, [S. l.], v. 40, n. 1, p. 165–203, 2017. DOI: 10.15446/rce.v40n1.56153. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/56153. Acesso em: 19 apr. 2024.

Chicago

Dey, Sanku, Enayetur Raheem, and Saikat Mukherjee. 2017. “Statistical properties and different methods of estimation of transmuted Rayleigh distribution”. Revista Colombiana De Estadística 40 (1):165-203. https://doi.org/10.15446/rce.v40n1.56153.

Harvard

Dey, S., Raheem, E. and Mukherjee, S. (2017) “Statistical properties and different methods of estimation of transmuted Rayleigh distribution”, Revista Colombiana de Estadística, 40(1), pp. 165–203. doi: 10.15446/rce.v40n1.56153.

IEEE

[1]
S. Dey, E. Raheem, and S. Mukherjee, “Statistical properties and different methods of estimation of transmuted Rayleigh distribution”, Rev. colomb. estad., vol. 40, no. 1, pp. 165–203, Jan. 2017.

MLA

Dey, S., E. Raheem, and S. Mukherjee. “Statistical properties and different methods of estimation of transmuted Rayleigh distribution”. Revista Colombiana de Estadística, vol. 40, no. 1, Jan. 2017, pp. 165-03, doi:10.15446/rce.v40n1.56153.

Turabian

Dey, Sanku, Enayetur Raheem, and Saikat Mukherjee. “Statistical properties and different methods of estimation of transmuted Rayleigh distribution”. Revista Colombiana de Estadística 40, no. 1 (January 1, 2017): 165–203. Accessed April 19, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/56153.

Vancouver

1.
Dey S, Raheem E, Mukherjee S. Statistical properties and different methods of estimation of transmuted Rayleigh distribution. Rev. colomb. estad. [Internet]. 2017 Jan. 1 [cited 2024 Apr. 19];40(1):165-203. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/56153

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2. Mahendra Saha, Harsh Tripathi, Sanku Dey. (2021). Single and double acceptance sampling plans for truncated life tests based on transmuted Rayleigh distribution. Journal of Industrial and Production Engineering, 38(5), p.356. https://doi.org/10.1080/21681015.2021.1893843.

3. Sanku Dey, Fernando A. Moala, Devendra Kumar. (2018). Statistical properties and different methods of estimation of Gompertz distribution with application. Journal of Statistics and Management Systems, 21(5), p.839. https://doi.org/10.1080/09720510.2018.1450197.

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