Slash Maxwell Distribution: Definition, Modified Maximum Likelihood Estimation and Applications

Year 2020, Volume: 33 Issue: 1, 249 - 263, 01.03.2020

Şükrü Acıtaş Talha Arslan Birdal Şenoğlu

https://doi.org/10.35378/gujs.539929

Cited By: 2

Abstract

In this study slash Maxwell (SM) distribution, defined as a ratio of a Maxwell random variate to a power of an independent uniform random variate, is introduced. Its stochastic representation and some distributional properties such as moments, skewness and kurtosis measures are provided. The maximum likelihood (ML) method is used for estimating the unknown parameters. However, closed forms of the ML estimators cannot be obtained since the likelihood equations include nonlinear functions of the unknown parameters. We therefore use Tiku's (1967,1968) modified maximum likelihood (MML) methodology which allows to obtain explicit forms of the estimators. Some asymptotic properties of the MML estimators are derived. A Monte-Carlo simulation study is also carried out to compare the performances of the ML and MML estimators. Two data sets taken from the literature are modelled using the SM distribution in application part of the study. 

Keywords

Maxwell, Slashing methodology, Modified maximum likelihood, Stochastic representation, Modified maximum likelihood

References

• Acitas, S., Kasap, P., Senoglu, B. and Arslan, O., 2013a. One-step M-esti- mators: Jones and Faddy’s skew t distribution. J Appl Stati 40(7), 1545–1560.
• Acitas, S., Kasap, P., Senoglu, B., Arslan, O., 2013b. Robust estimation with the skew t2 distribution. Pak J Stat 29(4), 409–430.
• Acitas, S., Arslan, T. and Senoglu, B., 2017. Scale mixture extension of the Maxwell distribution: Properties, estimation and application, 10th International Statistics Congress, Ankara, Turkey.
• Al-Baldawi, T.H.K., 2013. Comparison of maximum likelihood and some Bayes estimators for Maxwell distribution based on non-informative priors. Baghdad Science Journal 10(2), 480–488.
• Arslan, O., 2008. An alternative multivariate skew-slash distribution. Statist. Probab. Lett. 78, 2756–2761.
• Arslan, O., 2009. Maximum likelihood parameter estimation for the multivariate skew-slash distribution. Statist. Probab. Lett. 79, 2158–2165.
• Arslan, O. and Genc, A.I., 2009. A generalization of the multivariate slash distribution. J. Statist. plann. Inference 139, 1164–1170.
• Arslan, T., Acitas, S. and Senoglu, B., 2017. Estimating the location and scale parameters of the Maxwell distribution, Data Science, Statistics and Visualisation Conference, Lisboa.
• Astorga, J.M., Gomez, H.W. and Bolfarine, H., 2017. Slashed generalized exponential distribution, Communications in Statistics - Theory and Methods 46(5), 2091–2102.
• Barnett, V.D., 1966. Evaluation of the maximum likelihood estimator when the likelihood equation has multiple roots. Biometrika 53, 151–165.
• Bartlett, M.S., 1953. Approximate confidence intervals., Biometrika 40, 12–19.
• Bjerkedal, T., 1960. Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am. J. Hyg. 72, 130–148.
• Bowman, K.O. and Shenton, L.R., 2001. Weibull distributions when the shape parameter is defined. Computational Statistics and Data Analysis 36, 299-310.
• Dey, S., Maiti, S.S., 2010. Bayesian estimation of the parameter of Maxwell distribution under the different loss functions. Journal of Statistical Theory and Practice 4(2), 279–287.
• Dey, S., Dey, T., Ali, S. and Mulekar, M.S., 2016. Two parameter Maxwell distribution: Properties and different methods of estimation. Journal of Statistical Theory and Practice 10(2), 291–300.
• Fan, G., 2016. Estimation of the loss and risk functions of parameter of Maxwell distribution. Science Journal of Applied Mathematics and Statistics 4(4), 129–133.
• Genc, A.I., 2007. A generalization of the univariate slash by a scale- mixtured exponential power distribution. Communications in Statistics - Simulation and Computation 36(5), 937–947.
• Gomez, H.W., Quintana, F.A., and Torres, J., 2007. A new family of slash-distributions with elliptical contours. Statist. Probab. Lett. 77, 717–725.
• Gomez, H.W. and Venegas, O., 2008. Erratum to: A new family of slash-distributions with elliptical contours. Statist. Probab. Lett. 78, 2273–2274.
• Gomez, H.W., Olivares-Pacheco, J.F. and Bolfarine, H., 2009. An extension of the generalized Birnbaun- Saunders distribution. Stat Probab Lett. 79, 331–338.
• Hossain, A.M. and Huerta, G., 2016. Bayesian estimation and prediction for the Maxwell failure distribution based on Type II censored data. Open Journal of Statistics 6, 49–60.
• Iriarte, Y.A., Castillo, N.O., Bolfarine, J. and Gomez H.W., 2017. Modified slashed-Rayleigh distribution. Communications in Statistics - Theory and Methods, DOI: 10.1080/03610926.2017.1353621.
• Johnson, N. L., Kotz, S. and Balakrishnan. N., 1994. Continuous univariate distributions. Wiley, New York.
• Kafadar, K., 1982. A biweight approach to the one-sample problem. J. Amer. Statist. Assoc. 77, 416–424.
• Kantar, Y.M. and Senoglu, B., 2008. A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter. Computers and Geosciences 34(12), 1900–1909.
• Kazmi, S.M.A., Aslam, M. and Ali, S., 2011. A note on the maximum likelihood estimators for the mixture of Maxwell distributions using Type-I censored scheme. The Open Statistics and Probability Journal 3, 31–35.
• Kendall, M.G., Stuart, A., 1961. The Advanced Theory of Statistics, Vol. 2. Charles Griffin and Co., London.
• Korkmaz M.Ç., 2017. A generalized skew slash distribution via gamma- normal distribution. Communications in Statistics - Simulation and Computation 46(2), 1647–1660.
• Li, L., 2016. Minimax estimation of the parameter of Maxwell distribution under the different loss functions. American Journal of Theoretical and Applied Statistics 5(4), 202–207.
• Mahdavi, A. and Jabbari, L., 2017. An extended weighted exponential distribution. Journal of Modern Applied Statistical Methods 16(1), 296–307, DOI: 10.22237/jmasm/1493597760.
• Mosteller, F. and Tukey, J. W., 1977. Data analysis and regression. Addison-Wesley, Reading, MA.
• Olmos, N.M., Varela, H.H., Gomez, W. and Bolfarine, H., 2012. An extension of the half-normal distribution. Statistical Papers 53(4), 875–886.
• Olmos, N.M., Varela, H., Bolfarine, H., Gomez, H.W., 2014. An extension of the generalized half- normal distribution. Stat Pap. 55(4), 967–981.
• Punathumparambath, B., 2011. A new family of skewed slash distributions generated by the normal kernel. Statistica 71(3), 345–353.
• Punathumparambath, B., 2012. The multivariate asymmetric slash Laplace distribution and its applications. Statistica 72(2), 235–249.
• Punathumparambath, B., 2013. A New Familiy of Skewed Slash Distributions Generated by the Cauchy Kernel. Communications in Statistics-Theory and Methods, 42(13), 2351–2361.
• Puthenpura, S. and Sinha, N.K., 1986. Modified maximum likkelihood method for the robust estimation of system parameters from very noise data. Automatica, 22, 231–235.
• Rogers, W. H. and Tukey, J. W., 1972. Understanding some long- tailed symmetrical distributions. Statist. Neerlandica 26, 211–226.
• Senoglu, B., 2007. Estimating parameters in one-way analysis of covarinace model with short-tailed symmetric error distributions. Journal of Computational and Applied Mathematics 201, 275–283.
• Tahir, M. H., Cordeiro, G. M., Mansoor, M. and Zubair, M., 2015. The Weibull- Lomax distribution: properties and applications. Hacettepe Journal of Mathematics and Statistics 44(2), 455–474.
• Tiku, M. L., 1967. Estimating the mean and standard deviation from a censored normal sample. Biometrika 54, 155–165.
• Tiku, M. L., 1968. Estimating the parameters of Normal and Logistic distributions form censored samples. Aust. J. Stat. 10, 64–74.
• Tiku, M.L., 1982. Testing linear contrast of means in experimental design without assuming normality and homogeneity of variances. Biometrical Journal 24(6), 613–627.
• Tyagi, R.K., Battacharya, S.K., 1989a. Bayes estimation of the Maxwell’s velocity distribution function. Statistica 4, 563–567.
• Tyagi, R.K., Battacharya, S.K., 1989b. A Note on the MVU estimation of the Maxwell’s failure distribution. Estadistica 41, 73–79.
• Vaughan, D.C., 1992. On the Tiku-Suresh method of estimation. Commun. Stat.-Theory Meth. 21(2), 451–469.
• Wang, J. and Genton, M. G., 2006. The multivariate skew-slash distribu- tion. J. Statist. plann. Inference 136, 209–220.