Exponentiated Weibull Distribution for Modeling the Vehicle Headway Data based on Ranked Set Sampling

Year 2020, Volume: 33 Issue: 4, 892 - 902, 01.12.2020

Büşra Sevinç , Selma Gurler , Melek Esemen

https://doi.org/10.35378/gujs.681465

Cited By: 1

Abstract

Modeling the vehicle headway data is fundamental for intelligent transportation applications in traffic engineering. It is useful for the traffic signal optimization and flow modelling. Exponentiated Weibull (EW) is one of the best flexible model for characterizing uncertainty in various fields of data. In this study, we revisit EW distribution and propose to use of ranked set sampling as a useful sampling method for estimating the unknown parameters. We deal with the performance of ranked set sampling and simple random sampling methods by a simulation study in R-software in terms of mean squared errors. We estimate the parameters of EW distribution using the maximum likelihood method under the assumption that all parameters are unknown. We illustrate the flexibility and the usefulness of EW distribution by analysing generated data from a real application study in transportation field.

Keywords

Exponentiated Weibull, Ranked set sampling, Parameter estimation, Maximum likelihood, Headway modeling

References

• Mudholkar, G. S., & Srivastava D. K., “Exponentiated Weibull family for analyzing bathtub failure-rate data.” IEEE Transactions on Reliability 42(2):299–302, (1993).
• Pal M., Ali M. M., & Woo J., “Exponentiated Weibull distribution", Statistica, 66(2):13-147, (2006).
• Shittu, O. I., & Adepoju. K. A., “On the exponentiated Weibull distribution for modeling wind speed in south western Nigeria.” Journal of Modern Applied Statistical Methods 13(1):431–445, (2014).
• Elshahhat, A., “Parameters estimation for the exponentiated Weibull distribution based on generalized progressive hybrid censoring schemes.” American Journal of Applied Mathematics and Statistics 5(2):33–48, (2017).
• Ghnimi, S., & Gasmi. S., “Parameter estimations for some modifications of the Weibull distribution.” Open Journal of Statistics 4(8):597–610, (2014).
• Mudholkar, G. S., Srivastava, D. K., & Freimer, M., “The exponentiated Weibull family: A reanalysis of the bus-motor-failure data.” Technometrics 37(4):436–445, (1995).
• Achcar, J. A., Rodriguez, G. O., & Rodrigues, E. R., “Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model and a Metropolis–Hastings algorithm.” International Journal of Pure and Applied Mathematics 53(1):1–20, (2009).
• Barrios, R., & Dios, F., “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence.” Optics & Laser Technology 45:13–20, (2013).
• Cancho, V. G., & Bolfarine H., “Modeling the presence of immunes by using the exponentiated-Weibull model.” Journal of Applied Statistics 28(6):659–671, (2001).
• Jiang, R., “Discrete competing risk model with application to modeling bus-motor failure data.” Reliability Engineering & System Safety 95(9):981–988 (2010).
• Mudholkar, G. S., & Hutson, A. D., “The exponentiated Weibull family: Some properties and a flood data application.” Communications in Statistics–Theory and Methods 25(12):3059–3083, (1996).
• Nassar, M. M., & Eissa F. H., “On the exponentiated Weibull distribution.” Communications in Statistics-Theory and Methods 32(7):1317–1336, (2003).
• Almalki, S. J., & Nadarajah, S., “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety 124:32–55, (2014).
• Nadarajah, S., Cordeiro, G. M., & Ortega, E. M., “The exponentiated Weibull distribution: A survey.” Statistical Papers 54 (3): 839–877, (2013).
• McIntyre, G.A., “A method for unbiased selective sampling, using ranked sets.” Australian Journal of Agricultural Research 3(4): 385–390, (1952).
• Takahasi, K., &Wakimoto, K.,“On unbiased estimates of the population mean based on the sample stratified by means of ordering.” Annals of the Institute of Statistical Mathematics 20(1): 1–31, (1968).
• Lam, K., Sinha, B. K., & Wu, Z., “Estimation of parameters in a two-parameter exponential distribution using ranked set sample.” Annals of the Institute of Statistical Mathematics 46(4): 723–736, (1994).
• Hassan, A. S., “Maximum likelihood and Bayes estimators of the unknown parameters for exponentiated exponential distribution using ranked set sampling.” International Journal of Engineering Research and Applications 3:720–725, (2013).
• Esemen, M., & Gürler, S., “Parameter estimation of generalized Rayleigh distribution based on ranked set sample.” Journal of Statistical Computation and Simulation 88(4):615–628, (2018).
• Al-Omari, A. I., & Bouza, C. N., “Review of ranked set sampling: Modifications and applications.” Investigación Operacional 35(3):215–241, (2014).
• Barabesi, L., & El-Sharaawi, A., “The efficiency of ranked set sampling for parameter estimation.” Statistics & probability letters 53(2):189–199, (2001).
• Qian, W., Chen, W., & He, X., “Parameter estimation for the Pareto distribution based on ranked set sampling.” Statistical Papers 1–23, (2019).
• Shaibu, A. B., & Muttlak, H. A., “Estimating the parameters of the normal, exponential and gamma distributions using median and extreme ranked set samples.” Statistica, 64(1):75-98, (2004).
• Badhrudeen, M., Ramesh, V., & Vanajakshi, L., “Headway analysis using automated sensor data under Indian traffic conditions.” Transportation Research Procedia 17:331–339, (2016).
• Riccardo, R., & Massimiliano, G. “An empirical analysis of vehicle time headways on rural two-lane two-way roads.” Procedia-Social and Behavioral Sciences 54:865–874, (2012).
• Abtahi, S. M., Tamannaei, M., & Haghshenash, H., “Analysis and modeling time headway distributions under heavy traffic flow conditions in the urban highways: Case of Isfahan.” Transport 26(4):375–382, (2011).
• Al-Ghamdi, A. S., “Analysis of time headways on urban roads: Case study from Riyadh.” Journal of Transportation Engineering 127(4):289–294, (2001).
• Greenberg, L., “The log normal distribution of headways.” Australian Road Research 2(7):14–18, (1966).
• Yin, S., Li, Z., Zhang, Y., Yao, D., Su, Y., & Li L., “Headway distribution modeling with regard to traffic status.” In 2009 IEEE Intelligent Vehicles Symposium, 1057–1062, (2009).
• Dey, P. P., & Chandra, S., “Desired time gap and time headway in steady-state car-following on two-lane roads.” Journal of transportation engineering 135(10):687–693, (2009).
• Jang, J., “Analysis of time headway distribution on suburban arterial.” KSCE Journal of Civil Engineering 16(4):644–649, (2012).
• Panichpapiboon, S., “Time-headway distributions on an expressway: Case of Bangkok.” Journal of Transportation Engineering 141(1):1–8, (2014).
• Li, L., & Chen, X. M., “Vehicle headway modeling and its inferences in macroscopic/microscopic traffic flow theory: A survey.” Transportation Research Part C: Emerging Technologies 76:170–188, (2017).