Modification of weibull-rayleigh distributions and their applications

Abstract

The objectives of this thesis are to propose the length-biased Weibull-Rayleigh distribution and the mixed Weibull-Rayleigh distribution modified from the Weibull-Rayleigh distribution and investigation their properties and to apply the length-biased Weibull-Rayleigh distribution and the mixed Weibull-Rayleigh distribution for modeling rainfall and runoff data. The statistical properties of the length-biased Weibull-Rayleigh distribution and the mixed Weibull-Rayleigh distribution such as survival function, hazard rate function, the r th moment, mean, variance, skewness, and kurtosis were investigated and were presented in this thesis. Moreover, we derived the parameter estimations of the length-biased Weibull-Rayleigh and mixed Weibull-Rayleigh distributions based on the maximum likelihood estimation, method of moments, Bayesian approach, maximum product of spacing estimators, Anderson-Darling minimum distance estimators and Cramer-von Mises minimum distance estimators. A Monte Carlo simulation was used to compare the efficiency of parameter estimation methods for estimating parameters of the length-biased Weibull-Rayleigh and mixed Weibull-Rayleigh distributions. In most of the scenarios, the mean square error values of the sample mean of the parameter estimation based on the method of moments and Bayesian approach were smaller than other methods, while the bias values of the sample means of the parameter estimations based on the Anderson-Darling minimum distance estimators and Cramer-von Mises minimum distance estimators methods were closer to zero than other methods. Applications of the length-biased Weibull-Rayleigh distribution were conducted on hydrological datasets, that are the annual flood discharge rates data and rainy season rainfall data. Judging from the Kolmogorov-Smirnov statistic, Akaike information criterion and Bayesian information criterion values, the length-biased Weibull-Rayleigh distribution provided the best fit for two hydrological datasets. Additionally, the mixed Weibull-Rayleigh distribution was applied to October rainfall data. From the Kolmogorov-Smirnov statistic and Akaike information criterion values, the results showed that the mixed Weibull-Rayleigh distribution gave the best fit for October rainfall data. In addition, we used the annual maximum rainfall data from Mae Taeng, Muang Chiang Mai, Samoeng, and Muang Lamphun stations, as well as the annual maximum runoff data from Chiang Dao, Phrao, Mae Tha, and Chom Thong stations, to predict return levels and return periods based on the length-biased Weibull-Rayleigh and mixed Weibull-Rayleigh distributions. The return periods based on the length-biased Weibull-Rayleigh and mixed Weibull-Rayleigh distributions for the annual maximum rainfall data showed that all rain gauge stations were in a high hazard class of the flood. Therefore, all rain gauge stations could be a high risk of flooding. The return periods based on the length-biased Weibull-Rayleigh and mixed Weibull-Rayleigh distributions for the annual maximum runoff data showed that most runoff gauge stations had a moderate hazard class of the flood. Therefore, most runoff gauge stations could be a moderate risk of flooding.