Abstract
The Topp-Leone (TL) family is one of the methods used to generalize probability distributions to make them more flexible in application. This generalization was used to generalize the Exponentiated Rayleigh (ER) distribution, this distribution called Topp Leone Rayleigh (TLR). The statistical properties of the new distribution were studied represented by the r-order moments around zero, median, mode, quartiles, and moment generating function, skewness and kurtosis. The parameters of distribution were estimated by the maximum likelihood (ML) method. The results of the estimation method were applied to data generated with different sample sizes and values for the parameters. With the application of data represented by the lengths of the saturated carbon fibres, the research reached the most important conclusions, including that the scale parameter does not affect the skewness and kurtosis measures. While the additional parameter affected the two measures. The TLER distribution became more flexible than the ER distribution. On the practical side, it was found that the mean square error (MSE) decreases with increasing sample size. When the sample size is n = 25,50,100 and the scale parameter is fixed, and When λ = 0.5,1,2 it was found that the estimators when α = 0.75 are the best for having the smallest MSE. The lengths of carbon fibres follow the TLER distribution.
