Estimation of parameters of Alpha Logarithm Transformed Rayleigh distribution by using Maximum Likelihood Estimation method
Author(s):
M. Vijaya Lakshmi, K. V. Subrahmanyam, G. V. S. R. Anjaneyulu
Keywords:
ALTR, MLE, Average Estimate (AE), Variance (VAR), Mean Absolute Deviation (MAD), Mean Square Error (MSE), Relative Absolute Bias (RAB) and Relative Efficiency (RE), Asymptotic confidence bounds.
Abstract
In this paper, we review the maximum likelihood method for estimating the statistical parameters which specify a probabilistic model and show that it generally gives an optimal estimator with minimum mean square error asymptotically. Thus, for most applications in information sciences, the maximum likelihood estimation suffices. Fisher information matrix, which defines the orthogonality between parameters in a probabilistic model, naturally arises from the maximum likelihood estimation. The parameters involved in the model are estimated using Maximum Likelihood Estimation (MLE) method. Maximum Likelihood estimation method under some risk function to estimate parameter of Alpha Logarithm Transformed Rayleigh distribution to know the best method. As the inverse of the Fisher information matrix gives the covariance matrix for the estimation errors of the parameters, the orthogonalization of the parameters guarantees that the estimates of the parameters distribute independently from each other. Finally, the proposed extended model is applied on real data sets and the results are given which illustrate the superior performance of the ALTR distribution compared to some other well-known distributions. . We present MLE of the unknown parameters in this distribution using Newton-Raphson. We also computed Average Estimate (AE), Variance (VAR), Mean Absolute Deviation (MAD), Mean Square Error (MSE), Relative Absolute Bias (RAB) and Relative Efficiency (RE) for both the parameters under sample based on 1000 simulations to assess the performance of the estimators. Also we derive the asymptotic confidence bounds for unknown parameters.
