The method of maximum likelihood

This thesis is the comprehensive study of the method of maximum likelihood and its relative merit over other methods of estimation. This method of estimation, developed by R. A. Fisher in 1921 is the oldest method. Since that time, Fisher and some others have introduced wide successive developments which led the maximum likelihood method to be used in most practical applications. In chapters I and II (where single and several parameters are considered) it has been shown that the method of maximum likelihood has all the properties of the best method of estimation; that is, the estimators of the maximum likelihood method have the property of consistency, and they are asymptotically most efficient, having normal distribution and also they are unbiased estimators. Also it has been shown that if a sufficient estimator exists, then the method of maximum likelihood affords it. The inequality of Fisher has been discussed which supplies the maximum attainable variance when the equality holds, There has also been discussed the process of the successive approximations by which the maximum likelihood estimates can be obtained in cases when the maximum likelihood equations are difficult to be solved. She Wald technique and Lagrange multiplier technique are explained for estimating the unrestricted and the restricted parameters with their tests respectively. In chapter III there has been shown the practical applications of the method of maximum likelihood. In the field of genetics we applied some other methods in addition to the maximum likelihood method and we saw that the estimates of this method are the most efficient. In the field of bioassay we have shown the applications of the method of maximum likelihood for estimating the two parameters using the probit- transformation and the logistic formula. In the field of blood groups, the application of the maximum likelihood method has been shown for estimating the three parameters. We have mentioned the Bernstien method and applied both the Wald and the lagrange multiplier techniques for estimating the unrestricted and the restricted parameters. In chapter IV we discussed the likelihood ratio test which is frequently unbiased and based on a sufficient statistic and also it is the uniformly most powerful test. In virtue of the desirable properties mentioned above, this test becomes more accurate for testing the statistical hypothesis than the others.