Abstract
Statistical inference for an unknown single parameter θ of a population distribution often involves finding a solution to a non-linear equation f(θ)=0. Computation of the maximum likelihood estimate and calculation of confidence limits of a confidence interval are such examples. Instead of directly solving f(θ)=0 a promising iterative method is to solve an equivalent fixed-point equation θ=g(θ), which can be obtained by reformulating the original equation.
In this paper, we consider the iterative method θt+1=g(θt) in detail. A sufficient condition for unique convergence has been given in terms of Lipschitz condition, which essentially states that|g′(θ)| should be less than 1 in a closed interval I. An alternative sufficient condition which is some-what weaker than the condition above is given in this article, in which the slope of g(θ) is allowed to be greater than 1. A simple example numerically illustrates the behaviors of the present iterative method in some detail.
Interesting statistical examples are also shown to demonstrate that the present iterative method is useful and very powerful. An example comes from a classroom exercise and the others are taken from recent statistical literature. One of the messages of this paper is to emphasize the importance of numerical computation in everyday statistical data analysis and also in education and training of statistical methodologies.
