PARETO EXPONENT ESTIMATION USING ROLLING REGRESSION AND NONPARAMETRIC METHODS

Abstract

InthisstudywehavecheckedthevalidityofZipf’sLawforcitysizedataofofU.S,China, Pakistan and India. Zipf’s Law says that, the distribution of city sizes follows a Power Law distribution with shape parameter equal to 1. We have used two-step approach to check the validityofZipf’sLaw,whereinfirststepwetest(usinggoodnessoffittest)ifthedistribution of city sizes follow a Power Law distribution and in second step, we estimate the Power Law exponent whether its value is equal to unity or not. The HILL, OLS, MOLS, ML and MVU estimation techniques are considered for the estimation purposes. Graphical display is presented to overview the nature of the city size data sets. The Kolomogrove-Simirnov (KS) goodness of fit test is applied to check the distributions of the all the data sets, assuming the Power Law distribution under null hypothesis. The KS statistics is also used to estimate the minimum threshold values. Simulation study is carried out to point out an efficient estimator for the estimation of the Power Law exponent. Base on the bootstrap simulation we conclude that minimum variance unbiased estimator (MVUE) is more efficient and unbiased. The range, in which the exponent value is one, is to be found through rolling sampling technique under the considered estimation methods. A nonparametric analysis is carried out to give a more detailed description of the Power Law exponent. It will be shown, through kernel density plots (a nonparametric technique), that the Power Law exponent distribution is uni-modal.