PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATES OF THE EXPONENT OF PARETO DISTRIBUTION - I. S. Pulkin; A. Tatarintsev (Российский технологический журнал)
This paper investigates the statistical properties of maximum likelihood estimation index of the Pareto distribution. In recent years, power distribution laws such as Pareto distribution attract the attention of researchers in various fields of science and technology, from economics and linguistics to Internet traffic analysis. Therefore, the problem of determining the exponent of the power law for a given sample is of exceptional practical importance. It is analytically proved that this estimate is biased, although valid. A formula that eliminates the bias is proposed. Besides, a formula for the variance of the unbiased estimate is analytically derived. In addition, the problem of finding the distribution function and probability density of this estimate as a random variable is set and analytically solved. Next, a formula for mathematical expectation and dispersion based on previously determined probability density is found. The obtained results can be used in various fields of human activity, for example, to predict the intensity of natural and man-made disasters.