How to Estimate Statistical Characteristics Based on a Sample: Nonparametric Maximum Likelihood Approach Leads to Sample Mean, Sample Variance, etc

In many practical situations, we need to estimate different statistical characteristics based on a sample. In some cases, we know that the corresponding probability distribution belongs to a known finite-parametric family of distributions. In such cases, a reasonable idea is to use the Maximum Likelihood method to estimate the corresponding parameters, and then to compute the value of the desired statistical characteristic for the distribution with these parameters. In some practical situations, we do not know any family containing the unknown distribution. We show that in such nonparametric cases, the Maximum Likelihood approach leads to the use of sample mean, sample variance, etc. 1 Need to Estimate Statistical Characteristics Based on a Sample: Formulation of the Problem Need to estimate statistical characteristics. In many practical situations, we need to estimate statistical characteristic of a certain random phenomenon based on a given sample. For example, to check that for all the mass-produced gadgets from a given batch, the valued of the corresponding physical quantity are within the desired bounds, the ideal solution would be to measure the quantity for all the gadgets. This may be reasonable to do if these gadgets are intended for a spaceship, where a minor fault can lead to catastrophic results. However, in most applications, it is possible Vladik Kreinovich Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, Texas 79968, USA, e-mail: [email protected] Thongchai Dumrongpokaphan Department of Mathematics, Faculty of Science, Chiang Mai University, Thailand, e-mail: [email protected]