A simple criterion for the effectiveness of biased estimators

Abstract

Point estimation is one of the most common forms of statistical inference. This paper touches upon the concepts of unbiasedness and effectiveness of biased estimates. An attractive approximation for the true (quantitative) value of the estimated parameter t whose values belong to a certain numerical set t T is an estimate , for which the sum of the mathematical expectation of the square of the differences between the possible realization of and the estimated parameter t is minimal (least squares estimation). Another approach is to find the value of the parameter t based on the sum of the mathematical expectation of the difference between the possible realisation ϵ = xi and the estimated parameter t, provided that this sum is zero, so that the positive and negative differences are balanced ∑(xi – t) = 0. Implementations of N independent random values have a general distribution that depends on the estimated parameter t. There are other approaches, but at present there is no single convincing definition of the optimality of finding effective estimates. The well-known statistician Box wrote: “All models are wrong, some are useful.” It is not always easy to establish the adequacy of a model, so our choice is defined by its utility! The simple effectiveness criterion of biased estimates given below is too convenient, however this does not mean that the assumption of the effectiveness of biased estimates based on this criterion is true, yet the usefulness of this simple and convenient model is obvious. The Aim of the paper. The paper aims to construct a simple effectiveness criterion of biased estimates and obtain simple effective estimates of dependability indicators for a binomial test plan and a test plan with limited time and recovery using the constructed simple effectiveness criterion of biased estimates. Conclusions. A simple effectiveness criterion of biased estimates was obtained that compensates for the effect of unequal dispersion values and the square of the bias. New effective biased estimates of various dependability indicators were obtained for a binomial test plan and a plan with limited test time and recovery of failed items.