Efficiency criterion of biased estimates. A new take on old problems

The perfect case estimation scenario involves unbiased estimation with minimal variance, if such estimate exists. Currently, there are no means of obtaining unbiased estimates (if they do exist!). For instance, a maximum likelihood estimate (NBT test plan) of a mean time to failure T_mn = (total operation time)/(number of failures) is highly biased. Those involved in solving applied problems are not satisfied with the situation. Efficient unbiased estimates are used whenever such are available. If it is impossible to find an efficient unbiased estimate in terms of standard deviation, then biased estimate comparison is to be mastered. The vast majority of problems is associated with biased estimates. Within the class of biased estimates, estimates with minimal bias are to be sought, and, among the latter, those with minimal bias. Such estimates in the class of biased estimates should be called bias-efficient or simply efficient, which does not contradict the conventional definition, but only extends it. Such search process guarantees that the obtained estimates are highly accurate. However, with this definition of a bias-efficient estimate, there will always be a pair of compared estimates, in which the total bias of one estimate is slightly higher than that of the other, the same being the case with the total variances of such estimates, but in a different order. In this setting, a formal selection of a bias-efficient estimate becomes impossible and is arbitrary, i.e., the test engineer selects a bias-efficient estimate intuitively. In this case, the test engineer’s choice may prove to be incorrect. Thus arises the problem of constructing a criterion of efficiency that would enable a formal selection of a bias-efficient estimate. The Aim of the paper. The paper aims to build an efficiency criterion, using which the choice of a bias-efficient estimate is unambiguously defined through computation. Methods of research. To find the bias-efficient estimate, we used integral numerical characteristics of the accuracy of the estimate, namely, the total square of the offset of the expected implementation of a certain variant estimate from the examined parameters of the distribution laws, etc. Conclusions. 1) For the binomial plan and the test plan with recovery and limited test time, performance criteria were constructed that allow unambiguously identifying the bias-efficient estimate out of the submitted estimates. 2) Based on the constructed performance criteria for various test plans, bias-efficient estimates were selected out of the submitted ones.

1. Yasnogorodsky R.M. [Probability theory and mathematical statistics. Textbook]. Saint Petersburg: Naukoyomkiye tekhnologii; 2019. (in Russ.)

2. Mikhailov V.S., Yurkov N.K. [Integral estimation in the dependability theory. Introduction and main findings]. TEKHNOSFERA; 2020. (in Russ.)

3. Pugachiov V.S. [Probability theory and mathematical statistics: textbook: 2-nd edition, corrected and extended]. Moscow: FIZMATLIT; 2002. (in Russ.)

4. Borovkov A.A. [Probability theory]. Moscow: Editorial URSS; 1999. (in Russ.)

5. Shulenin V.P. [Mathematical statistics. Part 1. Parametric statistics]. Tomsk: Izdatelstvo NTL; 2012. (in Russ.)

6. Barzilovich E.Yu., Beliaev Yu.K., Kashtanov V.A. et al. Gnedenko B.V., editor. [Matters of mathematical dependability theory]. Moscow: Radio i sviaz; 1983. (in Russ.)

7. Gnedenko B.V., Beliaev Yu.K., Soloviev A.D. [Mathematical methods in the dependability theory. Primary dependability characteristics and their statistical analysis: Second edition, corrected and extended]. Moscow: Knizhny dom LIBROKOM; 2013. (in Russ.)

8. Mikhailov V.S. [Efficient estimation of mean time to failure]. Nadiozhnost i kontrol kachestva 1988;9:6-11. (in Russ.)