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Fisher information contained in incomplete observations
https://doi.org/10.26896/1028-6861-2024-90-7-84-92
Abstract
The accuracy of statistical estimates of unknown distribution parameters depends not only on the bulk of sampling data but also on the method of data acquisition. The information content of experimental data is one of the basic requirements. Problems of mathematical statistics, in particular parametric estimation based on censored observations, have specific features. Typical representatives of models of incomplete observations on a straight line are models of random censoring, competing risks of (single, multiple) random censoring. The purpose of this study is to show that censoring does not always lead to loss of (Fisher) information. It is shown that if censoring is informative, i.e., the distribution of censoring random variables depends on the same parameter, it is possible to specify a model where information can be preserved due to censoring. On the contrary, if the censoring is not informative, then the loss of information is inevitable. The Cramer – Rao efficiency was taken as a criterion for the quality of the assessment, whereas the Fisher information was taken as the criterion for information about the unknown parameter.
About the Authors
A. A. Abdushukurov
Tashkent Branch of the M. V. Lomonosov Moscow State University
Russian Federation
Russian Federation
Abdurakhim A. Abdushukurov
22, Amir Timur prosp., Tashkent, 100060
N. S. Nurmukhamedova
Mirzo Ulugbek National University of Uzbekistan
Russian Federation
Russian Federation
Nargiza S. Nurmukhamedova
4, Universitetskaya ul., Tashkent, 100174, Uzbekistan
S. A. Erisbaev
Berdakh Karakalpak State University
Russian Federation
Russian Federation
Sabitbek A. Erisbaev
1, Abdirova ul., Nukus, 742012, Uzbekistan
References
1. Fisher R. A. Theory of Statistical Estimation / Proceedings of the Cambridge Philosophical Society. 1925. P. 700 – 725.
2. Zacks Sh. The theory of statistical inference. — New York: Wiley, 1971.
3. Efron B., Zonstone I. M. Fisher Information in terms of the hazard rate / Ann. Statistics 1990. Vol. 18. N 1. P. 38 – 62.
4. Prakasa Rao B. L. S. On Cramer – Rao type integral inequalities / Calcutta Statistical Association Bulletin. 1991. Vol. 40. P. 183 – 205.
5. Prakasa Rao B. L. S. Cramer – Rao type integral inequalities for functions of multidimensional parameter / Sankhya: Ser. A. 1992. Vol. 54. P. 53.
6. Prakasa Rao B. L. S. Improved Cramer – Rao type integral inequalities or Bayesian Cramer – Rao lower-band / J. Indian Society for Probability and Statistics. 2018(a). Vol. 19. P. 1 – 7.
7. Prakasa Rao B. L. S. Improved sequential Cramer – Rao type integral inequality / Sequential Analysis. 2018(b). Vol. 37. P. 59 – 68.
8. Borovkov A. A., Sakhanenko A. I. On estimates of the average quadratic risk / Probability and Mathematical Statistics. 1980. Vol. 1. N 2. P. 185 – 195.
9. Abdushukurov A. A., Kim L. V. Lower Cramer – Rao and Bhattackarya bunds for randomly censored observation / J. Soviet. Math. 1987. Vol. 38. N 5. P. 2171 – 2185.
10. Prakasa Rao B. L. S. Remarks on Cramer – Rao type integral inequalities for random censored data / Analysis of censored data. 1995. Vol. 27. P. 163 – 175.
11. Zheng G., Gastwirth J. H. On the Fisher information in randomly censored data / Statistics Probability Letters. 2001. Vol. 52. P. 421 – 426.
12. Zeng G., Gastwirth J. H. Fisher information in ordered randomly censored data with application to characterization problems / Statistica Sinica. 2003. Vol. 13. P. 507 – 5017.
13. Prakasa Rao B. L. S. Improved Cramer – Rao inequality for randomly censored data / J. Iranian Statistical Society. 2018 (c). Vol. 17. P. 17 – 26.
14. Prakasa Rao B. L. S. Cramer – Rao Inequality Revisited for Randomly Censored Data / Proceedings of Conf. «Statistics» and its applications 2019. Tashkent. P. 19 – 28.
15. Abdushukurov A. A., Erisbaev S. A. Fisher information decomposition in terms of hazard rate functions under random censoring from the right / Electronic J. Science and education in Karakalpakstan. 2020. N 3 – 4. P. 36 – 43.
16. Abdushukurov A. A., Erisbaev S. A. Fisher information and Cramer – Rao inequality for Competing Risks Model / Bulletin of the Institute of Mathematics. 2022. Vol. 4. N 5. P. 50 – 59.
17. Abdushukurov A. A., Erisbaev S. A. On Fisher information function in competing risks model under random censoring from unobservation intervals and efficiency / Bulletin of the Institute of Mathematics. 2022. Vol. 5. N 3. P. 13 – 19.
18. Erisbaev S. A. Calculation of the Fisher information in Competing risks model in combined gibrid right random censoring / Bulletin of the Institute of Mathematics. 2022. Vol. 5. N 6. P. 178 – 181.
19. Erisbaev S. Semiparametric estimator of the conditional distribution function in the Cox regression model with partially informative random censoring on the right / Bulletin of the Institute of Mathematics. 2023. Vol. 6. N 3. P. 114 – 120.
20. Walker S. C. A self-improvement to the Cauchy – Swartz inequality / Statist. Probab. Lett. 2017. Vol. 122. P. 86 – 90.
21. Zheng G., Balakrishnan N., Parks. Fisher information in ordered data: A review / Statistics and its Interface. 2009. Vol. 2. P. 101 – 113.
22. Dorn M. F. Evaluating Fisher Information in Order Statistics / A research paper. — University of Chicago, 2011. — 52 p.
23. Nurmukhamedova N. S., Yusupov J. R. Fisher information in models of right random censoring / Modern problems of applied mathematics and information technology AL-KHORASMY-2016. P. 222 – 226.
24. Abdushukurov A. A., Nurmukhamedova N. S. Local asymptotic normality of statistical experiments and its role in estimation theory and hypothesis testing / Industr. lab. Mater. diagn. 2016. Vol. 82. N 3. P. 74 – 79.
Review
For citations:
Abdushukurov A.A., Nurmukhamedova N.S., Erisbaev S.A. Fisher information contained in incomplete observations. Industrial laboratory. Diagnostics of materials. 2024;90(7):84-92. (In Russ.) https://doi.org/10.26896/1028-6861-2024-90-7-84-92
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