Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


In this paper, we consider a sequential interval estimation by intervals of a fixed width of the asymptotic variance of rank estimates of the shift parameter. Reviewed the asymptotical properties of estimates of functionals of an unknown probability density and the conditions of the asymptotical consistency of a confidence interval of a fixed width and the asymptotical efficiency of the stopping time. The convergence rate of consistency of the fixed width interval for the asymptotic variance of rank estimates of the shift parameter is obtained.

First Page


Last Page



1. Hodges J.L., Lehman E.L. Estimates of location based on rank tests. Ann. Math. Statist., Vol. 34, 598–611 (1963).

2. Schweder~T. Window estimation of the asymptotic variance of rank estimators of location. Scand. J. Statist., Vol. 2, 113–126 (1975).

3. Schuster E.F. On the rate of convergence of an estimate of a functional of a probability density. Scand. Actuarial J., Vol. 1, 103–107 (1974).

4. Ahmad I.A. On asymptotic properties of an estimate of a functional of a probability density. Scand. Actuarial J., Vol. 3, 176–181 (1976).

5. Tursunov G.T. On estimating the asymptotic variance of rank estimates of the shift parameter from a sample of a random volume, Theory of random processes, "Naukovo Dumka", Kiev, Vol. 15, 97–102 (1987). [in Russian].

6. Csorgo M, Fischler R: Departure from independence: the strong law, standart and random–sum central limit theorems, Acta Math. Acad. Sci. Hung., Vol. 21, no.1–2, 105–114 (1970).

7. Dvoretsky A, Kiefer J, Wolkfowitz J. Asimptotical minimax character of the sample distribution function and of the multinominal estimator, Ann. Math. Statist., Vol. 27, 642–669 (1956).

8. Gikhman I.I, Skorohod A.V. Introduction to the theory of random processes, Moscow, Nauka, (1977). [in Russian]

9. Silvestrov D. S., Mirzakhmedov M. A., Tursunov G. T. On the application of limit theorems for complex random functions to some problems of statistics, Probability Theory and Mathematics. Statistics, Vol. 14, 124–137 (1976). [in Russian].

10. Sachs S. Theory of statistical conclusions, Moscow, Mir, (1975). [in Russian].

11. Gartsema J.C. Sequential confidence intervals based on rank tests, Ann. Math. Statist., Vol. 41, 1016–1926 (1970).

12. Bickel P.J., Yahav J.A. Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist., Vol. 39, 442–456 (1968).

13. Korolyuk V.S., Borovskikh Yu.V. Theory of U–statistics, Naukova Dumka, Kiev, (1989). [in Russian].

14. Michel R., Pfanzagl J. The accuracy of the normal approximation for minimum contrast estimates, Z. Wahr. verw. Geb., Vol. 18, 73–84 (1971).

15. Csenki A. A theorem on the departure of randomly indexed U–statistics from normality with an application in fixed–width sequential interval estimation, Sankhya, Indian J. Statist., Vol. 43, ser. A, 1, 84–99 (1981).



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.