Fourier methods for recursive estimating of distribution function in dose-effect relationship
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2018), pp. 31-49 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proposed a kernel Fourier methods for recursive estimating of the distribution function in dose-effect relationship where the entered doses are observed with errors. The asymptotic normality of the offered estimates is proved. The possibility for increasing the accuracy of the estimators through repeated measurements of the entered doses is discussed.
Keywords: dose-effect relationship, deconvolving kernel distribution function estimators, asymptotic normality.
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M. S. Tikhov. Fourier methods for recursive estimating of distribution function in dose-effect relationship. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2018), pp. 31-49. http://geodesic.mathdoc.fr/item/VTPMK_2018_4_a2/

[1] Krishtopenko S. V., Tikhov M. S., Popova E. B., Dose-Effect, Medicina Publ., Moscow, 2008, 228 pp. (in Russian)

[2] Tikhov M. S., “Statistical Estimation on the Basis of Interval-Censored Data”, Interuniversity Transactions on Statistical Method of Estimation and Testing Hypotheses, Perm University, Perm, 2000, 49–70 (in Russian)

[3] Tikhov M. S., “Statistical estimation based on interval censored data”, Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, Statistics for Industry and Technology, eds. N. Balakrishnan, M.S. Nikulin, M. Mesbah, N. Limnios, Birkhauser, Boston, MA, 2004, 211–218 | DOI | MR

[4] Tikhov M. S., “Statistical estimation based on interval censored data”, Journal of Mathematical Sciences, 119:3 (2004), 321–335 | DOI | MR

[5] Tikhov M. S., Dolgih I. S., “Asymptotically unbiased estimates of a distribution function in dose-effect relationships”, Journal of Mathematical Sciences, 205:1 (2015), 113–120 | DOI | MR | Zbl

[6] Tikhov M. S., Krishtopenko S. V., “Asymptotically unbiased estimates of a distribution function in dose-effect relationships”, 2nd All-Russian Colloquium on Stochastic Methods, TVP Publ., Moscow, 1995, 81–82 (in Russian)

[7] Tikhov M. S., Krishtopenko D. S., “Asymptotic distributions of integrated square errors of nonparametric estimation based on indirect observations under dose-effect dependence”, Interuniversity Transactions on Statistical Method of Estimation and Testing Hypotheses, Perm University, Perm, 2007, 82–97 (in Russian)

[8] Tikhov M., Borodina T., Ivkin M., “On reduction of measurement errors at estimation of distribution in dose-effect relationships”, Advances in Mathematics and Statistical Sciences: Proc. of the 3rd International Conference on Mathematical, Computational and Statistical Sciences, MCSS'15, 2015, 19–27

[9] Tikhov M., Ivkin M., “Goodness of fit tests on the basis of the kernel quantile estimators in dose-effect relationship”, International Journal of Mathematics and Computers in Simulation, 9 (2015), 127–135

[10] Fan J., “On optimal rates of convergence for nonparametric deconvolution problemes”, Annals of Statistics, 19:3 (1991), 1257–1272 | DOI | MR | Zbl

[11] Zu Y., “A note on the asymptotic normality of the kernel deconvolution density estimator with logarithmic chi-square noise”, Econometrics, 3:3 (2015), 561–576 | DOI

[12] Roussas G. G., Tran L. T., “Asymptotic normality of the recursive kernel regression estimate under dependence conditions”, Annals of Statistics, 20:1 (1992), 98–120 | DOI | MR | Zbl

[13] Graham R., Knuth D., Patashnik O., Concrete Mathematics, Addison-Wesley Publishing Company, New York, 1994, 657 pp. | MR | MR | Zbl

[14] Gnedenko B. V., The Theory of Probability, AMS Chelsea Publishing, Providence, Rhode Island, 2005, 529 pp. | MR | MR

[15] Petrov V. V., Limit theorems for sums of independent random variables, Nauka Publ., Moscow, 1987, 318 pp. (in Russian)

[16] Kolmogorov A. N., Foundations of The Theory of Probability, Nauka Publ., Moscow, 1974, 120 pp. (in Russian)

[17] Stein C., “A bound for the error in the normal approximation to the distribution of a sum of dependent random variables”, Proc. of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, v. 2, University of California Press, Berkeley, 1972, 583–602 | MR | Zbl

[18] Stein C., Approximate Computation of Expectations, Lecture Notes-Monograph Series, Institute of Mathematical Statistics, 1986, 164 pp. | MR

[19] Dey P., Stein's Method for Normal Approximation URL: http://math.uiuc.edu/ math561sp16/lec25.pdf

[20] Vaserstein L. N., “Markov processes on countable space products describing large systems of automata”, Problems of Information Transmission, 5:3 (1969), 47–52 | MR

[21] Kantorovich L. V., Rubinshtein S. G., “On a space of totally additive functions”, Vestnik of the St. Petersburg University: Mathematics, 13:7 (1958), 52–59 | MR | Zbl

[22] Kantorovich L. V., Akilov G. P., Functional Analysis in Normed Spaces, GIFML Publ., Moscow, 1959, 684 pp. (in Russian)

[23] Delaigle A., Hall P., Meister A., “On deconvolution with repeated measurements”, Annals of Statistics, 36:2 (2008), 665–685 | DOI | MR | Zbl

[24] Delaigle A., Hall P., “Estimation of observation-error variance in errors-in-variables regression”, Statistica Sinica, 21:3 (2011), 1023–1063 | DOI | MR | Zbl

[25] Comte F., Kappus J., “Density deconvolution from repeated measurements without symmetry assumption on the errors”, Journal of Multivariate Analysis, 140 (2015), 31–46 | DOI | MR | Zbl

[26] Delaigle A., “An alternative view of the deconvolution problem”, Statistica Sinica, 18:3 (2008), 1025–1045 | MR | Zbl