Negative binomial regression in dose-effect relationships

M. S. Tikhov

Ufa mathematical journal, Tome 14 (2022) no. 4, pp. 96-112

Résumé

This paper is devoted to problem on estimating the distribution function and its quantiles in the dose-effect relationships with nonparametric negative binomial regression. Most of the mathematical researches on dose-response relationships concerned models with binomial regression, in particular, models with binary data. Here we propose a kernel-based estimates for the distribution function, the kernels of which are weighted by a negative binomial random variable at each covariate. These covariates are quasirandom van der Corput and Halton low-discrepancy sequences. Our estimates are consistent, that is, they converge to their optimal values in probability as the number of observations $n$ grows to infinity. The proposed estimats are compared by their mean-square errors. We show that our estimates have a smaller asymptotic variance in comparison, in particular, with estimates of the Nadaraya-Watson type and other estimates. We present nonparametric estimates for the quantiles obtained by inverting a kernel estimate of the distribution function. We show that the asymptotic normality of these bias-adjusted estimates is preserved under some regularity conditions. We also provide a multidimensional generalization of the obtained results.

Détail

Keywords: negative binomial response model, effective dose level, nonparametric estimate.

Bibliographie

[1] D.J. Finney, Probit Analysis, Cambridge University Press, NY, 1971, 333 pp. | MR | Zbl

[2] M. Razzaghi, Statistical Models in Toxicology, Taylor Fransis Group, NY, 2020. 270

[3] S.V. Krishtopenko, M.S. Tikhov, E.B. Popova, Dose-effect, Medicina, M., 2008 (in Russian)

[4] R.L. Hayes, N.Mantel, “Procedures for computing the mean age of eruption of human teeth”, J. Dental Research, 35:5 (1958), 938–947 | DOI

[5] M.C. Bisi, R. Stagni, “Evalution of toddler different strategies during the first six-months of independent walking: A longitudinal study”, Gait Posture, 41:2 (2015), 574–579 | DOI

[6] M.S. Tikhov, K.N. Shkileva, “Nonparametric estimation for quantile in binary regression models”, Vestnik TvGU. Ser. Prikl. Matem., 1 (2020), 5–19

[7] M.S. Tikhov, “Statistical Estimation Based on Interval Censored Data”, Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, eds. N.Balakrishnan etc., Springer, NY, 2004, 211–218, 555 pp. | MR

[8] M.S. Tikhov, D.S. Krishtopenko, “Estimation of distribution under dose-effect dependence with fixed experiment plan”, J. Math. Scien., 220:6 (2017), 753–762 | DOI | MR | Zbl

[9] È.A. Nadaraya, P. Babilua, G.A. Sokhadze, “On the integral square deviation of one nonparametric estimation of the Bernoulli regession”, Theory Probab. Appl., 57:2 (2013), 265–278 | DOI | DOI | MR | MR | Zbl

[10] H. Okumura, K. Naito, “Weighted kernel estimators in nonparametric binomial regression”, J. Nonparametr. Statist., 16:1-2 (2004), 39–62 | DOI | MR | Zbl

[11] M.S. Tikhov, T.S. Borodina, “Kernel estimators of quantiles in dose-effect relationships”, Automatic Control and Computer Sciences, 2 (2013), 29–43

[12] M.S. Tikhov, “Negative binomial regression in dose-effect relationships”, Proc. of Int. Scien. Conf. Peoples Frindship University of Russia, The $5^{th}$ International Conference on Stochastic Methods (ICSM-5), M., 2020, 205–208 | MR

[13] M.G. Kendall, A. Stuart, The advanced theory of statistics, v. 1, Distribution theory, Griffin, New York, 1958 | MR

[14] W. Feller, An introduction to probability theory and its applications, v. I, John Wiley Sons, New York, 1950 | MR | MR | Zbl

[15] I.P. Natanson, Theory of functions of real variable, Ungar, New York, 1955 | MR | MR | Zbl

[16] H. Niederreiter, Random number generation and quasi-Monte Carlo method, SIAM, Philadelphia, 1992, 241 pp. | MR

[17] I.M. Sobol', Multidimensional quadrature formulas and Haar functions, Nauka, M., 1969

[18] T.M. Tovstik, “Calculation of the discrepancy of a finite set of points in the unit $n$-cube”, Vestn. St-Peterbg. Univ. Ser. I. Mat. Mekh. Astron., 40:3 (2007), 118–121 | MR | MR | Zbl

[19] L. Kuipers, H. Niederreiter, Uniform distribution of sequences, John Wiley Sons, New York, 1974 | MR | MR | Zbl

[20] J. Kiefer, “On large deviations of the empiric d.f. of vector chance variables and a law of the iterated logarithm”, Pacific. J. Math., 11:2 (1961), 649–660 | DOI | MR | Zbl

[21] J.H. Halton, “Algorithm 247: Radical-inverse quasi-random point sequence”, CACM, 7:12 (1964), 701–702 | DOI

[22] L.K. Hua, Y. Wang, Applications of number theory to numerical analysis, Springer-Verlag, NY, 1981, 241 pp. | MR | Zbl

[23] N.M. Kolobov, Theoretical-number methods in approximate analysis, MCCME, M., 2014

[24] E.L.Lehmann, Elements of Large-Sample Theory, Springer, NY, 1999, 632 pp. | MR | Zbl

[25] M.S. Tikhov, “Nonparametric estimation of effective doses at quantal response”, Ufa Math. J., 5:2 (2013), 94–108 | DOI | MR

[26] B.V. Gnedenko, Course of probability theory, URSS, M., 2005 | MR

[27] M.B. Priestly, M.T. Chao, “Nonparametric function fitting”, J. Royal Statist. Soc., Ser. B, 34 (1972), 385–392 | MR

[28] N.L. Johnson, S. Kotz, A.W. Kemp, Univariate discrete distributions, John Wiley Sons, New York, 1992 | MR | Zbl

[29] M. DeGroot, “Unbiased sequential estimation for binomial populations”, Ann. Math. Stat., 30:1 (1959), 80–101 | DOI | MR | Zbl

[30] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series, v. 1, Elementary functions, Gordon Breach Science Publ., New York, 1986 | MR | MR | Zbl

[31] R.S. Ellis, “Large deviations for a general class of random vectors”, Ann. Statist., 12:1 (1984), 1–12 | MR | Zbl