Borshchevskij, A. V.; Ivanov, A. V. Normal approximation of the distribution of the optimum point in the data processing problem by the method of least moduli. (English. Russian original) Zbl 0611.62055 Cybernetics 21, 831-839 (1985); translation from Kibernetika 1985, No. 6, 86-92 (1985). Consider the model \(x_ j=x_ j(\theta)=g(j,\theta)+\epsilon_ j\), \(j\geq 1\), where for each j, g(j,\(\theta)\) is a function of the vector \(\theta\in \Theta \subset {\mathbb{R}}^ q\). The least moduli estimate \({\check \theta}_ n\) of the parameter \(\theta\) is defined to minimize \(R_ n(\tau)=\sum^{n}_{j=1}| x_ j(\theta)-g(j,\tau)|\). Under certain conditions, the normal approximation of the distribution of \({\check \theta}_ n\) is given. Reviewer: Bai Zhidong MSC: 62H12 Estimation in multivariate analysis 62E20 Asymptotic distribution theory in statistics 62H10 Multivariate distribution of statistics Keywords:nonsmooth optimization problem; least absolute deviation; limiting distribution; least moduli estimate; normal approximation PDFBibTeX XMLCite \textit{A. V. Borshchevskij} and \textit{A. V. Ivanov}, Cybernetics 21, 831--839 (1985; Zbl 0611.62055); translation from Kibernetika 1985, No. 6, 86--92 (1985) Full Text: DOI References: [1] P. Bloomfield and W. Steiger, ?Lease absolute deviations curve-fitting,? SIAM J. Sci. Statist. Comput.,1, No. 2, 290?301 (1980). · Zbl 0471.65007 · doi:10.1137/0901019 [2] N. Z. Shor, Methods of Minimization of Nondifferentiable Functions and Their Applications [in Russian], Naukova Dumka, Kiev (1979). · Zbl 0524.49002 [3] A. B. Tsybakov, ?On a method of minimization of the empirical risk in identification problems,? Avtomat. Telemekh., No. 9, 77?85 (1981). [4] G. Basset, Jr., and R. Koenker, ?Asymptotic theory of least absolute error regression,? J. Am. Statist. Assoc.,73, No. 363, 618?622 (1979). [5] V. A. Gurevich, ?The method of least moduli for a nonlinear regression model,? in: Applied Statistics [in Russian], Nauka, Moscow (1983), pp. 355?361. [6] A. V. Ivanov and O. I. Kozlov, ?On the consistency of the minimal contrast estimates in the case of nonidentically distributed observations,? Teor. Veroyatn. Mat. Statist.,23, 59?68 (1980). · Zbl 0476.62030 [7] A. V. Borshchevskii and A. V. Ivanov, ?Asymptotic properties of the optimum point in a certain data processing problem by the method of least deviations,? in: Proc. Internat. Conf. on Stochastic Optimization, Abstract of Communications [in Russian], Vol. 1, Institute of Cybernetics, Academy of Sciences of the Ukrainian SSR, Kiev (1984), pp. 43?45. [8] A. V. Ivanov, ?On the consistency and the asymptotic normality of least absolute value estimates,? Ukr. Mat. Zh.,36, No. 3, 297?303 (1984). [9] A. V. Borshchevskii and A. V. Ivanov, ?A property of the optimum point in the data processing problem by the method of least deviations,? Dokl. Akad. Nauk USSR, No. 1, 53?56 (1985). [10] I. A. Ibragimov and Yu. A. Rozanov (Y. A. Rozanov), Gaussian Random Processes, Springer-Verlag, New York (1978). · Zbl 0451.60045 [11] Yu. A. Davydov, ?On the convergence of distributions generated by stationary random processes,? Teor. Veroyat. Ee Primen.,13, No. 4, 730?737 (1968). [12] P. J. Huber, ?The behavior of maximum likelihood estimates under nonstandard conditions,? in: Proc. Fifth Berkeley Symp. Math. Statist. Probab. Vol. 1, Univ. of California Press, Berkeley (1967), pp. 221?224. · Zbl 0212.21504 [13] A. S. Kholevo, ?On the asymptotic normality of estimates of regression coefficient,? Teor. Veroyatn. Ee Primen.,16, No. 4, 724?728 (1971). [14] R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, Wiley, New York (1976). · Zbl 0331.41023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.