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Toward the history of the St. Petersburg school of probability and statistics. IV: Characterization of distributions and limit theorems in statistics. (English. Russian original) Zbl 1433.01024

Vestn. St. Petersbg. Univ., Math. 52, No. 1, 36-53 (2019); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 6(64), No. 1, 53-80 (2019).
The authors present the following:
“This is the fourth article in a series of surveys devoted to the scientific achievements of the Leningrad-St. Petersburg School of Probability and Statistics from 1947 to 2017. It is devoted to studies on the characterization of distributions, limit theorems for kernel density estimators, and asymptotic efficiency of statistical tests. The characterization results are related to the independence and equidistribution of linear forms of sample values, as well as to regression relations, admissibility, and optimality of statistical estimators. When calculating the Bahadur asymptotic efficiency, particular attention is paid to the logarithmic asymptotics of large deviation probabilities of test statistics under the null hypothesis. Constructing new goodness-of-fit and symmetry tests based on characterizations is considered, and their asymptotic behavior is analyzed. Conditions of local asymptotic optimality of various nonparametric statistical tests are studied.”
In the first section, the main attention is given to characterization problems as a branch of analytical statistics. The authors note that results obtained in this field by Yu.V. Linnik’s school are presented in this study. The first section consists of the following items:
– Independence of linear forms of independent random variables
– Independence of nonlinear functions of independent random variables
– Constancy of regression
– Equidistribution of linear forms
– Admissibility and optimality of estimators
– Miscellanea.
In the second section “Construction and asymptotic comparison of statistical tests”, the authors note that the problem of a reasonable choice of a statistical test from the several tests at the statistician’s disposal, is one of the classical and most well-known problems of mathematical statistics. The authors consider the following items:
– Asymptotic efficiency of tests
– Large deviations of test statistics
– Large deviations of \(U\)-statistics and von Mises functionals
– Calculation of the Bahadur efficiency
– Local asymptotic optimality of tests and characterization of distributions
– Goodness-of-fit and symmetry tests based on characterizations.
The third section of this paper is on estimates for the rate of approximation in the central limit theorem for the \(L_1\)-norm of kernel density estimators.
A conclusion on the contribution of the Leningrad-St.Petersburg School of Probability to the characterization of distributions by various properties of statistics, is given, and some open problems are noted in sections.

MSC:

01A72 Schools of mathematics
62-03 History of statistics
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References:

[1] A. Yu. Zaitsev, A. A. Zinger, M. A. Lifshits, Ya. Yu. Nikitin, and V. V. Petrov, “Toward the history of the Saint Petersburg School of Probability and Statistics. I. Limit theorems for sums of independent random variables,” Vestn. St. Petersburg Univ.: Math. 51, 144-163 (2018). https://doi.org/10.3103/S1063454118020115 · Zbl 1401.60002 · doi:10.3103/S1063454118020115
[2] D. N. Zaporozhets, I. A. Ibragimov, M. A. Lifshits, and A. I. Nazarov, “On the history of St. Petersburg School of Probability and Mathematical Statistics: II. Random processes and dependent variables,” Vestn. St. Petersburg Univ. Math. 51, 213-236 (2018). https://doi.org/10.3103/S1063454118030123 · Zbl 1433.01023 · doi:10.3103/S1063454118030123
[3] A. N. Borodin, Yu. A. Davydov, and V. B. Nevzorov, “St. Petersburg School of Probability and Mathematical Statistics: III. Distribution of functionals of processes, stochastic geometry, and extrema,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh, Astron. 5(63) (4). C. 572-596 (2018). · Zbl 1509.01064
[4] S. N. Bernstein, “On a property characteristic of the law of Gauss,” Tr. Leningr. Politekh. Inst. 3-C, 21-22 (1941); S. N. Bernstein, “On a property characteristic of the law of Gauss,” in Collected Papers (Nauka, Moscow, 1964), Vol. 4, pp. 314-315 [in Russian].
[5] M. Kac, “On a characterization of the normal distribution,” Am. J. Math. 61, 726-728 (1939). · JFM 65.0569.01 · doi:10.2307/2371328
[6] V. P. Skitovitch, “On a property of the normal distribution,” Dok. Akad. Nauk SSSR 89, 217-219 (1953). · Zbl 0053.27401
[7] V. P. Skitovich, “Linear forms of independent random variables and the normal distribution law,” Izv. Akad. Nauk SSSR Ser. Mat. 18, 185-200 (1954). · Zbl 0055.36702
[8] G. Darmois, “Analyse générale des liaisons stochastiques,” Rev. Inst. Int. Stat. 21, 2-8 (1953). · Zbl 0051.36003 · doi:10.2307/1401511
[9] L. V. Mamay, “On the theory of characteristic functions,” Vestn. Leningr. Univ. 15, 85-99 (1960). · Zbl 0121.13502
[10] B. Ramachandran, Advanced Theory of Characteristic Functions (Statistical Publishing Society, Calcutta, 1967). · Zbl 0189.18102
[11] I. A. Ibragimov, “On the Skitovich—Darmois—Ramachandran theorem,” Theory Probab. Its Appl. (Engl. Transl.) 57, 368-374 (2013). · Zbl 1280.60015 · doi:10.1137/S0040585X97986059
[12] A. M. Kagan, “New classes of dependent random variables and a generalization of the Darmois—Skitovich theorem to several forms,” Theory Probab. Its Appl. (Engl. Transl.) 33, 305-314 (1988). · Zbl 0704.62010
[13] A. M. Kagan, Yu. V. Linnik, and C. R. Rao, Characterization Problems of Mathematical Statistics (Nauka, Moscow, 1972; Wiley, New York, 1973). · Zbl 0243.62009
[14] Yu. V. Linnik, “On the classical derivation of the Maxwell law,” Dokl. Acad. Nauk USSR 85, 1251-1254 (1951). · Zbl 0048.19803
[15] A. Kagan, R. G. Laha, and V. Rohatgi, “Independence of the sum and absolute difference of independent random variables does not imply their normality,” Math. Meth. Stat. 6, 263-265 (1997). · Zbl 0883.62015
[16] E. Lukacs, “A characterization of the normal distribution,” Ann. Math. Stat. 13, 91-94 (1942). · Zbl 0060.28509 · doi:10.1214/aoms/1177731647
[17] T. Kawata and H. Sakamoto, “On the characterization of independence of the sample mean and the sample variance,” J. Math. Soc. Jpn. 1, 111-115 (1949). · Zbl 0039.14105 · doi:10.2969/jmsj/00120111
[18] A. A. Zinger, “On independent samples from normal populations,” Usp. Mat. Nauk 6 (5), 172-175 (1951). · Zbl 0043.34107
[19] A. A. Zinger, “Independence of quasi-polynomial statistics and analytical properties of distributions,” Theory Probab. Its Appl. (Engl. Transl.) 3, 247-265 (1958). · Zbl 0089.36205 · doi:10.1137/1103022
[20] A. A. Zinger, “Distributions of polynomial statistics in samples from a normal population,” Dokl. Akad. Nauk SSSR 149, 20-21 (1963). · Zbl 0126.35301
[21] A. A. Zinger, “On the distribution of polynomial statistics in samples from normal and related populations,” Proc. Steklov Inst. Math., 79, 167-177 (1966). · Zbl 0171.40004
[22] A. A. Zinger and Y. V. Linnik, “On a class of differential equations with an application to certain questions of regression theory,” Vestn. Leningr. Univ. 12 (7), 121-130 (1957). · Zbl 0094.33501
[23] A. A. Zinger and Y. V. Linnik, “Polynomial statistics from a normal sample,” Dokl. Akad. Nauk SSSR 176, 766-767 (1967). · Zbl 0291.62025
[24] A. A. Zinger and Y. V. Linnik, “Nonlinear statistics and random linear forms,” Proc. Steklov Inst. Math. 111, 25-44 (1970). · Zbl 0266.62010
[25] A. M. Vershik, “Some characteristic properties of Gaussian stochastic processes,” Theory Probab. Its Appl. (Engl. Transl.) 9, 353-356 (1964). · Zbl 0141.15203 · doi:10.1137/1109053
[26] A. M. Kagan, Y. V. Linnik, and C. R. Rao, “On a characterization of the normal law based on a property of the sample average,” Sankhyā A27, 405-406 (1965). · Zbl 0168.40203
[27] J. Marcinkiewicz, “Sur une propriété de la loi de Gauss,” Math. Z. 44, 622-638 (1938). · Zbl 0019.31705
[28] G. Pölya, “Herleitung des Gaußschen Fehlergesetzes aus einer Funktionalgleichung,” Math. Z. 18, 96-108 (1923). · JFM 49.0366.01 · doi:10.1007/BF01192398
[29] Yu. V. Linnik, “On some equidistributed statistics,” Dokl. Acad. Nauk SSSR 89, 9-11 (1953). · Zbl 0053.27307
[30] Yu. V. Linnik, “Linear forms and statistical criteria. I,” Ukr. Mat. Zh. 5 (2), 207-243; Yu. V. Linnik, “Linear forms and statistical criteria. II,” Ukr. Mat. Zh. 5 (3), 247-290 (1953). · Zbl 0052.36701
[31] A. M. Kagan, “Generalized condition of the identity of distributions of random vectors in connection with the asymptotic theory of linear forms in independent random values,” Theory Probab. Its Appl. (Engl. Transl.) 34, 327-332 (1989). · Zbl 0693.62014 · doi:10.1137/1134029
[32] A. M. Kagan, “On the estimation theory of a location parameter,” Sankhyā A28, 335-352 (1966). · Zbl 0156.39207
[33] A. M. Kagan, “Fisher information contained in a finite-dimensional linear space, and a correctly posed version of the method of moments,” Probl. Inf. Transm. (Engl. Transl.) 12, 98-115 (1976).
[34] V. V. Petrov, “On the method of least squares and its extremal properties,” Usp. Mat. Nauk 9 (1), 41-62 (1954). · Zbl 0055.37605
[35] A. M. Kagan and O. V. Shalaevskii, “Admissiblity of the least squares estimator is a characteristic property of the normal law,” Mat. Zametki 6 (1), 81-89 (1969).
[36] A. A. Zinger, A. M. Kagan, L. B. Klebanov, “The sample mean as an estimator of the shift parameter in the presence of certain losses which differ from the quadratic,” Dokl. Akad. Nauk SSSR 189, 29-30 (1969).
[37] A. M. Kagan and A. A. Zinger, “Sample mean as an estimator of a location parameter. Case of nonquadratic loss functions,” Sankhyā A33, 351-358 (1971). · Zbl 0274.62008
[38] A. M. Kagan and A. A. Zinger, “Sample mean as an estimator of the location parameter in presence of the nuisance scale parameter,” Sankhyā A35, 447-454 (1973). · Zbl 0291.62019
[39] Kagan, A. M.; Melamed, I. A.; Zinger, A. A., A class of estimators of a location parameter in presence of a nuisance scale parameter, 359-368 (1982), Amsterdam · Zbl 0483.62022
[40] A. A. Zinger and A. M. Kagan, “A least squares estimate, nonquadratic losses, and Gaussian distributions,” Theory Probab. Its Appl. (Engl. Transl.) 36, 115-123 (1991). · Zbl 0744.62039 · doi:10.1137/1136009
[41] A. M. Kagan and A. L. Rukhin, “On the estimation of a scale parameter,” Theory Probab. Its Appl. (Engl. Transl.) 12, 672-678 (1967). · Zbl 0283.62041 · doi:10.1137/1112083
[42] A. M. Kagan, “On ε-admissibility of the sample mean as an estimator of location parameter,” Sankhyā A32, 37-40 (1970). · Zbl 0206.20103
[43] N. A. Sapogov, “The stability problem for a theorem of Cramér,” Izv. Akad. Nauk SSSR Ser. Mat. 15, 205-218 (1951). · Zbl 0043.13202
[44] N. A. Sapogov, “On independent terms of a sum of random variables which is distributed almost normally,” Vestn. Leningr. Univ. 19, 78-105 (1959).
[45] A. A. Zinger, “On a problem by A. N. Kolmogorov,” Vestn. Leningr. Univ. 1, 53-56 (1956).
[46] A. A. Zinger and Yu. V. Linnik, “A characteristic property of the normal distribution,” Theory Probab. Its Appl. (Engl. Transl.) 9, 624-626 (1964). · Zbl 0136.41101 · doi:10.1137/1109084
[47] F. M. Kagan, “An information property of the gamma-distribution,” Izv. Akad. Nauk Uzb. SSR. Ser. Fiz.-Mat. 5, 67-68 (1967). · Zbl 0173.20402
[48] L. Yu. Morozenskii, “A characterization of the normal law by an optimality property of a test based on the sample average,” Vestn. Leningr. Univ. 13, 61-63 (1971). · Zbl 0232.62003
[49] A. M. Kagan and O. V. Shalaevskii, “A characterization of the normal law by a property of the non-central chisquare distribution,” Lith. Math. J. 7, 57-58 (1969).
[50] E. J. G. Pitman, Lecture Notes on Nonparametric Statistical Inference (Univ. of N. Carolina, Chapel Hill, NC, 1948).
[51] R. R. Bahadur, Some Limit Theorems in Statistics (SIAM, Philadelphia, PA, 1971). · Zbl 0257.62015 · doi:10.1137/1.9781611970630
[52] J. L. Hodges and E. L. Lehmann, “The efficiency of some nonparametric competitors of the t-test,” Ann. Math. Stat. 27, 324-335 (1956). · Zbl 0075.29206 · doi:10.1214/aoms/1177728261
[53] H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493-507 (1952). · Zbl 0048.11804 · doi:10.1214/aoms/1177729330
[54] W. C. M. Kallenberg, “Intermediate efficiency, theory and examples,” Ann. Stat. 11, 170-182 (1983). · Zbl 0512.62057 · doi:10.1214/aos/1176346067
[55] A. A. Borovkov and A. A. Mogulskii, Large Deviations and Testing Statistical Hypotheses, Nauka, Novosibirsk, 1992) [in Russian]. · Zbl 0847.62012
[56] Ya. Yu. Nikitin, Asymptotic Efficiency of Nonparametric Tests (Nauka, Moscow, 1995) [in Russian]. · Zbl 0879.62045 · doi:10.1017/CBO9780511530081
[57] Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, 2nd ed. (Cambridge Univ. Press, Cambridgev, 2009). · Zbl 1171.62031
[58] I. G. Abrahamson, “The exact Bahadur efficiencies for the Kolmogorov-Smirnov and Kuiper one- and two-sample statistics,” Ann. Math. Stat. 38, 1475-1490 (1967). · Zbl 0157.48003 · doi:10.1214/aoms/1177698702
[59] Y. Y. Nikitin and A. G. Pankrashova, “Bahadur efficiency and local asymptotic optimality of certain nonpara-metric tests for independence,” J. Sov. Math. 52, 2942-2955 (1990). · Zbl 0900.62238 · doi:10.1007/BF01103751
[60] Ya. Yu. Nikitin, “Large deviations of U-empirical Kolmogorov-Smirnov tests, and their efficiency,” J. Non-parametric Stat. 22, 649-668 (2010). · Zbl 1210.62048 · doi:10.1080/10485250903118085
[61] I. N. Sanov, “On the probability of large deviations of random variables,” Mat. Sb. 42 (1), 11-44 (1957).
[62] P. Groeneboom, J. Oosterhoff, and F. H. Ruymgaart, “Large deviation theorems for empirical probability measures,” Ann. Probab. 7 (4), 553-586 (1979). · Zbl 0425.60021 · doi:10.1214/aop/1176994984
[63] A. D. Wentzell and M. I. Freidlin, Fluctuations in Dynamical Systems Caused by Small Random Perturbations (Nauka, Moscow, 1979) [in Russian]. · Zbl 0499.60053
[64] Y. Y. Nikitin, “Large deviations and asymptotic efficiency of statistics of integral type. I,” J. Sov. Math. 20, 2224-2231 (1982). · Zbl 0493.62046 · doi:10.1007/BF01240001
[65] Y. Y. Nikitin, “Large deviations and asymptotic efficiency of integral type statistics. II,” J. Sov. Math. 24, 585-603 (1980). · Zbl 0599.62060 · doi:10.1007/BF01702336
[66] M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations (Nauka, Moscow, 1969; Noordhoff, Leyden, 1974). · Zbl 0186.20805
[67] A. A. Mogul’skii, “Remarks on large deviations for the ω2 statistics,” Theory Probab. Its Appl. (Engl. Transl.) 22, 166-171 (1977). · Zbl 0375.60034 · doi:10.1137/1122018
[68] Y. Y. Nikitin, “Bahadur efficiency of integral type of symmetry,” Zap. Nauchn. Semin. LOMI 119, 181-194 (1982). · Zbl 0541.62032
[69] Ya. Yu. Nikitin, “Large deviations and asymptotic efficiency of integral statistics for testing independence,” J. Sov. Math. 38, 2382-2391 (1987). · Zbl 0663.62056 · doi:10.1007/BF01095081
[70] Ya. Yu. Nikitin, “Hodges-Lehmann and Chernoff efficiencies of linear rank tests,” J. Stat. Plann. Infer. 29, 309-323 (1991). · Zbl 0763.62028 · doi:10.1016/0378-3758(91)90006-Z
[71] Y. Y. Nikitin, “Local Chernoff and Hodges-Lehmann efficiencies of linear rank tests for symmetry,” J. Math. Sci. (N. Y.) 68, 551-559 (1994). · Zbl 0836.62035 · doi:10.1007/BF01254281
[72] Ya. Yu. Nikitin, “Hodges-Lehmann efficiency of nonparametric tests,” in Proc. 4th Vilnius Conf. on Probability Theory and Mathematical Statistics, Vilnius, June 24-29, 1985 (VNU Sci., Utrecht, 1987), pp. 391-408. · Zbl 0666.62049
[73] Y. Y. Nikitin, “On the Hodges-Lehmann Asymptotic Efficiency of Nonparametric Tests of Goodness of Fit and Homogeneity,” Theory Probab. Its Appl. (Engl. Transl.) 32, 77-85 (1988). · Zbl 0642.62032 · doi:10.1137/1132007
[74] G. Tusnády, “On asymptotically optimal tests,” Ann. Stat. 5, 385-393 (1977). · Zbl 0361.62034 · doi:10.1214/aos/1176343804
[75] W. Hoeffding, “A class of statistics with asymptotically normal distribution,” Ann. Math. Stat. 19, 293-325 (1948). · Zbl 0032.04101 · doi:10.1214/aoms/1177730196
[76] R. von Mises, “On the asymptotic distribution of differential statistical functions,” Ann. Math. Stat. 18, 309-348 (1947). · Zbl 0037.08401 · doi:10.1214/aoms/1177730385
[77] Y. Y. Nikitin and E. V. Ponikarov, “Large deviation of Chernoff type for U- and V-statistics,” Dokl. Math. 60, 316-318 (1999). · Zbl 1041.60027
[78] Nikitin, Ya Yu; Ponikarov, E. V.; Uraltseva, N. N (ed.), Rough large deviation asymptotics of Chernoff type for von Mises functionals, No. Ser. 2, 107-146 (2001), Providence, RI · Zbl 1213.60056
[79] Ya. Yu. Nikitin and E. V. Ponikarov, “On large deviations of non-degenerate two-sample U- and V-statistics with applications to Bahadur efficiency,” Math. Meth. Stat. 15, 103-122 (2006).
[80] V. V. Litvinova and Y. Y. Nikitin, “Asymptotic efficiency and local optimality of tests based on two-sample U-and V-statistics,” J. Math. Sci. (N. Y.) 152 (6), 921-927 (2008). · Zbl 1288.62070 · doi:10.1007/s10958-008-9108-1
[81] Y. Y. Nikitin and E. V. Ponikarov, “Asymptotic efficiency of Maesono statistics for testing of symmetry,” Ann. Inst. Stat. Math. 54, 382-390 (2002). · Zbl 1012.62050 · doi:10.1023/A:1022482220794
[82] A. Durio and Y. Y. Nikitin, “Local Bahadur efficiency of some goodness-of-fit tests under skew alternatives,” J. Stat. Plann. Inference 115, 171-179 (2003). · Zbl 1041.62036 · doi:10.1016/S0378-3758(02)00155-6
[83] Durio, A.; Nikitin, Y. Y.; Frosini, B. V (ed.); Magagnoli, U. (ed.); Boari, G. (ed.), On asymptotic efficiency of certain distribution-free symmetry tests under skew alternatives, 223-239 (2002), Milano · Zbl 1330.62212
[84] A. Durio and Y. Y. Nikitin, “Local efficiency of integrated goodness-of-fit tests under skew alternatives,” Stat. Probab. Lett. 117, 136-143 (2016). · Zbl 1345.62075 · doi:10.1016/j.spl.2016.05.016
[85] N. Henze, Ya. Nikitin, and B. Ebner, “Integral distribution-free statistics of Lp-type and their asymptotic comparison,” Comput. Stat. Data Anal. 53, 3426-3438 (2009). · Zbl 1453.62109 · doi:10.1016/j.csda.2009.02.018
[86] O. A. Podkorytova, “Rough large-deviation probability asymptotics of some functionals of the norm kind,” J. Math. Sci. (N. Y.) 99, 1161-1172 (2000). · Zbl 0954.60028 · doi:10.1007/BF02673639
[87] O. A. Podkorytova, “Large deviations and Bahadur efficiency of the Khmaladze—Aki statistic,” J. Math. Sci. (N. Y.) 68, 560-565 (1994). · Zbl 0836.62037 · doi:10.1007/BF01254282
[88] O. A. Podkorytova, “On tail asymptotics for L1-norm of centered brownian bridge,” Le Matematiche 53, 3-9 (1998). · Zbl 0947.60035
[89] R. S. Leontiev, “Asymptotics of the P-value for the omega-square statistic for goodness-of-fit,” Zap. Nauchn. Semin. LOMI 166, 67-71 (1988). · Zbl 0663.62054
[90] Y. Y. Nikitin and P. P. Sporysheva, “On asymptotic efficiency of tests of fit based on the Deheuvels empirical process,” J. Math. Sci. (N. Y.) 159, 317-323 (2009). · Zbl 1274.62325 · doi:10.1007/s10958-009-9443-x
[91] N. Henze and Y. Y. Nikitin, “A new approach to goodness-of-fit testing based on the integrated empirical process,” J. Nonparametric Stat. 12, 391-416 (2000). · Zbl 0945.62051 · doi:10.1080/10485250008832815
[92] N. Henze and Y. Y. Nikitin, “Watson-type goodness-of-fit tests based on the integrated empirical process,” Math. Methods Stat. 11, 183-202 (2002). · Zbl 1005.62050
[93] N. Henze and Y. Y. Nikitin, “Two-sample tests based on the integrated empirical process,” Commun. Stat. — Theory Methods 32, 1767-1788 (2003). · Zbl 1184.62068 · doi:10.1081/STA-120022708
[94] Y. Y. Nikitin and A. V. Tchirina, “Lilliefors test for exponentiality: Large deviations, asymptotic efficiency, and conditions of local optimality,” Math. Methods Stat. 16, 16-24 (2007). · Zbl 1283.62036 · doi:10.3103/S1066530707010024
[95] A. V. Tchirina, “Large deviations for a class of scale-free statistics under the gamma distribution,” J. Math. Sci. (N. Y.) 128, 2640-2655 (2005). · Zbl 1074.60031 · doi:10.1007/s10958-005-0212-1
[96] Y. Y. Nikitin and A. V. Tchirina, “Bahadur efficiency and local optimality of a test for the exponential distribution based on the Gini statistic,” Stat. Meth. Appl. 5, 163-175 (1996). · Zbl 1417.62117
[97] P. L. Conti and Y. Nikitin, “Asymptotic efficiency of independence tests based on Gini’s rank association coefficient, Spearman’s footrule and their generalizations,” Commun. Stat. — Theory Methods 28, 453-465 (1999). · Zbl 0927.62058 · doi:10.1080/03610929908832306
[98] S. Meintanis, Ya. Yu. Nikitin, and A. V. Tchirina, “A test of exponentiality against alternative NBRUE life distributions,” Int. J. Stat. Manage. Syst. 2, 207-219 (2007).
[99] S. Meintanis and Ya. Yu. Nikitin, “A class of count models and a new consistent test for the Poisson distribution,” J. Stat. Plann. Inference 138, 3722-3732 (2008). · Zbl 1146.62324 · doi:10.1016/j.jspi.2007.12.011
[100] N. A. Stepanova, “Multivariate rank tests for independence and their asymptotic efficiency,” Math. Methods Stat. 12, 197-217 (2003).
[101] A. Nazarov and N. Stepanova, “An extremal problem with applications to the problem of testing multivariate independence,” J. Nonparametric Stat. 24, 3-17 (2012). · Zbl 1241.62068 · doi:10.1080/10485252.2011.603831
[102] Y. Y. Nikitin and N. A. Stepanova, “Pitman efficiency of independence tests based on weighted rank statistics,” J. Math. Sci. (N. Y.) 118, 5596-5606 (2003). · Zbl 1106.62330 · doi:10.1023/A:1026190506310
[103] Burgio, G.; Nikitin, Y. Y.; Balakrishnan, N. (ed.); Ibragimov, I. (ed.); Nevzorov, V. (ed.), The combination of the sign and Wilcoxon tests of symmetry and their Pitman efficiency, 395-408 (2001), Boston · doi:10.1007/978-1-4612-0209-7_28
[104] G. Burgio and Y. Y. Nikitin, “On the combination of the sign and Maesono tests for symmetry and its efficiency,” Statistica 63 (2), 213-222 (2003). · Zbl 1116.62049
[105] L. Yu. Kopylev and Ya. Yu. Nikitin, “On conditions of Chernoff local asymptotic optimality of some nonparametric symmetry tests,” J. Math. Sci. (N. Y.) 81, 2424-2429 (1993). · Zbl 0852.62043 · doi:10.1007/BF02362347
[106] Y. Y. Nikitin, “Characterization of distributions by the local asymptotic optimality property of tests statistics,” J. Sov. Math. 25, 1186-1195 (1981). · Zbl 0528.62039 · doi:10.1007/BF01084797
[107] Y. Y. Nikitin, “Local asymptotic Bahadur optimality and characterization problems,” Theory Probab. Its Appl. (Engl. Transl.) 29, 79-92 (1985). · Zbl 0576.62033 · doi:10.1137/1129007
[108] L. Y. Kopylev and Y. Y. Nikitin, “On conditions of Chernoff local asymptotic optimality of some nonparametric symmetry tests,” J. Math. Sci. (N. Y.) 81, 2424-2429 (1996). · Zbl 0852.62043 · doi:10.1007/BF02362347
[109] Ya. Yu. Nikitin and I. Peaucelle, “Efficiency and local optimality of distribution-free tests based on U- and V-statistics,” Metron 62, 185-200 (2004). · Zbl 1416.62253
[110] Nazarov, A. I.; Nikitin, Y. Y., Some extremal problems for Gaussian and empirical random fields, No. Series 2, 189-202 (2002), Providence, RI · Zbl 1023.60031
[111] R. Helmers, P. Janssen, and R. Serfling, “Glivenko-Cantelli properties of some generalized empirical DF’s and strong convergence of generalized L-statistics,” Probab. Theory Relat. Fields 79, 75-93 (1988). · Zbl 0631.60032 · doi:10.1007/BF00319105
[112] J. Galambos and S. Kotz, Characterizations of Probability Distributions, in Ser.: Lecture Notes in Mathematics, Vol. 675 (Springer-Verlag, New York, 1978). · Zbl 0381.62011 · doi:10.1007/BFb0069530
[113] P. Muliere and Y. Nikitin, “Scale-invariant test of normality based on Polya’s characterization,” Metron 60 (1-2), 21-33 (2002). · Zbl 1020.62039
[114] V. Litvinova and Y. Nikitin, “Kolmogorov tests of normality based on some variants of Polya’s characterization,” J. Math. Sci. (N. Y.) 219, 782-788 (2016). · Zbl 1358.62047 · doi:10.1007/s10958-016-3146-x
[115] K. Y. Volkova and Y. Y. Nikitin, “On the asymptotic efficiency of normality tests based on the Shepp property,” Vestn. St. Petersburg Univ.: Math. 42, 256-261 (2009). · Zbl 1183.62080 · doi:10.3103/S1063454109040025
[116] Ya. Yu. Nikitin and K. Yu. Volkova, “Asymptotic efficiency of exponentiality tests based on order statistics characterization,” Georgian Math. J. 17, 749-763 (2010). · Zbl 1205.62052
[117] K. Y. Volkova and Y. Y. Nikitin, “Exponentiality tests based on Ahsanullah’s characterization and their efficiency,” J. Math. Sci. (N. Y.) 204, 42-54 (2015). · Zbl 1335.62082 · doi:10.1007/s10958-014-2185-4
[118] K. Y. Volkova, “On asymptotic efficiency of exponentiality tests based on Rossberg’s characterization,” J. Math. Sci. (N. Y.) 167, 486-494 (2010). · Zbl 1288.62017 · doi:10.1007/s10958-010-9934-9
[119] B. C. Arnold and J. A. Villasenor, “Exponential characterizations motivated by the structure of order statistics in samples of size two,” Stat. Probab. Lett. 83, 596-601 (2013). · Zbl 1266.60009 · doi:10.1016/j.spl.2012.10.028
[120] M. Jovanovic, B. Miloševic, Y. Y. Nikitin, M. Obradovic, and K. Volkova, “Tests of exponentiality based on Arnold-Villasenor characterization and their efficiencies,” Comput. Stat. Data Anal. 90, 100-113 (2015). · Zbl 1468.62095 · doi:10.1016/j.csda.2015.03.019
[121] V. Litvinova and Y. Nikitin, “Kolmogorov tests of normality based on some variants of Polya’s characterization,” J. Math. Sci. (N. Y.) 219, 6582-6588 (2016). · Zbl 1358.62047 · doi:10.1007/s10958-016-3146-x
[122] K. Y. Volkova and Y. Y. Nikitin, “Goodness-of-Fit Tests for the power function distribution based on the Puri-Rubin characterization and their efficiences,” J. Math. Sci. (N. Y.) 199, 130-138 (2014). · Zbl 1305.62184 · doi:10.1007/s10958-014-1840-0
[123] K. Y. Volkova, M. S. Karakulov, and Y. Y. Nikitin, “Goodness-of-fit tests based on the characterization of uniformity by the ratio of order statistics, and their efficiencies,” Zap. Nauchn. Semin. POMI 466, 67-80 (2017).
[124] K. Volkova, “Goodness-of-fit tests for the Pareto distribution based on its characterization,” Stat. Methods Appl. 25 (3), 1-23 (2015).
[125] L. Baringhaus and N. Henze, “A characterization of and new consistent tests for symmetry,” Commun. Stat. — Theory Methods 21 (6), 1555-1566 (1992). · Zbl 0800.62244 · doi:10.1080/03610929208830863
[126] V. V. Litvinova, “New nonparametric test for symmetry and its asymptotic efficiency,” Vestn. St. Petersburg Univ.: Math. 34 (4), 12-14 (2001).
[127] Ya. Yu. Nikitin, “On Baringhaus-Henze test for symmetry: Bahadur efficiency and local optimality for shift alternatives,” Math. Methods Stat. 5, 214-226 (1996). · Zbl 0859.62045
[128] Nikitin, Y. Y.; Ahsanullah, M., New U-empirical tests of symmetry based on extremal order statistics, and their efficiencies, 231-248 (2015), Cham · Zbl 1320.62103
[129] G. T. Bookiya and Ya. Yu. Nikitin, “Asymptotic efficiency of new distribution-free tests of symmetry for generalized skew alternatives,” J. Math. Sci. (N. Y.) 229, 651-663 (2018). · Zbl 1388.62126 · doi:10.1007/s10958-018-3704-5
[130] B. Milosevic and M. Obradovic, “Characterization based symmetry tests and their asymptotic efficiencies,” Stat. Probab. Lett. 119, 155-162 (2016). · Zbl 1349.62155 · doi:10.1016/j.spl.2016.07.007
[131] Ya. Nikitin, “Tests based on characterizations, and their efficiencies: A survey,” Acta Commentat. Univ. Tartuensis Math. 21, 3-24 (2017). · Zbl 1372.60022
[132] P. P. B. Eggermont and V. N. LaRiccia, Maximum Penalized Likelihood Estimation, Vol. 1: Density Estimation (Springer-Verlag, New York, 2001). · Zbl 0984.62026
[133] A. Yu. Zaitsev, “Estimates of the rate of approximation in a de-Poissonization lemma,” Ann. Inst. H. Poincaré. Probab. Stat. 38, 1071-1086 (2002). · Zbl 1019.60017 · doi:10.1016/S0246-0203(02)01140-8
[134] Zaitsev, A. Yu; Giné, E. (ed.); Marcus, M. (ed.); Wellner, J. A (ed.), Estimates of the rate of approximation in the Central Limit Theorem for L1-norm of kernel density estimators, 255-292 (2003), Basel · doi:10.1007/978-3-0348-8059-6_16
[135] A. Yu. Zaitsev, “Moderate deviations for L1-norm of kernel density estimates,” Vestn. S.-Peterb. Univ., Ser. 1: Mat. Mekh. Astron, No. 4, 21-33 (2005). · Zbl 1272.62032
[136] E. Giné, D. M. Mason, and A. Yu. Zaitsev, “The L1-norm density estimator process,” Ann. Probab. 31, 719-768 (2003). · Zbl 1031.62026 · doi:10.1214/aop/1048516534
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