Born in Saint-Petersburg (then Leningrad) in 1947. Graduated from Saint -Petersburg University, Dept. of Math. and Mech. Pupil of I.A.Ibragimov. Ph.D. in 1973, Dr. of Sci. in 1987, Professor since 1990. Fellow of the Institute of Math. Stat. Scientific interests: nonparametric statistics, large and small deviations, asymptotic efficiency of tests, stochastic geometry, characterization of distributions.
A survey of goodness-of-fit and symmetry tests based on the characterization properties of distri... more A survey of goodness-of-fit and symmetry tests based on the characterization properties of distributions is presented. This approach became popular in recent years. In most cases the test statistics are functionals of U-empirical processes. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and compared with each other as well as with diverse previously known tests. We also describe new directions of possible research in this domain.
We obtain logarithmic large deviation asymptotics for non-degenerate and weakly degenerate U- and... more We obtain logarithmic large deviation asymptotics for non-degenerate and weakly degenerate U- and V-statistics
—We study large deviations and Bahadur efficiency of the Lilliefors statistic for testing of expo... more —We study large deviations and Bahadur efficiency of the Lilliefors statistic for testing of exponentiality. This statistic belongs to the class of Kolmogorov–Smirnov type statistics with estimated parameters. Large deviation asymptotics of such statistic is found for the first time. We show that the test has relatively high local efficiency and construct the alternative for which it is locally optimal.
One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statist... more One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statistics for the testing of symmetry under certain regularity conditions, imposed on the labels and on the distribution of the observations under the alternative. It is revealed that these efficiencies coincide with the previously known Pitman and local Bahadur efficiencies. The asymptotic comparison of statistical tests is carried out usually on the basis of the concept of asymptotic relative efficiency (ARE). The most frequently encountered and thoroughly investigated is the Pitman ARE [1-3]. Two other fundamental approaches to the determination and computation of the ARE is due to Bahadur [4], [5], and also to Hodges and Lehmam~ [6], [7]. Both these approaches have definite merits and are developed in several investigations. However, sometimes one expresses objections to their use, connected with the "unbalancedness" of these two ARE concepts, since, at their computation, the probabilities of errors of the first or the second kind are kept fixed and the other one tends to zero. The Chernoff efficiency, introduced in [8], is devoid of this deficiency, since at its calculation both error probabilities tend to zero. The Chernoff ARE has been developed by Kallenberg [9], who has computed the Chernoff indices for a series of parametric tests. At the same thne, the Chernoff ARE of nonparametric tests remains little studied. The reason for this consists, apparently, in the difficulties of the determination of the asymptotic behavior of the probabilities of large deviations of the test statistics under the alternative when they lose the "distribution-free" property. Below we find the local Chernoff indices (i.e., the principal parts of these indices when the alternative., approaches the zero hypothesis) of the linear rank statistics for the testing of symmetry. Here we impose rather rigid regularity conditions on the label function and especially on the structure of the family of distributions of the observations in the case of the alternative. For the computation of the asymptotic behavior of the probabilities of large deviations under the alternative, we make use of the classical Chernoff-Savage [10] representation and then of the Sanov principle [11], [12]. Then, the problem of the minimization of the Kullback-Leibler information on a suitable set of distributions is solved by the me~ods of the calculus of variations and the theory of implicit operators. Some results on the Chernoff efficiency of the simplest statistics for testing symmetry have been obtained in [!.3] by another method, based on the theory of U-statistics. The results of the paper will enable us to find, incidentally, previously unknown expressions for the local Hodges-Lehmarm indices of the considered statistics. Comparing them with the previously known local Bahadur exact slopes and with the Pitman efficiency measure, we conclude that, under general conditions, the local ordering of the linear rank statistics for the testing of symmetry from their ARE does not depend on the type of the applied efficiency. It should be emphasized that for other nonparametric statistics (for example, Kolmogorov-Smirnov or J) this is not so [14], [15]. A brief summary of the results has been given in [t6], where one has also indicated analogous formulations in the case of linear rank statistics for the testing of homogeneity. Let X 1, X 2. .. .. X n be a sample from a population with a common distribution function (d.f.) F(x). We test the hypothesis of symmetry with respect to zero
One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statist... more One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statistics for the testing of symmetry under certain regularity conditions, imposed on the labels and on the distribution of the observations under the alternative. It is revealed that these efficiencies coincide with the previously known Pitman and local Bahadur efficiencies.
Linear rank statistics for the two-sample problem are considered. Under general conditions on the... more Linear rank statistics for the two-sample problem are considered. Under general conditions on the score function and the distribution of the observations large deviation asymptotics for these statistics under the alternative are obtained. After using the Sanov principle an extremal problem of minimization of the Kullback-Leibler information is solved by means of the calculus of variations and nonlinear analysis. As an application Hodges-Lehmann and Chernoff efficiencies are evaluated. It is
We define a class of count distributions which includes the Poisson as well as many alternative c... more We define a class of count distributions which includes the Poisson as well as many alternative count models. Then the empirical probability generating function is utilized to construct a test for the Poisson distribution, which is consistent against this class of alternatives. The limit distribution of the test statistic is derived in case of a general underlying distribution, and efficiency considerations are addressed. A simulation study indicates that the new test is comparable in performance to more complicated omnibus tests.
The efficiency of distribution-free integrated goodness-of-fit tests was studied by Henze and Nik... more The efficiency of distribution-free integrated goodness-of-fit tests was studied by Henze and Nikitin (2000, 2002) under location alternatives. We calculate local Ba-hadur efficiencies of these tests under more realistic generalized skew alternatives. They turn out to be unexpectedly high.
We investigate the logarithmic large deviation asymptotics for anisotropic norms of Gaussian rand... more We investigate the logarithmic large deviation asymptotics for anisotropic norms of Gaussian random functions of two variables. The problem is solved by the evaluation of the anisotropic norms of corresponding integral covariance operators. We find the exact values of such norms for some important classes of Gaussian fields. To cite this article: M. Lifshits et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé Nous étudions les grandes déviations logarithmiques pour les normes anisotropes des champs gaussiens aléatoires de deux variables. Le problème est résolu en calculant des normes anisotropes pour les opérateurs intégraux engendrés par les covariances. Nous trouvons des valeurs exactes de telles normes pour quelques classes importantes de champs gaussiens. Pour citer cet article : M. Lifshits et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 1631-073X/03/$ -see front matter 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
Generalizing the Cramér-von Mises and the Kolmogorov-Smirnov test, different integral statistics ... more Generalizing the Cramér-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L p -norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L p -norms should receive more attention. It is shown that, given a distribution function F 0 and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.
We construct integral and supremum type tests of exponentiality based on Ahsanullah's characteriz... more We construct integral and supremum type tests of exponentiality based on Ahsanullah's characterization of the exponential law. We discuss limiting distributions and large deviations of new test statistics under the null-hypothesis and calculate their local Bahadur efficiency under common parametric alternatives. Conditions of local optimality of the new statistics are given. Bibliography: 33 titles.
A survey of goodness-of-fit and symmetry tests based on the characterization properties of distri... more A survey of goodness-of-fit and symmetry tests based on the characterization properties of distributions is presented. This approach became popular in recent years. In most cases the test statistics are functionals of U-empirical processes. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and compared with each other as well as with diverse previously known tests. We also describe new directions of possible research in this domain.
We obtain logarithmic large deviation asymptotics for non-degenerate and weakly degenerate U- and... more We obtain logarithmic large deviation asymptotics for non-degenerate and weakly degenerate U- and V-statistics
—We study large deviations and Bahadur efficiency of the Lilliefors statistic for testing of expo... more —We study large deviations and Bahadur efficiency of the Lilliefors statistic for testing of exponentiality. This statistic belongs to the class of Kolmogorov–Smirnov type statistics with estimated parameters. Large deviation asymptotics of such statistic is found for the first time. We show that the test has relatively high local efficiency and construct the alternative for which it is locally optimal.
One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statist... more One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statistics for the testing of symmetry under certain regularity conditions, imposed on the labels and on the distribution of the observations under the alternative. It is revealed that these efficiencies coincide with the previously known Pitman and local Bahadur efficiencies. The asymptotic comparison of statistical tests is carried out usually on the basis of the concept of asymptotic relative efficiency (ARE). The most frequently encountered and thoroughly investigated is the Pitman ARE [1-3]. Two other fundamental approaches to the determination and computation of the ARE is due to Bahadur [4], [5], and also to Hodges and Lehmam~ [6], [7]. Both these approaches have definite merits and are developed in several investigations. However, sometimes one expresses objections to their use, connected with the "unbalancedness" of these two ARE concepts, since, at their computation, the probabilities of errors of the first or the second kind are kept fixed and the other one tends to zero. The Chernoff efficiency, introduced in [8], is devoid of this deficiency, since at its calculation both error probabilities tend to zero. The Chernoff ARE has been developed by Kallenberg [9], who has computed the Chernoff indices for a series of parametric tests. At the same thne, the Chernoff ARE of nonparametric tests remains little studied. The reason for this consists, apparently, in the difficulties of the determination of the asymptotic behavior of the probabilities of large deviations of the test statistics under the alternative when they lose the "distribution-free" property. Below we find the local Chernoff indices (i.e., the principal parts of these indices when the alternative., approaches the zero hypothesis) of the linear rank statistics for the testing of symmetry. Here we impose rather rigid regularity conditions on the label function and especially on the structure of the family of distributions of the observations in the case of the alternative. For the computation of the asymptotic behavior of the probabilities of large deviations under the alternative, we make use of the classical Chernoff-Savage [10] representation and then of the Sanov principle [11], [12]. Then, the problem of the minimization of the Kullback-Leibler information on a suitable set of distributions is solved by the me~ods of the calculus of variations and the theory of implicit operators. Some results on the Chernoff efficiency of the simplest statistics for testing symmetry have been obtained in [!.3] by another method, based on the theory of U-statistics. The results of the paper will enable us to find, incidentally, previously unknown expressions for the local Hodges-Lehmarm indices of the considered statistics. Comparing them with the previously known local Bahadur exact slopes and with the Pitman efficiency measure, we conclude that, under general conditions, the local ordering of the linear rank statistics for the testing of symmetry from their ARE does not depend on the type of the applied efficiency. It should be emphasized that for other nonparametric statistics (for example, Kolmogorov-Smirnov or J) this is not so [14], [15]. A brief summary of the results has been given in [t6], where one has also indicated analogous formulations in the case of linear rank statistics for the testing of homogeneity. Let X 1, X 2. .. .. X n be a sample from a population with a common distribution function (d.f.) F(x). We test the hypothesis of symmetry with respect to zero
One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statist... more One computes the local asymptotic Chernoff and Hodges-Lehmann efficiencies of linear rank statistics for the testing of symmetry under certain regularity conditions, imposed on the labels and on the distribution of the observations under the alternative. It is revealed that these efficiencies coincide with the previously known Pitman and local Bahadur efficiencies.
Linear rank statistics for the two-sample problem are considered. Under general conditions on the... more Linear rank statistics for the two-sample problem are considered. Under general conditions on the score function and the distribution of the observations large deviation asymptotics for these statistics under the alternative are obtained. After using the Sanov principle an extremal problem of minimization of the Kullback-Leibler information is solved by means of the calculus of variations and nonlinear analysis. As an application Hodges-Lehmann and Chernoff efficiencies are evaluated. It is
We define a class of count distributions which includes the Poisson as well as many alternative c... more We define a class of count distributions which includes the Poisson as well as many alternative count models. Then the empirical probability generating function is utilized to construct a test for the Poisson distribution, which is consistent against this class of alternatives. The limit distribution of the test statistic is derived in case of a general underlying distribution, and efficiency considerations are addressed. A simulation study indicates that the new test is comparable in performance to more complicated omnibus tests.
The efficiency of distribution-free integrated goodness-of-fit tests was studied by Henze and Nik... more The efficiency of distribution-free integrated goodness-of-fit tests was studied by Henze and Nikitin (2000, 2002) under location alternatives. We calculate local Ba-hadur efficiencies of these tests under more realistic generalized skew alternatives. They turn out to be unexpectedly high.
We investigate the logarithmic large deviation asymptotics for anisotropic norms of Gaussian rand... more We investigate the logarithmic large deviation asymptotics for anisotropic norms of Gaussian random functions of two variables. The problem is solved by the evaluation of the anisotropic norms of corresponding integral covariance operators. We find the exact values of such norms for some important classes of Gaussian fields. To cite this article: M. Lifshits et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé Nous étudions les grandes déviations logarithmiques pour les normes anisotropes des champs gaussiens aléatoires de deux variables. Le problème est résolu en calculant des normes anisotropes pour les opérateurs intégraux engendrés par les covariances. Nous trouvons des valeurs exactes de telles normes pour quelques classes importantes de champs gaussiens. Pour citer cet article : M. Lifshits et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 1631-073X/03/$ -see front matter 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
Generalizing the Cramér-von Mises and the Kolmogorov-Smirnov test, different integral statistics ... more Generalizing the Cramér-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L p -norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L p -norms should receive more attention. It is shown that, given a distribution function F 0 and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.
We construct integral and supremum type tests of exponentiality based on Ahsanullah's characteriz... more We construct integral and supremum type tests of exponentiality based on Ahsanullah's characterization of the exponential law. We discuss limiting distributions and large deviations of new test statistics under the null-hypothesis and calculate their local Bahadur efficiency under common parametric alternatives. Conditions of local optimality of the new statistics are given. Bibliography: 33 titles.
New goodness-of-fit tests for exponentiality based on a particular property of exponential law ar... more New goodness-of-fit tests for exponentiality based on a particular property of exponential law are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, the second one is a Kolmogorov type statistic. We show that the kernels corresponding to our statistics are non-degenerate. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various para-metric alternatives is calculated and is compared with simulated powers of new tests. Conditions of local optimality of new statistics in Bahadur sense are discussed and examples of " most favorable " alternatives are given. New tests are applied to reject the hypothesis of exponentiality for the length of reigns of Roman emperors which was intensively discussed in recent years.
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Papers by Yakov Nikitin