TY - JOUR

T1 - Toward the History of the St.Petersburg School of Probability and Statistics. IV. Characterization of Distributions and Limit Theorems in Statistics.

AU - Zaitsev, A. Yu.

AU - Kagan, Abram

AU - Nikitin, Ya. Yu.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - U-statistics

KW - asymptotic efficiency

KW - characterization of distributions

KW - empirical process

KW - equidistribution

KW - kernel density estimator

KW - large deviations

KW - normal approximation

KW - regression

UR - http://www.scopus.com/inward/record.url?scp=85064889941&partnerID=8YFLogxK

U2 - 10.3103/S106345411901014X

DO - 10.3103/S106345411901014X

M3 - Article

VL - 52

SP - 36

EP - 53

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -