Asymptotic Formulas for the Moments of U-Statistics with Degenerate Kernel

1982 ◽  
Vol 27 (1) ◽  
pp. 49-58 ◽  
Author(s):  
A. F. Ronzhin
Keyword(s):  
Asymptotic Formulas ◽  
Degenerate Kernel ◽  
U Statistics

scholarly journals Quasi U-statistics of innite order and applications to the subgroup decomposition of some diversity measures

2014 ◽  
Vol 8 (2) ◽  
pp. 285
Author(s):  
Alusio Pinheiro ◽  
Pranab Kumar Sen
Keyword(s):  
Categorical Data ◽  
Hamming Distance ◽  
Regularity Conditions ◽  
Degenerate Kernel ◽  
Central Tendency ◽  
Asymptotically Normal ◽  
U Statistics ◽  
Diversity Measures ◽  
Common Framework ◽  
Order Degenerate

In several applications, information is drawn from quali- tative variables. In such cases, measures of central tendency and dis- persion may be highly inappropriate. Variability for categorical data can be correctly quantied by the so-called diversity measures. These measures can be modied to quantify heterogeneity between groups (or subpopulations). Pinheiro et al. (2005) shows that Hamming distance can be employed in such way and the resulting estimator of hetero- geneity between populations will be asymptotically normal under mild regularity conditions. Pinheiro et al. (2009) proposes a class of weighted U-statistics based on degenerate kernels of degree 2, called quasi U-statistics, with the property of asymptotic normality under suitable conditions. This is generalized to kernels of degree m by Pinheiro et al. (2011). In this work we generalize this class to an innite order degenerate kernel. We then use this powerful tools and the reverse martingale nature of U-statistics to study the asymptotic behavior of a collection of trans- formed classic diversity measures. We are able to estimate them in a common framework instead of the usual individualized estimation procedures. MSC 2000: primary - 62G10; secondary - 62G20, 92D20.

Asymptotics for rank moments of overpartitions

2014 ◽  
Vol 10 (08) ◽  
pp. 2011-2036 ◽  
Author(s):  
Renrong Mao
Keyword(s):  
Asymptotic Formulas ◽  
Error Terms ◽  
Cranks Of Partitions

Bringmann, Mahlburg and Rhoades proved asymptotic formulas for all the even moments of the ranks and cranks of partitions with polynomial error terms. In this paper, motivated by their work, we apply the same method and obtain asymptotics for the two rank moments of overpartitions.

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