scholarly journals RATES OF CONVERGENCE IN DISTRIBUTION OF A LINEAR COMBINATION OF U-STATISTICS FOR NON-DEGENERATE KERNEL

10.5109/13514 ◽  
2002 ◽  
Vol 34 (2) ◽  
pp. 133-141
Author(s):  
Koichiro Toda ◽  
Hajime Yamato
Keyword(s):  
Linear Combination ◽  
Rates Of Convergence ◽  
Degenerate Kernel ◽  
Convergence In Distribution ◽  
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.

scholarly journals Rates of convergence for U-Statistics with varying kernels

1980 ◽  
Vol 21 (1) ◽  
pp. 1-5 ◽  
Author(s):  
N.C. Weber
Keyword(s):  
Distribution Function ◽  
Rates Of Convergence ◽  
U Statistics ◽  
Standard Normal Variable ◽  
Normal Variable ◽  
Standard Normal

Let Un be a U-statistic whose kernel depends on the size n of the sample under consideration. It is shown that when Un is suitably normalised its distribution function differs in Lp norm from the distribution function of a standard normal variable by a term of O(n-½).

Uniform rates of convergence in extreme-value theory

1982 ◽  
Vol 14 (03) ◽  
pp. 600-622 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Extreme Value Theory ◽  
Error Term ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Convergence In Distribution ◽  
Principal Error ◽  
Close Parallel

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

scholarly journals Rates of strong convergence for U-statistics in finite populations

1991 ◽  
Vol 50 (3) ◽  
pp. 468-480
Author(s):  
P. N. Kokic ◽  
N. C. Weber
Keyword(s):  
Strong Convergence ◽  
Random Sample ◽  
Finite Population ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Simple Random Sample ◽  
Finite Populations ◽  
Strong Law ◽  
U Statistics ◽  
Convergence Results

AbstractLet UNn be a U-statistic based on a simple random sample of size n selected without replacement from a finite population of size N. Rates of convergence results in the strong law are obtained for UNn, which are similar to those known for classical U-statistics based on samples of independent and identically distributed (iid) random variables.

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