scholarly journals On the Order Statistics of Standard Normal-Based Power Method Distributions

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Normal Distribution ◽  
Order Statistics ◽  
Remainder Term ◽  
Standard Normal Distribution ◽  
Power Method ◽  
Sample Sizes ◽  
Elementary Functions ◽  
Standard Normal

This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution () and its powers of order three and five ( and ). The procedure is demonstrated for sample sizes of . It is shown that and have expectations of order statistics that are functions of the expectations for and can be expressed in terms of explicit elementary functions for sample sizes of . For sample sizes of the expectations of the order statistics for , , and only require a single remainder term.

scholarly journals A New Linear Regression Kalman Filter with Symmetric Samples

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2139
Author(s):  
Xiuqiong Chen ◽  
Jiayi Kang ◽  
Mina Teicher ◽  
Stephen S.-T. Yau
Keyword(s):  
Kalman Filter ◽  
Linear Regression ◽  
Particle Filter ◽  
Normal Distribution ◽  
Extended Kalman Filter ◽  
Nonlinear Filtering ◽  
Unscented Kalman Filter ◽  
Standard Normal Distribution ◽  
Leibler Divergence ◽  
Standard Normal

Nonlinear filtering is of great significance in industries. In this work, we develop a new linear regression Kalman filter for discrete nonlinear filtering problems. Under the framework of linear regression Kalman filter, the key step is minimizing the Kullback–Leibler divergence between standard normal distribution and its Dirac mixture approximation formed by symmetric samples so that we can obtain a set of samples which can capture the information of reference density. The samples representing the conditional densities evolve in a deterministic way, and therefore we need less samples compared with particle filter, as there is less variance in our method. The numerical results show that the new algorithm is more efficient compared with the widely used extended Kalman filter, unscented Kalman filter and particle filter.

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