There is still another old version in this room published in March 2008. I would like to keep it in its original status. Although this version published in May 2009 is not the best one of the author, it will be never revised from today so that you may find differences between the two versions and how I improved it. Maybe you can find a better way to improve it just like what Pearson did on the work done by Galton.

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My personal permanent email address is: chenlgyq@hotmail.com

Thank you very much for your visiting!

Have a good journey with my most sincerely blessing!

2009/08/23 / 陈立功 (Chen  Ligong)=========================================================

The Fundamentals of Statistics for Rebuilding the Methodology of

Piecewise Regression Analysis

Based on a Functionalized General Trichotomy

Ligong Chen^{1, 2, 3 *}, Yongmei Chen³

1. Department of Epidemiology and Health Statistics, Tongji Medical College,

School of Public Health, Huazhong University of Science & Technology,

13 Hangkong Road, Wuhan, Hubei 430030, China

2. Center for Prostate Disease Research (CPDR), Department of Surgery,

Uniformed Services University of the Health Sciences,

1530 East Jefferson Street, Rockville, MD 20852, USA

E-mail: * chenlgyq@hotmail.com

Please Note: If you meet "Insert FGTR_pic_xx here" in the text, please go to the album "FGTR_2009_5" in the "Photos" at the left side of the main page of this website (http://cotglti.spaces.live.com) to find them.

Of course, you can download the pictures if you would like. 

Abstract

       We discussed the fundamentals of Statistics and point out that the optimizations and the enforced continuity assumption in the current methods of piecewise regresion analysis violate the fundamentals. Based on constructing a random threshold variable (RTV) and a convergence rate (CRR) in data iterations in regression for searching thresholds, we introduced a functionalized general trichotomic regression (FGTR) as the most general segmentation in either symmetric or asymmetric with full parametric estimates and hypothesis. We proved that the probability for establishing a trichotmic model is sufficiently large as long as the sample size is satisfied. According to the properties of sample size n in all regular statistics, we used the RTV and the CRR per iteration to estimate two basic thresholds in weighted means and weighted confidence intervals (WCI), which may be adjusted by a discrete coefficient. We used weighted tests to infer differences between any two adjacent thresholds; an asymptotic Chi-square statistic to infer differences among full and piecewise models; several probabilistic statistic to infer continuity; and a statistic to evaluate the goodness-of-fit or similarity of the piecewise models. In our opinion, the current smoothing techniques may be applied based on the weighted mean threshold and the continuity test, while an improved bootstrap method is limited for large samples to decreas overwhelming computation works. A 500-time simulation strongly supported our idea. A sample and logistic regresison were used to illustrate the methodology.